19Birefringent Ray Trace

19.1Ray Tracing in Birefringent Materials

The optical properties of birefringent materials depend on the direction of the light’s polarization, as opposed to isotropic materials, which have identical properties in all directions. Isotropic and anisotropic materials are characterized by 3 × 3 dielectric tensors and 3 × 3 gyrotropic tensors. Ray tracing through birefringent materials is different from tracing through isotropic materials. Rays refracting into anisotropic media are decomposed into two rays with different propagation directions and orthogonal polarizations. These two rays are eigen-modes and propagate without change of polarization state. The ray tracing details are different for each type of birefringent materials: uniaxial, biaxial, and optically active materials (Figure 19.1).

Figure 19.1Classes of isotropic and anisotropic media. Isotropic optically active materials are both isotropic and birefringent.

This chapter describes the interaction of light with birefringent materials, with algorithms relating the light fields before and after birefringent interfaces and a method to track the multiple rays generated due to double refraction/ray doubling. The ray tracing algorithm for birefringent ray intercepts tracks the light field, amplitude, and direction change through birefringent interfaces using the polarization ray tracing matrix. This integrates the birefringent ray trace with the polarization ray tracing methods of Chapter 10, maintaining a global three-dimensional matrix representation. Additional details on light propagation in uniaxial materials such as calcite and uniaxial devices such as waveplates are found in Chapter 21. An example polarization aberration analysis of a common uniaxial optical element, the Glan–Taylor polarizer, an application of this chapter’s algorithms, is presented in Chapter 22 (Crystal Polarizers).

To see why ray tracing birefringent materials is complicated, consider a real ray trace with the Polaris-M software1 for a ray propagating through an example anisotropic system shown in Figure 19.2. First, a ray refracts from air into a biaxial KTP (potassium titanyl phosphate, KTiOPO4) crystal, where the light divides into two modes, labeled fast (f1) and slow (s1), due to double refraction. These two modes then refract into a crystal of aragonite. The aragonite’s crystal axes (CA) are not aligned parallel to the KTP’s axes. Therefore, the f1 ray couples into two modes, f2 and s2, and similarly, the s1-mode couples into f2- and s2-modes. The collective mode labels after propagating through KTP and aragonite are fast–fast(f1 f2), fast–slow (f1s2), slow–fast (s1f2), and slow–slow (s1s2). This ray doubling continues into the third biaxial crystal, mica. When the incident light exits the three crystals, eight modes emerge, labeled as fff, ffs, fsf, fss, sff, sfs, ssf, and sss. Each f and s represent distinct electric field orientations along separate ray segments. The exiting polarization state and phase are found from the addition (superposition) of eight waves. The optical path length (OPL) has eight different values for the eight partial waves, a term for the division of an incident wave into multiple waves.

Figure 19.2A normally incident ray propagates through three blocks of anisotropic materials (potassium titanyl phosphate [KTP], aragonite, and mica), each with different crystal axis orientations shown as three lines inside each block. One incident ray results in eight exiting rays each with different sequences of polarizations and different OPL.

Depending on the type of anisotropic material, different symbols and subscripts label the types eigenmodes as tabulated in Tables 19.1 and 19.2. Isotropic materials are a special case with degenerate modes. When refracting into an isotropic material, both refracted modes, s and p, share the same Poynting vector direction $\widehat{\mathbf{S}}$, the same propagation direction $\widehat{\mathbf{k}}$, and the same refractive index; hence, these modes are degenerate. Thus, for isotropic refraction, the s- and p-modes can be combined and treated as a single mode labeled i, denoting an isotropic mode. In uniaxial materials, the two modes are labeled o for the ordinary and e for the extraordinary modes. In biaxial materials, the two modes are distinguished by the associated refractive index of the ray, the mode with the higher index being the slow-mode and the other mode being the fast-mode. In isotropic optically active materials, the two modes are the right and left circularly polarized modes. Note the symbols for modes are in lowercase. A list of parameters needed by the polarization ray trace for each ray segment is presented in Table 19.2. Many of these parameters were first introduced in Chapters 9 and 10, but birefringent interfaces need additional parameters.

Table 19.1Mode Labeling for Rays in Different Types of Birefringent Interfaces

Isotropic/Anisotropic Material	Descriptions of Eigenmodes	Mode Label
Biaxial	Mode with smaller n	f-mode; fast-mode
	Mode with larger n	s-mode; slow-mode
Uniaxial	Ordinary ray	o-mode
	Extraordinary ray	e-mode
Optically active	Left circularly polarized	l-mode; left-mode
	Right circularly polarized	r-mode; right-mode
Isotropic	Polarized in plane of incidence Polarized out of the plane of incidence Combined s- and p-states	p-polarization s-polarization i-mode
Eigenpolarizations in isotropic/anisotropic material		Labeling
Incidence; inc	Two incident modes	m, n
Exiting	Two exiting modes	v, w
Transmission; t	Two transmitted modes	ta, tb
Reflection; r	Two reflected modes	rc, rd

Note: The s- and p-polarizations in isotropic material have the same propagation directions and are grouped to one mode, isotropic.

Table 19.2Parameters to Be Calculated for Each Birefringent Ray Intercept to Characterize Each Exiting Ray

Parameter	Symbol
Ray intercept coordinates	r
Propagation vector	$\widehat{\mathbf{k}}$
Poynting vector	$\widehat{\mathbf{S}}$
Normal to surface	$\widehat{\mathbf{\eta }}$
Mode label	f, s, o, e, l, r, i
Mode refractive index	nf, ns, no, ne, nl, nr, n
Optical path length	OPL
Electric field vector	E
Magnetic field vector	H
Ray status	e.g., active or missed aperture
Surface order	e.g., (1, 2, 3, 4)
Polarization ray tracing matrix for interface	P
Polarization ray tracing matrix from object space	Pcumulative
Geometrical transformation for interface	Q
Geometrical transformation from object space	Qcumulative
Amplitude reflection or transmission coefficient	a

To simulate the propagation of a wavefront through optical systems with birefringent optical elements, a large number of rays are usually traced, as shown in Figure 19.3. For one birefringent element, one incident wavefront yields two exiting wavefronts. Each of these wavefronts focuses in different locations and has differing amounts of astigmatism and other aberrations.

Figure 19.3A converging beam focuses through a KTP plate. Because of ray doubling, two foci are observed, one for the fast mode and another one for the slow mode.

19.2Description of Electromagnetic Waves in Anisotropic Media

The description of light fields of a ray used by the polarization ray tracing algorithm is provided in this section. Light is a transverse electromagnetic wave characterized by its electric field E, its magnetic field H, its displacement field D, and its induction field B by Maxwell’s equation.1 The electromagnetic fields of a monochromatic plane wave in space r and time t with wavelength λ are

$$\begin{array}{l}\mathbf{E}(r,t)=\mathbf{ℛ}\mathbf{e}\left\{\mathbf{E}\text{exp}\left[i\left(\frac{2\pi n}{\lambda }\widehat{\mathbf{k}}\cdot \mathbf{r}-\omega t\right)\right]\right\},\\ \mathbf{H}(\mathbf{r},t)=\mathbf{ℛ}\mathbf{e}\left\{\mathbf{H}\text{exp}\left[i\left(\frac{2\pi n}{\lambda }\widehat{\mathbf{k}}\cdot \mathbf{r}-\omega t\right)\right]\right\},\\ \mathbf{D}(\mathbf{r},t)=\mathbf{ℛ}\mathbf{e}\left\{\mathbf{D}\text{exp}\left[i\left(\frac{2\pi n}{\lambda }\widehat{\mathbf{k}}\cdot \mathbf{r}-\omega t\right)\right]\right\},\text{and}\\ \mathbf{B}(r,t)=\mathbf{ℛ}\mathbf{e}\left\{\mathbf{B}\text{exp}\left[i\left(\frac{2\pi n}{\lambda }\widehat{\mathbf{k}}\cdot \mathbf{r}-\omega t\right)\right]\right\}.\end{array}$$19.1

19.1

The normalized propagation vector in a medium with refractive index n is $\widehat{\mathbf{k}}$ with wavenumber $\frac{2\pi n}{\lambda }$. In an absorptive material, the complex refractive index is n + iκ, so the magnitude of the fields in an absorptive material decays exponentially as light propagates.

The polarization vector describes the light’s polarization state in 3D as a 3 × 1 vector2

$$\mathbf{E}=E_o{e}^{i{\varphi }_o}\widehat{\mathbf{E}}=E_o{e}^{i{\varphi }_o}\left(\begin{array}{l}{E}_x\\ E_y\\ E_z\end{array}\right)=E_o{e}^{i{\varphi }_o}\left(\begin{array}{l}\left|E_x\right|e^{i{\varphi }_x}\\ \left|E_y\right|e^{i{\varphi }_y}\\ \left|E_z\right|e^{i{\varphi }_z}\end{array}\right).$$19.2

19.2

The field has an absolute complex magnitude $E_o{e}^{i{\varphi }_o}$ and complex components (Ex, Ey, Ez), where $\widehat{\mathbf{E}}\cdot \widehat{\mathbf{E}}*={\left|E_x\right|}^2+{\left|E_y\right|}^2+{\left|E_z\right|}^2=1$.

19.3Defining Birefringent Materials

For the purpose of polarization ray tracing, birefringent materials, including biaxial, uniaxial, and optically active materials, are described by a dielectric tensorε and a gyrotropic tensorG.

In an isotropic material, light with wavelength λ experiences the same refractive index regardless of propagation direction and polarization state. Optical glasses are isotropic, as are air, water, and vacuum. In birefringent materials, the refractive index experienced by the light varies with the direction of the light’s electric field. Many crystals, such as calcite and rutile, are anisotropic. Materials also become birefringent due to stress, strain, or applied electric or magnetic fields. In an anisotropic material, an optic axis is a direction of propagation in which light experiences zero birefringence. When light propagates along the optic axis, the refractive index is the same for all electric field components in the transverse plane. For propagation near the optic axis, the birefringence, the difference in refractive indices between two modes with the same k vector, is small. A biaxial material has three distinct principal indices, and there are four directions, plus and minus along two lines within the material having degenerate eigenpolarizations, as is explained in Section 19.5. Unlike uniaxial material, biaxial material has two optic axes, thus the label biaxial.

The dielectric tensorε relates the variation of refractive index with the light’s polarization state by relating E to D1–3:

$$\mathbf{D}=\mathbf{\epsilon }\mathbf{E}=\left(\begin{array}{l}{D}_x\\ D_y\\ D_z\end{array}\right)=\left(\begin{array}{lll}{\epsilon }_{XX}& {\epsilon }_{XY}& {\epsilon }_{XZ}\\ {\epsilon }_{YX}& {\epsilon }_{YY}& {\epsilon }_{YZ}\\ {\epsilon }_{ZX}& {\epsilon }_{ZY}& {\epsilon }_{ZZ}\end{array}\right)\left(\begin{array}{l}{E}_x\\ E_y\\ E_z\end{array}\right).$$19.3

19.3

When light’s oscillating E field is propagating through a crystal, the response of the crystal changes its orientation with respect to atomic configuration of the crystal and directions of the different molecular bonds. Under the influence of the light field, the charges oscillate at optical frequencies that contribute to the E field. The result is the D field, which includes the light’s field and a contribution from dipoles induced in the material. This relationship is described by the 3 × 3 dielectric tensor. The tensor ε can always be rotated into a diagonal form

$$\mathbf{\epsilon }=\left(\begin{array}{lll}{\epsilon }_X& 0& 0\\ 0& {\epsilon }_Y& 0\\ 0& 0& {\epsilon }_Z\end{array}\right)=\left(\begin{array}{lll}{(n_X+i{\kappa }_X)}^2& 0& 0\\ 0& {(n_Y+i{\kappa }_Y)}^2& 0\\ 0& 0& {(n_Z+i{\kappa }_Z)}^2\end{array}\right),$$19.4

19.4

where nX, nY, and nZ in uppercase subscripts are the principal refractive indices associated with three orthogonal principal axes or crystal axes (CA) and κX, κY, and κZ are the associated absorption coefficients along those three axes. Biaxial materials, such as mica and topaz, have three different principal refractive indices—nS for the largest index, nM, and nF for the smallest index.

An isotropic material can be considered a special case of an anisotropic material with εX = εY = εZ = ε,

$$\mathbf{\epsilon }=\left(\begin{array}{lll}\epsilon & 0& 0\\ 0& \epsilon & 0\\ 0& 0& \epsilon \end{array}\right)=\left(\begin{array}{lll}{(n+i\kappa )}^2& 0& 0\\ 0& {(n+i\kappa )}^2& 0\\ 0& 0& {(n+i\kappa )}^2\end{array}\right).$$19.5

19.5

The dielectric tensor of an isotropic material is proportional to the identity matrix; light experiences the same refractive index n + iκ regardless of propagation direction and polarization state. A uniaxial material has two equal principal refractive indices on the diagonal, nO as the principal ordinary index, and nE as the principal extraordinary index,

$$\mathbf{\epsilon }=\left(\begin{array}{lll}{\epsilon }_O& 0& 0\\ 0& {\epsilon }_O& 0\\ 0& 0& {\epsilon }_E\end{array}\right)=\left(\begin{array}{lll}{(n_O+i{\kappa }_O)}^2& 0& 0\\ 0& {(n_O+i{\kappa }_O)}^2& 0\\ 0& 0& {(n_E+i{\kappa }_E)}^2\end{array}\right).$$19.6

19.6

The principal index nO is associated with a plane and the principal index nE is associated with the principal axis orthogonal to that plane as shown in the middle of Table 19.3. The principal axis related to nE is the optic axis, which specifies the orientation of a uniaxial crystal. By definition, a negative uniaxial crystal, such as calcite, has nO > nE, while a positive uniaxial crystal has nO < nE.

Table 19.3Properties of Biaxial, Uniaxial, and Isotropic Materials When the Principal Axes Are Aligned to the Global Coordinate System

Material	Principal Label	Principal Refractive Index	Diagonal Dielectric Tensor εD
Biaxial	Slow Medium Fast	(nS, nM, nF)	$\left(\begin{array}{ccc}{\epsilon }_S& 0& 0\\ 0& {\epsilon }_M& 0\\ 0& 0& {\epsilon }_F\end{array}\right)=\left(\begin{array}{ccc}{n}_S^2& 0& 0\\ 0& n_M^2& 0\\ 0& 0& n_F^2\end{array}\right)$
Uniaxial	Ordinary Extraordinary	(nO, nE)	$\left(\begin{array}{ccc}{\epsilon }_O& 0& 0\\ 0& {\epsilon }_O& 0\\ 0& 0& {\epsilon }_E\end{array}\right)=\left(\begin{array}{ccc}{n}_O^2& 0& 0\\ 0& n_O^2& 0\\ 0& 0& n_E^2\end{array}\right)$
Isotropic	Isotropic	n	$\left(\begin{array}{ccc}\epsilon & 0& 0\\ 0& \epsilon & 0\\ 0& 0& \epsilon \end{array}\right)=\left(\begin{array}{ccc}{n}^2& 0& 0\\ 0& n^2& 0\\ 0& 0& n^2\end{array}\right)$

Note: A biaxial material has three principal refractive indices (nF, nM, nS). Their associated principal axes are oriented orthogonally. The uniaxial material has two principal refractive indices, the ordinary refractive index nO and extraordinary index nE. The isotropic material has one refractive index n.

The refractive index experienced by a light ray depends on the electric field orientation (the polarization state) relative to the principal axes of the material. Light linearly polarized with its electric field along each of the three principal axes is depicted in Figure 19.4, which demonstrates how the refractive index of the light depends on its polarization, not the direction of propagation. The refractive index characterizes how strongly the electrons in a material oscillate in response to an electromagnetic wave, which governs how fast the mode propagates.4

Figure 19.4(Left) Light propagating along z and polarized along x experiences the refractive index ns. (Middle) Light propagating along z and polarized along y experiences refractive index nM. (Right) Light propagating along x and polarized along z experiences refractive index nF. Thus, the refractive index depends on the light polarization, not the direction of propagation.

The birefringence Δn is the refractive index difference of two eigenpolarizations propagating in the same direction in a birefringent material. The maximum Δn of biaxial and uniaxial materials are nS − nF and nE − nO. Birefringence is dispersive and changes with wavelength. The maximum birefringence of various biaxial and uniaxial material as a function of wavelength is shown in Figures 19.5 through 19.7.5–9

Figure 19.5Birefringence spectra of common negative uniaxial materials.

Figure 19.6Birefringence spectra of common positive uniaxial materials.

Figure 19.7Birefringence spectra of common biaxial materials.

Optically active materials have a molecular structure, typically helical, that induces a rotation of the plane of the electric field oscillations as light passes through the material. The effect of optical activity is described by a gyrotropic tensor G in the constitutive relation:

$$\begin{array}{l}\mathbf{D}=\mathbf{\epsilon }\mathbf{E}+i\mathbf{G}\mathbf{H},\\ \mathbf{B}=\mathbf{\mu }\mathbf{H}-i\mathbf{G}\mathbf{E},\end{array}$$19.7

19.7

where µ is the magnetic permeability tensor and B is the magnetic induction.2,10 Organic liquids such as glucose and sucrose solutions are common examples of isotropic optically active liquids that induce birefringence, or a phase shift, between left and right circularly polarized light. The two circularly polarized eigenmodes have slightly different refractive index nR and nL. Thus, optical activity is a source of circular birefringence. The difference between these two indices is often characterized by the optical rotatory power α, which is related to the gyrotropic constant g,11–14

$$\alpha =\frac{2\pi }{\lambda }g=\frac{\pi }{\lambda }\left|n_R-n_L\right|.$$19.8

19.8

In general, G is a symmetric tensor with six independent coefficients,

$$\mathbf{G}=\left(\begin{array}{lll}{g}_{11}& g_{12}& g_{13}\\ g_{12}& g_{22}& g_{23}\\ g_{13}& g_{23}& g_{33}\end{array}\right).$$19.9

19.9

For an isotropic optically active liquid, such as a sugar solution, G is a diagonal tensor with only one parameter,

$$\mathbf{G}=\left(\begin{array}{lll}g& 0& 0\\ 0& g& 0\\ 0& 0& g\end{array}\right).$$19.10

19.10

The majority of biaxial and uniaxial materials have no optical activity, so G is 0. A few crystals combine uniaxial or biaxial properties with optical activity such as mercury sulfide. In general, molecules that lack mirror symmetry are optically active, that is, a molecule that cannot be superposed on its mirror image, similar to a left shoe and a right shoe.

Crystalline quartz has both uniaxial and optically active characteristics. In quartz, the optical activity is only significant when the light propagates near the uniaxial optical axis, so G for quartz has two dependent values,15

$$\mathbf{G}=\left(\begin{array}{lll}{g}_O& 0& 0\\ 0& g_O& 0\\ 0& 0& g_E\end{array}\right),$$19.11

19.11

where gO = ½ (nR − nL) = 3 × 10−5 and gE = −1.92gO at 589 nm. G is also a function of externally applied magnetic fields.16 Magnetically induced circular birefringence is known as the Faraday effect.

A schematic of the electric field for the two circularly polarized modes propagating in an optically active material is shown in Figure 19.8. When linearly polarized light passes through the optically active material, its plane of polarization rotates steadily through the medium, as shown in Figure 19.9. When it centers the medium, it decomposes into left and right circular components with equal amplitudes that propagate at different speeds, and it exits the material with rotated orientation but still linearly polarized. Optical activity is readily observed under a polariscope. In Figure 19.10, a bottle of concentrated sugar solution is placed between a pair of linear polarizers. The transmission between the polarizers depends on the amount of optical activity (nR − nL), the orientation of the polarizers, and the wavelength.

Figure 19.8The side and front view of the left (red) and right (blue) circularly polarized electric field propagating through an optically active material. The left circularly polarized beam propagates for three wavelengths, and the right propagates for three and a half wavelengths through the material, yielding one-half wave of circular retardance.

Figure 19.9The plane of polarization of linearly polarized light rotates at a uniform rate when propagating in an optically active medium. This one-and-a-half wave circular retarder generates 270° of optical rotation.

Figure 19.10Light propagating through corn syrup (concentrated sugar water solution) in a polariscope. The angle between the polarizer axes rotates 180° from the left figure to the right figure. The polarizer axes are crossed in the left-hand image, parallel in the middle image, and crossed again in the right-hand image. The colors arise from the dispersion of the polarization state, being rotated by a larger angle in the blue and a smaller angle in the red. For this specific jar, the blue–purple light has been rotated by more than 180° while the red light has been rotated by about 135°.

To trace rays through optical elements formed from birefringent materials, the dielectric tensor and gyrotropic tensor are expressed in the optical system’s global xyz Cartesian coordinates. Arbitrarily oriented tensors are obtained by rotating the material’s diagonal tensor, which is tabulated in optical materials tables.17 For principal refractive indices (nA, nB, nC) with principal axis orientations specified by the unit vectors (νA, νB, νC), the diagonal dielectric tensor is

$${\mathbf{\epsilon }}_{\mathbf{D}}=\left(\begin{array}{lll}{n}_A^2& 0& 0\\ 0& n_B^2& 0\\ 0& 0& n_C^2\end{array}\right),$$19.12

19.12

and the dielectric tensor in the optical system’s global coordinate is

$$\mathbf{\epsilon }=\left(\begin{array}{lll}{v}_{Ax}& v_{Bx}& v_{Cx}\\ v_{Ay}& v_{By}& v_{Cy}\\ v_{Az}& v_{By}& v_{Cy}\end{array}\right)\cdot {\mathbf{\epsilon }}_{\mathbf{D}}\cdot {\left(\begin{array}{lll}{v}_{Ax}& v_{Bx}& v_{Cx}\\ v_{Ay}& v_{By}& v_{Cy}\\ v_{Az}& v_{By}& v_{Cy}\end{array}\right)}^{-1}.$$19.13

19.13

This rotation is another example of the orthogonal transformation matrices introduced in Chapter 9. Gyrotropic tensors are rotated in the same manner.

Example 19.1Rotation from Diagonal ε to Global ε

Rotate the diagonal dielectric tensor ${\mathbf{\epsilon }}_{\mathbf{D}}=\left(\begin{array}{ccc}{\epsilon }_x& 0& 0\\ 0& {\epsilon }_y& 0\\ 0& 0& {\epsilon }_z\end{array}\right)$ with principal axes $v_x=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)$, $v_y=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$, and $v_z=\left(\begin{array}{c}-1\\ 0\\ 0\end{array}\right)$ into global coordinates (Figure 19.11).

Figure 19.11Three orthogonal principal axes shown in red, green, and blue on the left in its principal coordinate are rotated to the global coordinate shown on the right.

Using Equation 19.13,

$$\mathbf{\epsilon }=\left(\begin{array}{lll}0& 0& -1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)\cdot {\mathbf{\epsilon }}_{\mathbf{D}}\cdot {\left(\begin{array}{lll}0& 0& -1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)}^{-1}=\left(\begin{array}{lll}{\epsilon }_z& 0& 0\\ 0& {\epsilon }_y& 0\\ 0& 0& {\epsilon }_x\end{array}\right).$$

This operation, for example, can rotate a C-plate, a waveplate with the optic axis normal to the surfaces and no retardance on-axis, to a conventional A-plate waveplate (Figure 19.12).3

Figure 19.12The optic axis of a C-plate can rotate to the optic axis of an A-plate via the rotation operation of Figure 19.11.

19.4Eigenmodes of Birefringent Materials

Ray tracing through birefringent materials results in multiple rays with different polarizations. These polarization states are the eigenpolarizations or eigenmodes. When light refracts into a birefringent material, its energy divides into orthogonal polarized eigenmodes in a process called double refraction or ray splitting. These eigenmodes are the only polarization states that can propagate in a given direction without change of polarization. Figure 19.13 shows the image of the text “POLARIS” seen through a calcite crystal. The two images have orthogonal linear polarization states, so either image can be selected by rotating a linear polarizer in front of the crystal.

Figure 19.13Double refraction through a calcite rhomb.

Figure 19.14 shows the schematic of a ray refracting into a plate of the biaxial crystal ulexite, where it splits into the fast- and slow-modes, which refract into different directions due to their higher and lower refractive indices, propagating to the back face, and exiting the crystal. Here, the fast ray has its polarization coming out of the page and the slow ray has its polarization in the plane of the page. Any incident ray divides into two orthogonal polarization modes within the crystal. Unpolarized incident light divides its flux equally into both the fast- and slow-modes; other states divide unequally. The energy distribution between the two modes depends on the polarization state of the incident light, that is, electric field oscillation with respect to the three crystal axes. Similarly, for refraction into uniaxial materials, two eigenmodes emerge after the uniaxial interface with orthogonal polarizations labeled ordinary and extraordinary, while for optically active materials, the modes are labeled left and right.

Figure 19.14A ray splits into two modes through a ulexite biaxial block. Two of the crystal axes are shown in the bottom left corner and the third is out of the plane of the page. In this case, the fast-mode oscillates in and out of the page and the slow-mode oscillates within the plane of the page.

In general, ray doubling occurs each time a beam enters into or reflects toward a birefringent medium, unless the polarization is aligned exactly such that only one mode is excited with energy. An incident ray refracting through N birefringent interfaces results in a potential of 2N separate exiting rays with 2N different mode sequences. Each of these modes takes a different path and has its own amplitude, polarization, and OPL. To retrieve the properties of each of these modes, a list of ray parameters, presented in Table 19.2, are calculated for each ray segment during the polarization ray trace through an optical system. Mode label is an additional ray parameter needed for tracking ray doubling at each birefringent interface. The collective mode label of a resultant ray at the exit pupil describes the evolution of polarization along that specific ray path. Polarization ray tracing a grid of incident rays emerging from an incident wavefront through N birefringent surfaces produces 2N separate wavefronts at the exit pupil. The exiting rays with the same mode label represent one of these wavefronts. By studying the properties of these rays at the exit pupil, one mode at a time, the wavefront aberrations (amplitude aberration, defocus, spherical aberration, coma, astigmatism, etc.) of each mode sequence can be analyzed and the effects of overlapping these wavefronts can be calculated.

19.5Reflections and Refractions at Birefringent Interface

When light is incident at an optical interface, part of its energy refracts and another part reflects. At a birefringent intercept, the energy of the light divides into eigenmodes. The exiting ray parameters and the beam’s amplitudes are shown in Table 19.2 before and after the intercept are calculated from the incident ray parameters, as well as the material properties before and after the intercept. The algorithms to calculate the exiting ray parameters for polarization ray tracing in refraction and reflection for each exiting modes at an uncoated birefringent intercept are described in this section. These resultant ray parameters will be used in Section 19.7 to calculate their polarization ray tracing matrices, P, which represent the polarization properties of one or a series of ray intercepts. The type of uncoated birefringent interfaces described in this section includes isotropic/birefringent, birefringent/isotropic, and birefringent/birefringent interfaces. Depending on the type of the interface, the number of exiting rays varies, the resultant fields behave differently, but the calculations are similar and can be generalized. The incident, refracted, and reflected ray parameters are distinguished by their subscripts inc, t, and r, respectively, in the following discussion.

An isotropic interface is a special case of birefringent interfaces. In isotropic materials, k and S are aligned in the same direction, and the polarization of light describes the oscillation direction of E field. At an isotropic intercept, E is decomposed into its s- and p-components, Es and Ep. These are defined by the surface normal $\widehat{\mathbf{\eta }}$ and$\widehat{\mathbf{k}}$, where Es is in the direction of $(\widehat{\mathbf{k}}\times \widehat{\mathbf{\eta }})$ and Ep is in the direction of $(\widehat{\mathbf{k}}\times {\widehat{\mathbf{E}}}_s)$,4 as shown in Figure 19.15. The transmitted electric field inside the isotropic material is the superposition of the transmitted s- and p-components,2

Figure 19.15At isotropic interface, Es is in the direction of $({\widehat{\mathbf{k}}}_{inc}\times \widehat{\mathbf{\eta }})$ and Einc,p is in the direction of $(\widehat{\mathbf{k}}\times {\widehat{\mathbf{E}}}_{\mathrm{inc},s})$. They form a right-handed (s, p, k) basis.

$${\mathbf{E}}_t(\mathbf{r},t)=\mathbf{ℛ}\mathbf{e}\left[E_{\mathrm{inc}}\left(a_{ts}{\widehat{\mathbf{E}}}_{ts}+a_{tp}{\widehat{\mathbf{E}}}_{tp}\right)e^{i\frac{2\pi }{\lambda }n{\widehat{\mathbf{k}}}_t\cdot \mathbf{r}}{e}^{-i\omega t}\right],$$19.14

19.14

where Einc is the incident electric field amplitude, and ats and atp are the complex electric field amplitude transmission coefficients. Regardless of the propagation direction, the two refracted components experience the same refractive index n, and ${\widehat{\mathbf{k}}}_t={\widehat{\mathbf{S}}}_t$.

Inside birefringent materials, however, the refracted rays experience different refractive indices depending on the incident polarization and the direction of kinc relative to the crystal axis orientations. Einc splits into two orthogonal eigenpolarizations and propagate in different directions. As an example, consider a ray normally incident onto a biaxial crystal block as shown in Figure 19.16. In transmission, the incident ray splits into two orthogonal modes with different refractive indices that lie between the largest and smallest of the three principal indices of the biaxial material. Inside the crystal, each mode’s constant phase wavefront is not perpendicular to the energy propagation direction; kt is not aligned with St. Et is oscillating perpendicular to St, but not kt. The energy of the two modes propagates along different paths; the fast-mode’s energy propagates along Stf and the slow-mode’s energy propagates along Sts. The calculation of these ray parameters is shown later in this section. After the first interface and assuming plane waves, the refracted electric field in the biaxial material is the sum of the two modes having different propagation directions:

Figure 19.16A ray at normal incidence propagates into a block of ulexite. The direction of energy flow is along the Poynting vector S, which is shown as the black arrows passing through the crystal block. The ulexite’s principal axes associated with nF, nM, and nS are represented by the yellow, blue, and red arrows inside the block and change in each cases. The fast-mode is light blue and the slow-mode is magenta. The boxes in the right column show the locations of the two exiting modes at the exit surface; the gray axes mark the center of the exit surface in-line with the incident ray. One mode generally has a larger shift than the other as the crystal axes rotate, but neither of them stays fixed in a biaxial material.

$${\mathbf{E}}_t(\mathbf{r},t)=\mathbf{ℛ}\mathbf{e}\left\{E_{\mathrm{inc}}\left[a_{tf}{\widehat{\mathbf{E}}}_{tf}{e}^{i\frac{2\pi }{\lambda }{n}_f{\widehat{\mathbf{k}}}_{tf}\cdot \mathbf{r}}+a_{ts}{\widehat{\mathbf{E}}}_{ts}{e}^{i\frac{2\pi }{\lambda }{n}_s{\widehat{\mathbf{k}}}_{ts}\cdot \mathbf{r}}\right]e^{-i\omega t}\right\},$$19.15

19.15

where atf and ats are the complex electric field amplitude coefficients, and nf and ns are the refractive indices encountered by each mode.

In the following discussion, the transmitted and reflected modes are identified as ta, tb, rc, and rd for all types of materials. The four combinations of isotropic and birefringent interfaces along with the corresponding ray splitting configurations are depicted in Figure 19.17.15

Figure 19.17The four configurations of birefringent interfaces are shown with corresponding reflected rays toward the incident medium and refracted rays into the transmitting medium for a given incident ray. The black arrow represents S, the Poynting vector direction, which is the direction of energy flow and is not necessarily the same as k, the propagation vector direction. The transmitted and reflected modes are identified by subscripts ta, tb, rc, and rd. The three gray parallel lines along each ray represent the wavefronts (direction of D), which are perpendicular to k, but not necessarily perpendicular to S. (a) A ray propagating from an isotropic medium to another isotropic medium results in one reflected ray and one refracted ray. (b–d) If the incident medium or/and the transmitted medium is/are birefringent, ray splitting occurs and the two resultant rays in each birefringent material have different k and S directions.

In an isotropic material, the refractive index remains constant and is independent of the polarization of the ray. The k and S vectors of the s- and p-polarizations are parallel and one incident ray produces one reflected and one refracted ray. In this case, the reflected and refracted constant phase wavefronts are perpendicular to the energy propagation direction. In birefringent materials, the wavefronts are not generally perpendicular to the direction of energy propagation; k and S are not aligned in the same direction. At birefringent intercepts, one incident ray may result in up to four exiting rays, two reflected and two refracted.

The ray tracing algorithm used with isotropic materials derived from Snell’s law must be generalized to compute refraction and reflection at birefringent interfaces. Also, the corresponding Fresnel amplitude transmission and reflection coefficients are more complex for birefringent intercept, since the refractive index varies with E. The necessary set of parameters for ray tracing comprises six 3 × 1 vectors: (k, S, E, D, B, and H) and the associated refractive index of the ray. These are calculated by solving Maxwell’s equations with appropriate boundary values.15 The normalized refracted or reflected vector k at a birefringent material is

$$\widehat{\mathbf{k}}=\frac{n{\widehat{\mathbf{k}}}_{\mathrm{inc}}+\left(-n{\widehat{\mathbf{k}}}_{\mathrm{inc}}\cdot \widehat{\mathbf{\eta }}\pm \sqrt{n^2{({\widehat{\mathbf{k}}}_{\mathrm{inc}}\cdot \widehat{\mathbf{\eta }})}^2+({n^{\prime }}^2-n^2)}\right)\widehat{\mathbf{\eta }}}{\Vert n{\widehat{\mathbf{k}}}_{\mathrm{inc}}+\left(-n{\widehat{\mathbf{k}}}_{\mathrm{inc}}\cdot \widehat{\mathbf{\eta }}\pm \sqrt{n^2{({\widehat{\mathbf{k}}}_{\mathrm{inc}}\cdot \widehat{\mathbf{\eta }})}^2+({n^{\prime }}^2-n^2)}\right)\widehat{\mathbf{\eta }}\Vert },$$19.16

19.16

where n is the refractive index for the incident ray, n′ is the refractive index of the exiting ray, and the sign of the square root is + for refraction and − for reflection. Equation 19.16 is the general form of the refraction and reflection equations in Chapter 10. The solution is complicated by the fact that $\widehat{\mathbf{k}}$ is a function of n′, which is not specified at the beginning of the calculation. Hence, the reflection and refraction algorithm must simultaneously solve for $\widehat{\mathbf{k}}$ and n′. By combining the constitutive relations in Equation 19.7 with Maxwell’s equations, the eigenvalue equations for the E fields are formed:

$$\left[{\mathbf{\epsilon }}^{\prime }+{(n^{\prime }{\mathbf{K}}_t+i{\mathbf{G}}^{\prime })}^2\right]{\mathbf{E}}_t=0\text{and}\left[\mathbf{\epsilon }+{(n{\mathbf{K}}_r+i\mathbf{G})}^2\right]{\mathbf{E}}_r=0$$19.17

19.17

for refraction and reflection, respectively, where

$$K\mathbf{=}\left(\begin{array}{lll}0& -k_z& k_y\\ k_z& 0& -k_x\\ -k_y& k_x& 0\end{array}\right)$$19.18

19.18

and $\widehat{\mathbf{k}}=(k_x,k_y,k_z)$. For a non-zero exiting Et and Er, the determinant of Equation 19.19 has to be zero,

$$\left|{\mathbf{\epsilon }}^{\prime }+{(n^{\prime }{\mathbf{K}}_t+i{\mathbf{G}}^{\prime })}^2\right|=0\text{and}\left|\mathbf{\epsilon }+{(n^{\prime }{\mathbf{K}}_r+i\mathbf{G})}^2\right|=0.$$19.19

19.19

Equations 19.16 and 19.19 are solved simultaneously for n′ and k. The exiting E fields are calculated by Equation 19.17 through singular value decomposition. The exiting H and S fields are calculated with Equations 19.20 and 19.21.

$${\mathbf{H}}_t=(n_t{\mathbf{K}}_t+i{\mathbf{G}}^{\prime }){\mathbf{E}}_t\text{and}{\mathbf{H}}_r=(n_r{\mathbf{K}}_r+i\mathbf{G}){\mathbf{E}}_r$$19.20

19.20

$${\mathbf{S}}_t=\frac{\mathbf{ℛ}\mathbf{e}\left[{\mathbf{E}}_t\times {\mathbf{H}}_t^{*}\right]}{\left|\mathbf{ℛ}\mathbf{e}\left[{\mathbf{E}}_t\times {\mathbf{H}}_t^{*}\right]\right|}\text{and}{\mathbf{S}}_r=\frac{\mathbf{ℛ}\mathbf{e}\left[{\mathbf{E}}_r\times {\mathbf{H}}_r^{*}\right]}{\left|\mathbf{ℛ}\mathbf{e}\left[{\mathbf{E}}_r\times {\mathbf{H}}_r^{*}\right]\right|}.$$19.21

19.21

Then, E, D, B, H, and S fields are calculated for all exiting modes. Since highly transparent materials are preferred in optical systems, the absorption is assumed negligibly small. Extension of these methods to absorbing and dichroic materials is included in Refs. [18–21].

The equation for optical path length (OPL) in birefringent materials is generalized from its definition for isotropic materials in Equation 19.22. OPL describes the phase accumulated along a ray path between optical element interfaces and is calculated separately for each mode. The physical ray path ℓ is along S, which is the direction of energy flow, while the phase increases along k. Therefore, the OPL for the ray segment is

$$\mathrm{OPL}=n\ell \widehat{\mathbf{k}}\cdot \widehat{\mathbf{S}}$$19.22

19.22

as shown in Figure 19.18.

Figure 19.18The calculation of the OPL for a ray propagates through a birefringent material where S (orange) and k (gray) are not aligned. The wavefronts (gray parallel lines) are perpendicular to the k vector. The energy propagation direction is along S, which determines the location of ray intercept at the next surface. The OPL is the number of wavelengths between the two intercepts, multiplied by the projection of the ray path ℓ onto k, the refractive index of the mode, and the wavelength.15

The fraction of the incident energy that couples into each of the four modes is described by the amplitude transmission coefficients (ata and atb) and the amplitude reflection coefficients (arc and ard). These coefficients at an isotropic-to-isotropic intercept are the conventional Fresnel coefficients for the s- and p-polarizations. By matching the boundary conditions at the interface for E and H fields, all four exiting amplitude coefficients at intercept are calculated using Equations 19.23 through 19.26,2,15,18,19

$$\mathbf{A}={\mathbf{F}}^{-1}\cdot \mathbf{C},$$19.23

19.23

where

$$\mathbf{A}={(a_{ta},a_{tb},a_{rc},a_{rd})}^T,$$19.24

19.24

$$\mathbf{F}=\left(\begin{array}{llll}{\mathbf{s}}_1\cdot {\widehat{\mathbf{E}}}_{ta}& {\mathbf{s}}_1\cdot {\widehat{\mathbf{E}}}_{tb}& -{\mathbf{s}}_1\cdot {\widehat{\mathbf{E}}}_{rc}& -{\mathbf{s}}_1\cdot {\widehat{\mathbf{E}}}_{rd}\\ {\mathbf{s}}_2\cdot {\widehat{\mathbf{E}}}_{ta}& {\mathbf{s}}_2\cdot {\widehat{\mathbf{E}}}_{tb}& -{\mathbf{s}}_2\cdot {\widehat{\mathbf{E}}}_{rc}& -{\mathbf{s}}_2\cdot {\widehat{\mathbf{E}}}_{rd}\\ {\mathbf{s}}_1\cdot {\overrightarrow{\mathbf{H}}}_{ta}& {\mathbf{s}}_1\cdot {\overrightarrow{\mathbf{H}}}_{tb}& -{\mathbf{s}}_1\cdot {\overrightarrow{\mathbf{H}}}_{rc}& -{\mathbf{s}}_1\cdot {\overrightarrow{\mathbf{H}}}_{rd}\\ {\mathbf{s}}_2\cdot {\overrightarrow{\mathbf{H}}}_{ta}& {\mathbf{s}}_2\cdot {\overrightarrow{\mathbf{H}}}_{tb}& -{\mathbf{s}}_2\cdot {\overrightarrow{\mathbf{H}}}_{rc}& -{\mathbf{s}}_2\cdot {\overrightarrow{\mathbf{H}}}_{rd}\end{array}\right),$$19.25

19.25

$$\mathbf{C}={\left({\mathbf{s}}_1\cdot {\widehat{\mathbf{E}}}_{\mathrm{inc}},{\mathbf{s}}_2\cdot {\widehat{\mathbf{E}}}_{\mathrm{inc}},{\mathbf{s}}_1\cdot {\overrightarrow{\mathbf{H}}}_{\mathrm{inc}},{\mathbf{s}}_2\cdot {\overrightarrow{\mathbf{H}}}_{\mathrm{inc}}\right)}^T,$$19.26

19.26

${\mathbf{s}}_1={\widehat{\mathbf{k}}}_{\mathrm{inc}}\times \widehat{\mathbf{\eta }}$, and ${\mathbf{s}}_2=\widehat{\mathbf{\eta }}\times {\mathbf{s}}_1$. s1•V and s2•V operate on vector V to extract the tangential and normal components of V. Therefore, the transmitted electric field is the superposition of the two transmitted modes:

$${\mathbf{E}}_t(\mathbf{r},t)=\mathbf{ℛ}\mathbf{e}\left[E_{\mathrm{inc}}\left(a_{ta}{\widehat{\mathbf{E}}}_{ta}{e}^{i\frac{2\pi }{\lambda }{n}_{ta}{\widehat{\mathbf{k}}}_{ta}\cdot \mathbf{r}}+a_{tb}{\widehat{\mathbf{E}}}_{tb}{e}^{i\frac{2\pi }{\lambda }{n}_{tb}{\widehat{\mathbf{k}}}_{\mathbf{tb}}\cdot \mathbf{r}}\right)e^{-i\omega t}\right],$$19.27

19.27

and the reflected electric field is

$${\mathbf{E}}_r(\mathbf{r},t)=\mathbf{ℛ}\mathbf{e}\left[E_{inc}\left(a_{rc}{\widehat{\mathbf{E}}}_{rc}{e}^{i\frac{2\pi }{\lambda }{n}_{rc}{\widehat{\mathbf{k}}}_{rc}\cdot \mathbf{r}}+a_{rd}{\widehat{\mathbf{E}}}_{rd}{e}^{i\frac{2\pi }{\lambda }{n}_{rd}{\widehat{\mathbf{k}}}_{rd}\cdot \mathbf{r}}\right)e^{-i\omega t}\right].$$19.28

19.28

The algorithms above explain the calculation for uncoated birefringent interfaces. The calculations of the amplitude coefficients for layered birefringent slabs and birefringent coatings are discussed by Mansuripur,22 Abdulhalim,23,24 and others.25,26

The light intensity I of a given E is calculated by multiplying |E|2 with the cross section scaling factor, n2cosθs2/n1cosθs1, at a ray intercept as

$$I=\frac{n_2\text{cos}{\theta }_{s2}}{n_1\text{cos}{\theta }_{s1}}{\left|\mathbf{E}\right|}^2,$$19.29

19.29

where θs is the angle of the Poynting vector, subscript 1 for parameters before the interface, and subscript 2 for parameters after the interface.

For rays refracting into a uniaxial medium, the a- and b-modes are the o- and e-modes in Equation 19.27. For rays refracting into an optically active medium, the resultant two modes are left- and right-modes. It is important to note that Equations 19.27 and 19.28 must be repeated for the orthogonally polarized incident modes to yield the four refracting or four reflecting modes, as shown in Section 19.7. For biaxial-to-biaxial interface, there are fast–fast-, fast–slow-, slow–fast-, and slow–slow-modes; for uniaxial-to-uniaxial interface, there are oo-, oe-, eo-, and ee-modes. Further, at birefringent-to-birefringent interfaces, s is no longer refracting into s, or p into p, since s and p are not eigenpolarizations. At an uncoated isotropic interface, refraction (or reflection) can be represented in s–p coordinates as a diagonal Jones matrix. At a birefringent interface, the Jones matrix is not diagonal; s couples into p, and p also couples into s.

Example 19.2Calculate Ray Parameters at an Air/KTP Ray Intercept

This example shows the step-by-step procedure to propagate a ray through the birefringent interface of Figure 19.19. Given the incident ray parameters, the exiting ray parameters are calculated. A light ray propagates from an isotropic medium (air, n = 1) into a biaxial material (KTP), (nF, nM, nS) = (1.78559, 1.79718, 1.90206) with crystal axes along $\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)$, $\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$, $\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)$.

Figure 19.19A ray propagates into a KTP block of crystal and splits into two rays. Note the normal points away from the incident interface.

In air, the light direction is ${\mathbf{k}}_{\mathrm{inc}}=\left(\begin{array}{c}0\\ \text{sin}\theta \\ \text{cos}\theta \end{array}\right)=\left(\begin{array}{c}0\\ \text{sin}35°\\ \text{cos}35°\end{array}\right)$ and ${\mathbf{S}}_{\mathrm{inc}}=\left(\begin{array}{c}0\\ \text{sin}\theta \\ \text{cos}\theta \end{array}\right)=\left(\begin{array}{c}0\\ \text{sin}35°\\ \text{cos}35°\end{array}\right)$.

The interface has $\mathbf{\eta }=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)$, $\mathbf{\epsilon }=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$, ${\mathbf{\epsilon }}^{\prime }=\left(\begin{array}{ccc}{n}_F^2& 0& 0\\ 0& n_M^2& 0\\ 0& 0& n_S^2\end{array}\right)$, and $\mathbf{G}={\mathbf{G}}^{\prime }=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$. Note the normal points away from the incident medium.

With Equations 19.16 and 19.18, the resultant propagation parameters are

$$\widehat{\mathbf{k}}=\frac{1}{n^{\prime }}\left(\begin{array}{l}0\\ \text{sin}\theta \\ \pm \sqrt{{n^{\prime }}^2-{\text{sin}}^2\theta }\end{array}\right)\text{and}\mathbf{K}=\frac{1}{n^{\prime }}\left(\begin{array}{lll}0& \mp \sqrt{{n^{\prime }}^2-{\text{sin}}^2\theta }& \text{sin}\theta \\ \pm \sqrt{{n^{\prime }}^2-{\text{sin}}^2\theta }& 0& 0\\ -\text{sin}\theta & 0& 0\end{array}\right)$$

for transmission and reflection.

In transmission, based on Equation 19.17,

$$\left(\begin{array}{lll}{n}_F^2-{n^{\prime }}^2& 0& 0\\ 0& {\text{sin}}^2\theta +n_M^2-{n^{\prime }}^2& \sqrt{{n^{\prime }}^2-{\text{sin}}^2\theta }\text{sin}\theta \\ 0& \sqrt{{n^{\prime }}^2-{\text{sin}}^2\theta }\text{sin}\theta & n_S^2-{\text{sin}}^2\theta \end{array}\right){\mathbf{E}}_t=0.$$

For non-zero Et, the determinant of the matrix is zero:

$$\frac{1}{2}\left(n_F^2-{n^{\prime }}^2\right)\left[n_M^2+n_S^2\left(-1-2n_M^2+2{n^{\prime }}^2\right)+\left(n_S^2-n_M^2\right)\text{cos}2\theta \right]=0.$$

Since n′ > 0, n′ = nF, or $n^{\prime }=\sqrt{n_M^2+\left(1-\frac{n_M^2}{n_S^2}\right){\text{sin}}^2\theta }$, nf = 1.78559 and ns = 1.80697.

• The transmitted fast-mode satisfies

$$\left(\begin{array}{lll}{n}_F^2-n_f^2& 0& 0\\ 0& {\text{sin}}^2\theta -n_f^2+n_M^2& \sqrt{n_f^2-{\text{sin}}^2\theta }\text{sin}\theta \\ 0& \sqrt{n_f^2-{\text{sin}}^2\theta }\text{sin}\theta & n_S^2-{\text{sin}}^2\theta \end{array}\right){\mathbf{E}}_{tf}=0,$$

$$\left(\begin{array}{lll}0& 0& 0\\ 0& 0.371& 0.970\\ 0& 0.970& 3.289\end{array}\right){\mathbf{E}}_{tf}=0.$$

By singular value decomposition,

$$\left(\begin{array}{lll}0& 0& 0\\ 0& 0.371& 0.970\\ 0& 0.970& 3.289\end{array}\right)=\left(\begin{array}{lll}0& 0& 1\\ -0.289& -0.957& 0\\ -0.957& 0.289& 0\end{array}\right)\left(\begin{array}{lll}3.582& 0& 0\\ 0.& 0.078& 0\\ 0& 0& 0\end{array}\right){\left(\begin{array}{lll}0& 0& 1\\ -0.289& -0.957& 0\\ -0.957& 0.289& 0\end{array}\right)}^{†}.$$

The exiting state corresponds to the singular value of zero with electric field: ${\mathbf{E}}_{tf}=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)$.

The H and S fields are

$$\begin{array}{l}{\mathbf{H}}_{tf}=(n_f{\mathbf{K}}_{tf}+i{\mathbf{G}}^{\prime }){\widehat{\mathbf{E}}}_{tf}=\left(\begin{array}{lll}0& -1.691& 0.574\\ 1.691& 0& 0\\ -0.574& 0& 0\end{array}\right)\left(\begin{array}{l}1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{l}0\\ 1.691\\ -0.574\end{array}\right)\text{and}\\ {\widehat{\mathbf{S}}}_{tf}=\frac{\mathbf{ℛ}\mathbf{e}\left[{\widehat{\mathbf{E}}}_{tf}\times {\widehat{\mathbf{H}}}_{tf}*\right]}{\left|\mathbf{ℛ}\mathbf{e}\left[{\widehat{\mathbf{E}}}_{tf}\times {\widehat{\mathbf{H}}}_{tf}*\right]\right|}=\left(\begin{array}{l}0\\ 0.321\\ 0.947\end{array}\right).\end{array}$$

• The transmitted slow-mode satisfies

$$\left(\begin{array}{lll}{n}_F^2-n_s^2& 0& 0\\ 0& {\text{sin}}^2\theta +n_M^2-n_s^2& \sqrt{n_s^2-{\text{sin}}^2\theta }\text{sin}\theta \\ 0& \sqrt{n_s^2-{\text{sin}}^2\theta }\text{sin}\theta & n_S^2-{\text{sin}}^2\theta \end{array}\right){\mathbf{E}}_{ts}=0$$

$$\left(\begin{array}{lll}-0.077& 0& 0\\ 0& 0.294& 0.983\\ 0& 0.983& 3.289\end{array}\right){\mathbf{E}}_{ts}=0.$$

By singular value decomposition,

$$\left(\begin{array}{lll}-0.077& 0& 0\\ 0& 0.294& 0.983\\ 0& 0.983& 3.289\end{array}\right)=\left(\begin{array}{lll}0& -1& 0\\ -0.286& 0& -0.958\\ -0.958& 0& 0.286\end{array}\right)\left(\begin{array}{lll}3.583& 0& 0\\ 0& 0.077& 0\\ 0& 0& 0\end{array}\right){\left(\begin{array}{lll}0& 1& 0\\ -0.286& 0& -0.958\\ -0.958& 0& 0.286\end{array}\right)}^{†}.$$

The exiting state corresponds to the singular value of zero with electric field: ${\mathbf{E}}_{\mathbf{ts}}=\left(\begin{array}{c}0\\ -0.958\\ 0.286\end{array}\right)$.

The H and S fields are

$${\mathbf{H}}_{ts}=(n_s{\mathbf{K}}_{ts}+i{\mathbf{G}}^{\prime }){\widehat{\mathbf{E}}}_{ts}=\left(\begin{array}{lll}0& -1.714& 0.574\\ 1.714& 0& 0\\ -0.574& 0& 0\end{array}\right)\left(\begin{array}{l}0\\ -0.958\\ 0.286\end{array}\right)=\left(\begin{array}{l}1.806\\ 0\\ 0\end{array}\right)$$

and

$${\widehat{\mathbf{S}}}_{ts}=\frac{\mathbf{ℛ}\mathbf{e}\left[{\widehat{\mathbf{E}}}_{ts}\times {\widehat{\mathbf{H}}}_{ts}*\right]}{\left|\mathbf{ℛ}\mathbf{e}\left[{\widehat{\mathbf{E}}}_{ts}\times {\widehat{\mathbf{H}}}_{ts}*\right]\right|}=\left(\begin{array}{l}0\\ 0.286\\ 0.958\end{array}\right).$$

• In reflection, [ε + (nrKr + iG)2] Er = 0, where nr = 1:

$$\left(\begin{array}{lll}1-n_r^2& 0& 0\\ 0& 1-n_r^2+{\text{sin}}^2\theta & -\sqrt{n_r^2-{\text{sin}}^2\theta }\text{sin}\theta \\ 0& -\sqrt{n_r^2-{\text{sin}}^2\theta }\text{sin}\theta & {\text{cos}}^2\theta \end{array}\right){\mathbf{E}}_r=0$$

$$\left(\begin{array}{lll}0& 0& 0\\ 0& 0.329& -0.470\\ 0& -0.470& 0.671\end{array}\right){\mathbf{E}}_r=0.$$

By singular value decomposition,

$$\left(\begin{array}{lll}0& 0& 0\\ 0& 0.329& -0.470\\ 0& -0.470& 0.671\end{array}\right)=\left(\begin{array}{lll}0& 0& 1\\ -0.574& 0.819& 0\\ 0.819& 0.574& 0\end{array}\right)\left(\begin{array}{lll}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right){\left(\begin{array}{lll}0& 0& 1\\ -0.574& 0.819& 0\\ 0.819& 0.574& 0\end{array}\right)}^{†}.$$

The exiting states correspond to the singular value of zero, which are ${\mathbf{E}}_{rs}=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)$ and ${\mathbf{E}}_{rp}=\left(\begin{array}{c}0\\ 0.819\\ 0.574\end{array}\right)$.

Their H and S fields are

$${\mathbf{H}}_{rs}=(n_r{\mathbf{K}}_r+i\mathbf{G}){\widehat{\mathbf{E}}}_{rs}=\left(\begin{array}{lll}0& 0.819& 0.574\\ -0.819& 0& 0\\ -0.574& 0& 0\end{array}\right)\left(\begin{array}{l}1\\ 0\\ 0\end{array}\right)=\left(\begin{array}{l}0\\ -0.819\\ -0.574\end{array}\right),$$

$${\mathbf{H}}_{rp}=(n_r{\mathbf{K}}_r+i\mathbf{G}){\widehat{\mathbf{E}}}_{rp}=\left(\begin{array}{lll}0& 0.819& 0.574\\ -0.819& 0& 0\\ -0.574& 0& 0\end{array}\right)\left(\begin{array}{l}0\\ 0.819\\ 0.574\end{array}\right)=\left(\begin{array}{l}1\\ 0\\ 0\end{array}\right),$$

and

$${\widehat{\mathbf{S}}}_r=\frac{\mathbf{ℛ}\mathbf{e}\left[{\widehat{\mathbf{E}}}_r\times {\widehat{\mathbf{H}}}_r*\right]}{\left|\mathbf{ℛ}\mathbf{e}\left[{\widehat{\mathbf{E}}}_r\times {\widehat{\mathbf{H}}}_r*\right]\right|}=\left(\begin{array}{l}0\\ 0.574\\ -0.819\end{array}\right).$$

Now, we will construct the F matrix with ${\mathbf{s}}_1={\widehat{\mathbf{k}}}_{inc}\times \widehat{\mathbf{\eta }}=\left(\begin{array}{c}\text{sin}35°\\ 0\\ 0\end{array}\right)$ and ${\mathbf{s}}_2=\widehat{\mathbf{\eta }}\times {\mathbf{s}}_1=\left(\begin{array}{c}0\\ \text{sin}35°\\ 0\end{array}\right)$:

$$\mathbf{F}=\left(\begin{array}{llll}{\mathbf{s}}_1\cdot {\widehat{\mathbf{E}}}_{tf}& {\mathbf{s}}_1\cdot {\widehat{\mathbf{E}}}_{ts}& -{\mathbf{s}}_1\cdot {\widehat{\mathbf{E}}}_{rs}& -{\mathbf{s}}_1\cdot {\widehat{\mathbf{E}}}_{rp}\\ {\mathbf{s}}_2\cdot {\widehat{\mathbf{E}}}_{tf}& {\mathbf{s}}_2\cdot {\widehat{\mathbf{E}}}_{ts}& -{\mathbf{s}}_2\cdot {\widehat{\mathbf{E}}}_{rs}& -{\mathbf{s}}_2\cdot {\widehat{\mathbf{E}}}_{rp}\\ {\mathbf{s}}_1\cdot {\overrightarrow{\mathbf{H}}}_{tf}& {\mathbf{s}}_1\cdot {\overrightarrow{\mathbf{H}}}_{ts}& -{\mathbf{s}}_1\cdot {\overrightarrow{\mathbf{H}}}_{rs}& -{\mathbf{s}}_1\cdot {\overrightarrow{\mathbf{H}}}_{rp}\\ {\mathbf{s}}_2\cdot {\overrightarrow{\mathbf{H}}}_{tf}& {\mathbf{s}}_2\cdot {\overrightarrow{\mathbf{H}}}_{ts}& -{\mathbf{s}}_2\cdot {\overrightarrow{\mathbf{H}}}_{rs}& -{\mathbf{s}}_2\cdot {\overrightarrow{\mathbf{H}}}_{rp}\end{array}\right)=\left(\begin{array}{llll}0.574& 0& -0.574& 0\\ 0& -0.550& 0& -0.470\\ 0& 1.036& 0& -0.574\\ 0.970& 0& 0.470& 0\end{array}\right),$$

$${\mathbf{F}}^{-1}=\left(\begin{array}{llll}0.569& 0& 0& 0.695\\ 0& -0.715& 0.586& 0\\ -1.174& 0& 0.& 0.695\\ 0& -1.292& -0.685& 0.\end{array}\right).$$

With s-polarized incident light,

$${\widehat{\mathbf{E}}}_{inc,s}=\frac{{\widehat{\mathbf{k}}}_{inc}\times \widehat{\mathbf{\eta }}}{\Vert {\widehat{\mathbf{k}}}_{inc}\times \widehat{\mathbf{\eta }}\Vert }=\left(\begin{array}{l}1\\ 0\\ 0\end{array}\right),{\mathbf{H}}_{inc,s}=(n{\mathbf{K}}_{inc}+i\mathbf{G}){\widehat{\mathbf{E}}}_{inc,s}=\left(\begin{array}{l}0\\ \text{cos}35°\\ -\text{sin}35°\end{array}\right),$$

$${\mathbf{C}}_s=\left(\begin{array}{l}{\mathbf{s}}_1\cdot {\widehat{\mathbf{E}}}_{inc,s}\\ {\mathbf{s}}_2\cdot {\widehat{\mathbf{E}}}_{inc,s}\\ {\mathbf{s}}_1\cdot {\overrightarrow{\mathbf{H}}}_{inc,s}\\ {\mathbf{s}}_2\cdot {\overrightarrow{\mathbf{H}}}_{inc,s}\end{array}\right)=\left(\begin{array}{l}\text{sin}35°\\ 0\\ 0\\ \text{cos}35°\text{sin}35°\end{array}\right),$$

and

$${\mathbf{A}}_s={\mathbf{F}}^{-1}\cdot {\mathbf{C}}_s=\left(\begin{array}{llll}0.569& 0& 0& 0.695\\ 0& -0.715& 0.586& 0\\ -1.174& 0& 0.& 0.695\\ 0& -1.292& -0.685& 0.\end{array}\right)\left(\begin{array}{l}\text{sin}35°\\ 0\\ 0\\ \text{cos}35°\text{sin}35°\end{array}\right)=\left(\begin{array}{l}0.653\\ 0\\ -0.347\\ 0\end{array}\right)=\left(\begin{array}{l}{a}_{s\to tf}\\ a_{s\to ts}\\ a_{s\to rs}\\ a_{s\to rp}\end{array}\right).$$

With p-polarized incident light,

$${\widehat{\mathbf{E}}}_{inc,p}=\frac{{\widehat{\mathbf{k}}}_{inc}\times {\widehat{\mathbf{E}}}_{inc,s}}{\Vert {\widehat{\mathbf{k}}}_{inc}\times {\widehat{\mathbf{E}}}_{inc,s}\Vert }=\left(\begin{array}{l}0\\ \text{cos}35°\\ -\text{sin}35°\end{array}\right),{\mathbf{H}}_{inc,p}=(n{\mathbf{K}}_{inc}+i\mathbf{G}){\widehat{\mathbf{E}}}_{inc,p}=\left(\begin{array}{l}-1\\ 0\\ 0\end{array}\right),$$

$${\mathbf{C}}_p=\left(\begin{array}{l}{\mathbf{s}}_1\cdot {\widehat{\mathbf{E}}}_{inc,p}\\ {\mathbf{s}}_2\cdot {\widehat{\mathbf{E}}}_{inc,p}\\ {\mathbf{s}}_1\cdot {\overrightarrow{\mathbf{H}}}_{inc,p}\\ {\mathbf{s}}_2\cdot {\overrightarrow{\mathbf{H}}}_{inc,p}\end{array}\right)=\left(\begin{array}{l}0\\ 0.470\\ -0.574\\ 0\end{array}\right),$$

and

$${\mathbf{A}}_p={\mathbf{F}}^{-1}\cdot {\mathbf{C}}_p=\left(\begin{array}{llll}0.569& 0& 0& 0.695\\ 0& -0.715& 0.586& 0\\ -1.174& 0& 0& 0.695\\ 0& -1.292& -0.685& 0\end{array}\right)\left(\begin{array}{l}0\\ 0.470\\ -0.574\\ 0\end{array}\right)=\left(\begin{array}{l}0\\ -0.672\\ 0\\ -0.214\end{array}\right)=\left(\begin{array}{l}{a}_{p\to tf}\\ a_{p\to ts}\\ a_{p\to rs}\\ a_{p\to rp}\end{array}\right).$$

In summary, the amplitude coefficients are $\left(\begin{array}{c}{a}_{s\to tf}\\ a_{s\to ts}\\ a_{s\to rs}\\ a_{s\to rp}\end{array}\right)=\left(\begin{array}{c}0.653\\ 0\\ -0.347\\ 0\end{array}\right)$ and $\left(\begin{array}{c}{a}_{p\to tf}\\ a_{p\to ts}\\ a_{p\to rs}\\ a_{p\to rp}\end{array}\right)=\left(\begin{array}{c}0\\ -0.672\\ 0\\ -0.214\end{array}\right)$.

When Equation 19.16 yields a ray with complex k, the corresponding mode is an evanescent wave. The mode is total internally reflected for a complex kt and all the energy is reflected with zero transmission.27 In a birefringent material, inhibited reflection occurs when the reflected ray has a complex kr; then, all the energy transmits with zero reflection.28

Conical refraction is a complex phenomenon in biaxial materials where light refracts into a continuous cone of light, not just into two modes in two directions. It only happens when light propagates along one of the two optic axes2,29,30 in a biaxial material and the two orthogonal modes experience the same refractive index.31 Biaxial materials have two optic axes, as shown in Figure 19.20 (left); the optic axes do not correspond to any of the principal axes. When conical refraction occurs, the solution to Maxwell’s equations becomes degenerate with a family of k and E pairs. The associated refracted energy forms a hollow cone of light as shown in Figure 19.20 (right). The distribution of the energy depends on the incident polarization; the corresponding polarization state rotates around the cone through 180° and forms a ring on the next interface.32,33

Figure 19.20(Left) Orientations of the two optic axes of a biaxial material are perpendicular to the two circular cross sections through the index ellipsoid. (Right) A ray refracts into the direction of the biaxial optic axis. The incident polarizations distribute their Poynting vectors into a cone (the solid angle of the cone shown is exaggerated) and propagate as a cone through the biaxial crystal. The associated polarization (shown as the purple lines) rotates by π around the refracted cone. The distribution of flux around the cone depends on the distribution of incident polarization states. The fast, medium, and slow crystal axes are shown as the red, green, and blue arrows.

Figure 19.21 shows measurements of conical refraction in a KTP crystal. The distribution of the energy depends on the incident polarization; the corresponding polarization state rotates around the cone through 180° and forms a ring on the next interface. Thus, because of conical refraction, special care is required when ray tracing near the optic axes. For example, the cone of light can be modeled as a cone of a large number of rays.

Figure 19.21A biaxial KTP crystal is set up to display conical refraction and its refracted cone is measured in an imaging polarimeter. The Mueller matrix image (left) and diattenuation orientation image (right) of the refracted cone are shown.

Under conoscope (described in Section 7.7.1.5), the two optic axes of the biaxial material are easily visible as shown in Figures 19.22 and 19.23.

Figure 19.22An aragonite rotates under conoscope.

Figure 19.23A piece of muscovite rotates under conoscope.

19.6Data Structure for Ray Doubling

A data structure in the form of a tree is developed to keep track of the multiplicity of rays generated by birefringent elements when ray tracing optical systems. When a ray propagates through several birefringent interfaces, the number of resultant rays or modes typically doubles at each birefringent interface. In general, a polarization ray tracing program should not impose any assumptions and should automate the handling of ray doubling in order to properly simulate multiple wavefronts exiting the optical system. The optical designer may request a particular subset of partial rays to be traced, but otherwise, the program should generate and keep track of the entire tree of rays generated. Often, reflected rays have such a small effect that only refracted rays are traced.

Treating each split ray separately without assumptions allows us to observe polarization aberration easily by tracing a grid of rays with different positions or incident angles. In systems with only isotropic components, one incident wavefront produces one exiting wavefront. When birefringent components are involved, however, multiple wavefronts are produced as seen in Figure 19.3. The resultant wavefronts can be assembled after the ray trace by collecting and sorting rays using their sequence of eigenmodes, such as shown in the detailed analysis of a Glan-type polarizer in Chapter 22 (Crystal Polarizers). One ray passing through two Glan-type polarizers results in as many as 16 exiting rays, due to the sequence of four birefringent interfaces. Manually setting up and keeping track of 16 ray paths is tedious, so the birefringent ray tracing program should automate this.

Consider a ray propagating from one uniaxial material into another uniaxial material whose optic axes are not aligned, as shown in Figure 19.24. The eigenmodes (o1 and e1) in the first uniaxial material have different polarization orientations compared to the second material’s eigenmodes (o2 and e2). The o1 ray from the first uniaxial material splits into o2- and e2-modes in the second uniaxial material; these resultant modes will be labeled the oo- and oe-modes. This mode labeling provides a method of useful abbreviation for tracking the history of each partial ray.

Figure 19.24A ray splits into four modes after refracting through two uniaxial blocks with unaligned optic axes.

Figure 19.25 shows a ray incident on a uniaxial plate and an air space followed by a uniaxial block with an angled back surface. The two optic axes are configured, such that the normal incident ray produces four resultant rays with eie-, eio-, oie-, and oio-modes incident on the angled back surface. At each interface entering a uniaxial medium, considering only refraction, a maximum of two rays are produced due to the ray doubling inside the uniaxial material. A ray trace program traces all the ray branches in the ray tree shown in Figure 19.25, from left to right and top to bottom, to ensure that all daughter rays are accounted for. It starts at the first interface, calculates the e-mode, exits the first plate to air, calculates the ei-mode, entering the uniaxial block, and calculates the eie-mode. The eie-mode enters the last surface at an incidence angle that is greater than the critical angle for the e-mode. Hence, the eiei-mode is evanescent and the refraction branch stops with its energy being totally internally reflected back to the second uniaxial block as eiee- and eieo-modes. After the first exiting ray stopped, the program moves to the next branch where the eio-mode enters the last surface less than the critical angle for the o-mode. The eio-mode refracts out of the second plate and stops as the eioi-mode, since there are no more possible ray intercepts for this ray. The program repeats the process, moving through all the branches until all the possible ray splitting paths are calculated and all the rays terminate. The user of the ray trace program can set up ray stop criteria, such as (1) all rays stop at a certain surface, (2) all rays stop below a certain energy level, or (3) all rays stop after a certain number of surface interactions. In the system of Figure 19.25, one incident ray couples to two refracted rays through the first uniaxial plate, and four refracted rays after the first surface of the second uniaxial block. The orientation of the back surface selects the two o-modes out of four modes to refract out into air. The other two modes, eie and oie, are lost due to the evanescence. Section 19.9 contains a complex example of a ray splitting calculation and data structure, where the number of daughter rays is not simply 2N due to evanescence in reflection as well as refraction.

Figure 19.25(Left) A normal incident ray propagates through two uniaxial components with different optic axis (purple lines) as shown. (Right) A ray tree describes the resultant modes from each interface interaction through the system.

19.7Polarization Ray Tracing Matrices for Birefringent Interfaces

This section develops the 3 × 3 polarization ray tracing matrix P for each resultant mode. The P matrix keeps track of the resultant electric field direction, amplitude coefficients, and the mode direction in the global coordinates that are preferred for polarization ray tracing. As discussed in earlier chapters, Jones matrices with associated local coordinate systems can be used for ray tracing birefringent materials as well. However, our experience has convinced us that keeping the calculation in global coordinates has many advantages, and the P matrix described in Chapter 9 (Polarization Ray Tracing Calculus) is the ideal tool to accurately keep track of the polarization changes throughout complex systems.

The P matrix describes polarization interactions by generalizing Jones matrices into a 3 × 3 matrix formalism to handle arbitrary propagation directions in 3D coordinates; this avoids carrying along local coordinates, which is necessary with Jones matrices. A formalism that calculates the P matrix directly from the 3D orthonormal basis is presented in this section. This formalism is the basis for calculating the P matrix through birefringent components using only the ray parameters (S, E, a, and OPL) calculated from the birefringent ray tracing algorithms shown in Section 19.5. The resultant P matrices are used to study the polarization aberrations of birefringent elements, such as the angle dependence of crystal retarders and polarizers. For a pair of incident and exiting modes through a birefringent intercept, the P matrix relates the electric field amplitude and phase across the interface as well as the change of propagation direction. For example, part of the energy of an e-mode refracts to a fast-mode at a uniaxial/biaxial interface, or the p-polarized component couples to right circularly polarized light through an isotropic/optically active interface.

The electromagnetic fields of a pair of incident and exiting modes propagating through an isotropic interface and a birefringent interface are depicted in Figure 19.26. ${\widehat{\mathbf{E}}}_s\perp {\widehat{\mathbf{E}}}_p\perp \widehat{\mathbf{S}}=\widehat{\mathbf{k}}$ within the isotropic medium, and ${\widehat{\mathbf{E}}}_m\perp {\widehat{\mathbf{E}}}_n\perp \widehat{\mathbf{S}}\ne \widehat{\mathbf{k}}$ within the birefringent medium, where ${\widehat{\mathbf{E}}}_m$ and ${\widehat{\mathbf{E}}}_n$ are orthogonal electric fields in the transverse plane of $\widehat{\mathbf{S}}$.

Figure 19.26The incident and refracted E, D, k, and S field orientations of (left) an isotropic-to-isotropic and (right) a birefringent-to-birefringent interface. Only p-polarization coupling to p-polarization is shown refracting through the isotropic interface. Only e-mode coupling to e-mode is shown refracting through the birefringent interface. In isotropic media, the k and S are in the same direction, and the E and D are in the same direction. These fields are no longer aligned in anisotropic materials, but E stays orthogonal to S, and k stays orthogonal to D.

In a birefringent ray trace, two P matrices are needed to represent the two refracted or reflected modes at a birefringent ray intercept because the two exiting rays take different paths. Since the incident and exiting eigenmodes are generally not aligned, one of the incident eigenmodes can couple light into both exiting eigenmodes. The P matrix maps the three incident orthonormal basis vectors $({\widehat{\mathbf{E}}}_m,{\widehat{\mathbf{E}}}_n,\widehat{\mathbf{S}})$ into three exiting vectors $\left({{\mathbf{E}}^{\prime }}_m,{{\mathbf{E}}^{\prime }}_n,{\widehat{\mathbf{S}}}^{\prime }\right)$ associated with one eigenstate in the exiting medium along ${\widehat{\mathbf{S}}}^{\prime }$. Thus, the conditions defining the two exiting P matrices at a birefringent ray intercept are

$$\{\begin{array}{l}{\mathbf{P}}_v{\widehat{\mathbf{E}}}_m={{\mathbf{E}}^{\prime }}_{mv}=a_{mv}{\widehat{\mathbf{E}}}_v,\\ {\mathbf{P}}_v{\widehat{\mathbf{E}}}_n={{\mathbf{E}}^{\prime }}_{nv}=a_{nv}{\widehat{\mathbf{E}}}_v,\\ {\mathbf{P}}_v\widehat{\mathbf{S}}={{\widehat{\mathbf{S}}}^{\prime }}_v,\end{array}\text{and}\{\begin{array}{l}{\mathbf{P}}_w{\widehat{\mathbf{E}}}_m={{\mathbf{E}}^{\prime }}_{mw}=a_{mw}{\widehat{\mathbf{E}}}_w,\\ {\mathbf{P}}_w{\widehat{\mathbf{E}}}_n={{\mathbf{E}}^{\prime }}_{nw}=a_{nw}{\widehat{\mathbf{E}}}_w,\\ {\mathbf{P}}_w\widehat{\mathbf{S}}={{\widehat{\mathbf{S}}}^{\prime }}_w,\end{array}$$19.30

19.30

for refraction/reflection, where (amv, amw, anv, anw) are the complex amplitude coefficients associated with the E field coupling between each pair of incident and exiting states. These coefficients are calculated in Section 19.5.

The exiting v- and w-modes of Equation 19.30 are the transmitted ta- and tb-modes or the reflected rc- and rd-modes in Figure 19.17. In general, the incident ${\widehat{\mathbf{E}}}_m$ couples to both ${{\mathbf{E}}^{\prime }}_{mv}$ and ${{\mathbf{E}}^{\prime }}_{mw}$, and ${\widehat{\mathbf{E}}}_n$ couples to both ${{\mathbf{E}}^{\prime }}_{nv}$ and ${{\mathbf{E}}^{\prime }}_{nw}$. These exiting E fields ${{\mathbf{E}}^{\prime }}_{mv}$ and ${{\mathbf{E}}^{\prime }}_{nv}$ are in the transverse plane of ${{\widehat{\mathbf{S}}}^{\prime }}_v$; ${{\mathbf{E}}^{\prime }}_{mw}$ and ${{\mathbf{E}}^{\prime }}_{nw}$ are in the transverse plane of ${{\widehat{\mathbf{S}}}^{\prime }}_w$ in the exiting medium. The refracted/reflected exiting electric fields resulting from ${\widehat{\mathbf{E}}}_m$ and ${\widehat{\mathbf{E}}}_n$ at the ray intercept are

$${{\mathbf{E}}^{\prime }}_m(\mathbf{r},t)=\mathbf{ℛ}\mathbf{e}\left\{{\mathbf{E}}_{inc}\left[a_{mv}{\widehat{\mathbf{E}}}_v{e}^{i\frac{2\pi }{\lambda }{n}_{mv}{\widehat{\mathbf{k}}}_{mv}\cdot \mathbf{r}}+a_{mw}{\widehat{\mathbf{E}}}_w{e}^{i\frac{2\pi }{\lambda }{n}_{mw}{\widehat{\mathbf{k}}}_{mw}\cdot \mathbf{r}}\right]e^{-i\omega t}\right\}$$19.31

19.31

and

$${{\mathbf{E}}^{\prime }}_n(\mathbf{r},t)=\mathbf{ℛ}\mathbf{e}\left\{E_{inc}\left[a_{nv}{\widehat{\mathbf{E}}}_v{e}^{i\frac{2\pi }{\lambda }{n}_{nv}{\widehat{\mathbf{k}}}_{nv}\cdot \mathbf{r}}+a_{nw}{\widehat{\mathbf{E}}}_w{e}^{i\frac{2\pi }{\lambda }{n}_{nw}{\widehat{\mathbf{k}}}_{nw}\cdot \mathbf{r}}\right]e^{-i\omega t}\right\}.$$19.32

19.32

Consider the o- and e-modes with two different ${\widehat{\mathbf{S}}}^{\prime }$ emerging from an isotropic/uniaxial intercept: they each have their own P matrix because ${{\widehat{\mathbf{S}}}^{\prime }}_o\ne {{\widehat{\mathbf{S}}}^{\prime }}_e$. In this case, (m, n, v, w) in Equation 19.30 are (s, p, o, e); the detailed calculations of Po and Pe are described in Section 19.7.2. In general, ${\widehat{\mathbf{E}}}_m$ can couple to both ${\widehat{\mathbf{E}}}_v$ and ${\widehat{\mathbf{E}}}_w$, and (amv, amw, anv, anw) are all non-zero. In some situations, the amplitude coefficients are set to zero for certain properties of the interface. For example, at an uncoated isotropic ray intercept, the coupling between s- and p-polarizations is zero; (m, n, v, w) = (s, p, s′, p′) and (amw, anv) = (asp′, aps′) = (0, 0). Also, if a ray is incident at a uniaxial/isotropic intercept polarized in only one eigenmode for a given incident $\widehat{\mathbf{S}}$, then (m, n, v, w) = (e, e⊥, s, p) and (anv, anw) = (ae⊥s, ae⊥p) = (0, 0), because e⊥-mode carries zero energy to begin with. This will be further explained in Section 19.7.3. Since each exiting mode is in a particular eigenpolarization, the corresponding P matrix of the exiting mode is a singular matrix; that is, it has the form of a polarizer.

By placing the three pairs of incident and exiting 3 × 1 vectors in matrix form, the P matrix is calculated as

$$\begin{array}{l}\mathbf{P}=\left(\begin{array}{lll}{{\mathbf{E}}^{\prime }}_m& {E^{\prime }}_n& {\widehat{\mathbf{S}}}^{\prime }\end{array}\right)•{\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_m& {\widehat{\mathbf{E}}}_n& \widehat{\mathbf{S}}\end{array}\right)}^{-1}\\ \phantom{P}=\left(\begin{array}{lll}{E^{\prime }}_{m,x}& {E^{\prime }}_{n,x}& {S^{\prime }}_x\\ {E^{\prime }}_{m,y}& {E^{\prime }}_{n,y}& {S^{\prime }}_y\\ {E^{\prime }}_{m,z}& {E^{\prime }}_{n,z}& {S^{\prime }}_z\end{array}\right)•{\left(\begin{array}{lll}{E}_{m,x}& E_{n,x}& S_x\\ E_{m,y}& E_{n,y}& S_y\\ E_{m,z}& E_{n,z}& S_z\end{array}\right)}^{-1}\\ \phantom{P}=\left(\begin{array}{lll}{E^{\prime }}_{m,x}& {E^{\prime }}_{n,x}& {S^{\prime }}_x\\ {E^{\prime }}_{m,y}& {E^{\prime }}_{n,y}& {S^{\prime }}_y\\ {E^{\prime }}_{m,z}& {E^{\prime }}_{n,z}& {S^{\prime }}_z\end{array}\right)•{\left(\begin{array}{lll}{E}_{m,x}& E_{n,x}& S_x\\ E_{m,y}& E_{n,y}& S_y\\ E_{m,z}& E_{n,z}& S_z\end{array}\right)}^T,\end{array}$$19.33

19.33

where $\left(\begin{array}{ccc}{\widehat{\mathbf{E}}}_m& {\widehat{\mathbf{E}}}_n& \widehat{\mathbf{S}}\end{array}\right)$ is a real unitary matrix; hence, its inverse equals its transpose. The amplitude coefficients for the interface are contained in $\left(\begin{array}{ccc}{{\mathbf{E}}^{\prime }}_m& {{\mathbf{E}}^{\prime }}_n& {\widehat{\mathbf{S}}}^{\prime }\end{array}\right)$.

The multiple exiting modes from birefringent interfaces need to be described by multiple P matrices. The derivations of the P matrices for the four cases of uncoated birefringent interfaces—(1) isotropic/isotropic, (2) isotropic/birefringent, (3) birefringent/birefringent, and (4) birefringent/birefringent interfaces in refraction and reflection—are shown in the following subsections. In the derivations, the (m, n, v, w) states used in Equation 19.30 for each of the cases are shown in Table 19.4. In an exiting isotropic medium, the exiting s′- and p′-modes share the same S; thus, the associated P matrices can be reduced to one P matrix (see Sections 19.7.1 through 19.7.3). In an incident birefringent medium, the incident (m, n) modes are (o, o⊥), (e, e⊥), (fast, fast⊥), (slow, slow⊥), (right, right⊥), or (left, left⊥) modes. In an exiting birefringent medium, the exiting (v, w) are (o, e), (fast, slow), or (right, left) modes.

Table 19.4The Polarization States for Calculating the P Matrices for an Uncoated Interface

Interface	Reflected(m, n)→v=rc and (m, n)→w=rd		Refracted(m, n)→v=ta and (m, n)→w=tb
Isotropic/isotropic	(s, p)→s′ (s, p)→p′	⇒ (s, p)→(s′, p′)	(s, p)→s′ (s, p)→p′	⇒ (s, p)→(s′, p′)
Isotropic/birefringent	(s, p)→s′ (s, p)→p′	⇒ (s, p)→(s′, p′)	(s, p)→v (s, p)→w
Birefringent/isotropic	(m, n)→ v (m, n)→ w		(m, n)→s′ (m, n)→p′	⇒ (m, n)→(s′, p′)
Birefringent/birefringent	(m, n)→v (m, n)→w		(m, n)→v (m, n)→w

Note: (m, n, v, w) are defined in Equation 19.30. ′ indicates an exiting mode. The split eigenmodes in a birefringent material are described by two P matrices. However, the two P matrices for s′ and p′ exiting states can be combined (⇒) to one P matrix.

19.7.1Case I: Isotropic-to-Isotropic Intercept

For the isotropic-to-isotropic interface, the incident eigenstates are s- and p-polarizations,

$$\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}=\frac{{\widehat{\mathbf{S}}}_{inc}\times \widehat{\mathbf{\eta }}}{\left|{\widehat{\mathbf{S}}}_{inc}\times \widehat{\mathbf{\eta }}\right|}\hfill & \text{and}\hfill & {\widehat{\mathbf{E}}}_{inc,p}={\widehat{\mathbf{S}}}_{inc}\times {\widehat{\mathbf{E}}}_{inc,s},\hfill \end{array}$$19.34

19.34

as shown in Figure 19.27, where s and p are m and n in Equation 19.33. In this case, the four exiting modes are ts, tp, rs, and rp associated with four P matrices, Pts, Ptp, Prs, and Prp.

Figure 19.27A p-polarized (left) and an s-polarized (right) incident ray are launched into the interface to calculate E, S, a, and the P matrix of all exiting modes. The planes of incidence are shown in both cases. The arrows indicate the S directions. (Left) The triple parallel lines (blue and green) shown in the plane of incidence are the direction of Ep. (Right) The triple parallel lines (red and magenta) shown orthogonal to the plane of incidence are the direction of Es.

Equation 19.34 is for the isotropic interface, where $\widehat{\mathbf{k}}=\widehat{\mathbf{S}}$. No ray splitting occurs; the two reflected rays are degenerate, as are the two transmitted rays. Although the figure shows the amplitude coefficients for a reflected and transmitted p-polarized ray for the incident s-polarized ray to provide the most general description of the P matrix calculation, the coupling between s and p is zero for an uncoated isotropic interface.

Their electric fields ${\widehat{\mathbf{E}}}_{ts}$, ${\widehat{\mathbf{E}}}_{tp}$, ${\widehat{\mathbf{E}}}_{rs}$, and ${\widehat{\mathbf{E}}}_{rp}$, and propagation vectors are calculated by the method of Section 19.5. The amplitude coefficients as and ap are calculated by Equation 19.23 as

$$\begin{array}{lll}\left(\begin{array}{l}{a}_{inc,s\to ts}\\ a_{inc,s\to tp}\\ a_{inc,s\to rs}\\ a_{inc,s\to rp}\end{array}\right)={\mathbf{F}}^{-1}\cdot \left(\begin{array}{l}{\overrightarrow{\mathbf{s}}}_1\cdot {\widehat{\mathbf{E}}}_{inc,s}\\ {\overrightarrow{\mathbf{s}}}_2\cdot {\widehat{\mathbf{E}}}_{inc,s}\\ {\overrightarrow{\mathbf{s}}}_1\cdot {\overrightarrow{\mathbf{H}}}_{inc,s}\\ {\overrightarrow{\mathbf{s}}}_2\cdot {\overrightarrow{\mathbf{H}}}_{inc,s}\end{array}\right)\hfill & \text{and}\hfill & \left(\begin{array}{l}{a}_{inc,p\to ts}\\ a_{inc,p\to tp}\\ a_{inc,p\to rs}\\ a_{inc,p\to rp}\end{array}\right)={\mathbf{F}}^{-1}\cdot \left(\begin{array}{l}{\overrightarrow{\mathbf{s}}}_1\cdot {\widehat{\mathbf{E}}}_{inc,p}\\ {\overrightarrow{\mathbf{s}}}_2\cdot {\widehat{\mathbf{E}}}_{inc,p}\\ {\overrightarrow{\mathbf{s}}}_1\cdot {\overrightarrow{\mathbf{H}}}_{inc,p}\\ {\overrightarrow{\mathbf{s}}}_2\cdot {\overrightarrow{\mathbf{H}}}_{inc,p}\end{array}\right)\hfill \end{array},$$19.35

19.35

where the electric field amplitude from incident s-polarization coupled into the transmitted s-polarization is denoted as ainc,s→ts. For uncoated dielectric interfaces, this yields the Fresnel amplitude coefficients. With these amplitude coefficients and applying Equation 19.30, the P matrices for the exiting s- and p-modes should satisfy

$$\begin{array}{ll}\{\begin{array}{l}{\mathbf{P}}_{rs}{\widehat{\mathbf{E}}}_{inc,s}=a_{inc,s\to rs}{\widehat{\mathbf{E}}}_{rs}\\ {\mathbf{P}}_{rs}{\widehat{\mathbf{E}}}_{inc,p}=a_{inc,p\to rs}{\widehat{\mathbf{E}}}_{rs},\\ {\mathbf{P}}_{rs}{\widehat{\mathbf{S}}}_{inc}={\widehat{\mathbf{S}}}_{rs}\end{array}& \{\begin{array}{l}{\mathbf{P}}_{rp}{\widehat{\mathbf{E}}}_{inc,s}=a_{inc,s\to rp}{\widehat{\mathbf{E}}}_{rp}\\ {\mathbf{P}}_{rp}{\widehat{\mathbf{E}}}_{inc,p}=a_{inc,p\to rp}{\widehat{\mathbf{E}}}_{rp},\\ {\mathbf{P}}_{rp}{\widehat{\mathbf{S}}}_{inc}={\widehat{\mathbf{S}}}_{rp}\end{array}\\ \{\begin{array}{l}{\mathbf{P}}_{ts}{\widehat{\mathbf{E}}}_{inc,s}=a_{inc,s\to ts}{\widehat{\mathbf{E}}}_{ts}\\ {\mathbf{P}}_{ts}{\widehat{\mathbf{E}}}_{inc,p}=a_{inc,p\to ts}{\widehat{\mathbf{E}}}_{ts},\\ {\mathbf{P}}_{ts}{\widehat{\mathbf{S}}}_{inc}={\widehat{\mathbf{S}}}_{ts}\end{array}& \{\begin{array}{l}{\mathbf{P}}_{tp}{\widehat{\mathbf{E}}}_{inc,s}=a_{inc,s\to tp}{\widehat{\mathbf{E}}}_{tp}\\ {\mathbf{P}}_{tp}{\widehat{\mathbf{E}}}_{inc,p}=a_{inc,p\to tp}{\widehat{\mathbf{E}}}_{tp}.\\ {\mathbf{P}}_{tp}{\widehat{\mathbf{S}}}_{inc}={\widehat{\mathbf{S}}}_{tp}\end{array}\end{array}$$19.36

19.36

Using Equation 19.33, the four P matrices are calculated as

$$\begin{array}{l}{\mathbf{P}}_{ts}=\left(\begin{array}{lll}{a}_{inc,s\to ts}{\widehat{\mathbf{E}}}_{ts}& a_{inc,p\to ts}{\widehat{\mathbf{E}}}_{ts}& {\widehat{\mathbf{S}}}_{ts}\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T,\\ {\mathbf{P}}_{tp}=\left(\begin{array}{lll}{a}_{inc,s\to tp}{\widehat{\mathbf{E}}}_{tp}& a_{inc,p\to tp}{\widehat{\mathbf{E}}}_{tp}& {\widehat{\mathbf{S}}}_{tp}\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T,\\ {\mathbf{P}}_{rs}=\left(\begin{array}{lll}{a}_{inc,s\to rs}{\widehat{\mathbf{E}}}_{rs}& a_{inc,p\to rs}{\widehat{\mathbf{E}}}_{rs}& {\widehat{\mathbf{S}}}_{rs}\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T,\text{and}\\ {\mathbf{P}}_{rp}=\left(\begin{array}{lll}{a}_{inc,s\to rp}{\widehat{\mathbf{E}}}_{rp}& a_{inc,p\to rp}{\widehat{\mathbf{E}}}_{rp}& {\widehat{\mathbf{S}}}_{tp}\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T.\end{array}$$19.37

19.37

As depicted in Figure 19.17a, inside isotropic materials, ${\widehat{\mathbf{S}}}_{ts}={\widehat{\mathbf{S}}}_{tp}$ and ${\widehat{\mathbf{S}}}_{rs}={\widehat{\mathbf{S}}}_{rp}$. The result can be obtained by combining the two exiting modes. The electric fields can always be added, but care is needed in adding P matrices.5 Since the condition of ${\widehat{\mathbf{S}}}_t={\mathbf{P}}_t{\widehat{\mathbf{S}}}_{inc}$ is present for both modes, Pts and Ptp can be combined as follows:

$${\mathbf{P}}_t=\left(\begin{array}{lll}{a}_{inc,s\to ts}{\widehat{\mathbf{E}}}_{ts}+a_{inc,s\to tp}{\widehat{\mathbf{E}}}_{tp}& a_{inc,p\to ts}{\widehat{\mathbf{E}}}_{ts}+a_{inc,p\to tp}{\widehat{\mathbf{E}}}_{tp}& {\widehat{\mathbf{S}}}_t\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T.$$19.38

19.38

Similarly, Prs and Prp can be combined as

$${\mathbf{P}}_r=\left(\begin{array}{lll}{a}_{inc,s\to rs}{\widehat{\mathbf{E}}}_{rs}+a_{inc,s\to rp}{\widehat{\mathbf{E}}}_{rp}& a_{inc,p\to rs}{\widehat{\mathbf{E}}}_{rs}+a_{inc,p\to rp}{\widehat{\mathbf{E}}}_{rp}& {\widehat{\mathbf{S}}}_r\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T.$$19.39

19.39

Equation 19.37 shows the most general derivation for P matrices, which aids in understanding the procedures for non-isotropic interfaces introduced in Section 19.7.2 through 19.7.4. For an isotropic interface, ainc,s→tp, ainc,s→rp, ainc,p→ts, and ainc,p→rs are zero since the s-component only couples to s-polarization and the p-component only couples to p-polarization. Therefore, Equations 19.38 and 19.39 become

$$\begin{array}{l}{\mathbf{P}}_t=\left(\begin{array}{lll}{a}_{inc,s\to ts}{\widehat{\mathbf{E}}}_{ts}& a_{inc,p\to tp}{\widehat{\mathbf{E}}}_{tp}& {\widehat{\mathbf{S}}}_t\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T\text{and}\\ {\mathbf{P}}_r=\left(\begin{array}{lll}{a}_{inc,s\to rs}{\widehat{\mathbf{E}}}_{rs}& a_{inc,p\to rp}{\widehat{\mathbf{E}}}_{rp}& {\widehat{\mathbf{S}}}_r\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T.\end{array}$$19.40

19.40

19.7.2Case II: Isotropic-to-Birefringent Interface

The isotropic-to-birefringent interface case uses the same s- and p-incident basis as in the isotropic interface case in Equation 19.34. As shown in Figure 19.17b, the four exiting modes are rs, rp, ta, and tb; the two reflected modes reflect back to the incident isotropic medium and two refracted modes refract into the birefringent medium. The two reflected s- and p-modes share the same ${\widehat{\mathbf{S}}}^{\prime }$, while the two refracted birefringent modes, labeled with subscripts v and w in Equation 19.30, split into two directions. Hence, (m, n, v, w) = (s, p, s′, p′) in reflection and (s, p, v, w) for refraction. If the refracting medium is biaxial, the two refracted modes are fast- and slow-modes; (m, n, v, w) = (s, p, fast, slow). The refraction of this case is depicted in Figure 19.28.

Figure 19.28Mode coupling in refraction through an isotropic-to-birefringent interface. The incident ray with orthogonal modes, labeled as n (red) and m (blue), splits into two exiting modes as v = ta (pink) and w = tb (green) in two directions. In biaxial and uniaxial materials, Ev and Ew are linearly polarized. In optically active materials, Ev and Ew are circularly polarized. In birefringent and optically active materials like quartz, Ev and Ew are elliptically polarized. Given a ray with a specific pair of ${{\mathbf{k}}^{\prime }}_v$ and ${{\mathbf{S}}^{\prime }}_v$ in a birefringent material, the ray can only be polarized in Ev; thus, the orthogonal state Ev⊥ (dashed arrow) has zero amplitude.

The two refracted modes propagate in two different directions with ${\widehat{\mathbf{k}}}_{ta}\ne {\widehat{\mathbf{k}}}_{tb}\ne {\widehat{\mathbf{S}}}_{ta}\ne {\widehat{\mathbf{S}}}_{tb}$. From Equation 19.35, the amplitude coefficients of the exiting modes from each incident mode are

$$\begin{array}{lll}\left(\begin{array}{l}{a}_{inc,s\to ta}\\ a_{inc,s\to tb}\\ a_{inc,s\to rs}\\ a_{inc,s\to rp}\end{array}\right)={\mathbf{F}}^{-1}\cdot \left(\begin{array}{l}{\overrightarrow{\mathbf{s}}}_1\cdot {\widehat{\mathbf{E}}}_{inc,s}\\ {\overrightarrow{\mathbf{s}}}_2\cdot {\widehat{\mathbf{E}}}_{inc,s}\\ {\overrightarrow{\mathbf{s}}}_1\cdot {\overrightarrow{\mathbf{H}}}_{inc,s}\\ {\overrightarrow{\mathbf{s}}}_2\cdot {\overrightarrow{\mathbf{H}}}_{inc,s}\end{array}\right)\hfill & \text{and}\hfill & \left(\begin{array}{l}{a}_{inc,p\to ta}\\ a_{inc,p\to tb}\\ a_{inc,p\to rs}\\ a_{inc,p\to rp}\end{array}\right)={\mathbf{F}}^{-1}\cdot \left(\begin{array}{l}{\overrightarrow{\mathbf{s}}}_1\cdot {\widehat{\mathbf{E}}}_{inc,p}\\ {\overrightarrow{\mathbf{s}}}_2\cdot {\widehat{\mathbf{E}}}_{inc,p}\\ {\overrightarrow{\mathbf{s}}}_1\cdot {\overrightarrow{\mathbf{H}}}_{inc,p}\\ {\overrightarrow{\mathbf{s}}}_2\cdot {\overrightarrow{\mathbf{H}}}_{inc,p}\end{array}\right).\hfill \end{array}$$19.41

19.41

Using Equations 19.16 and 19.19, ${\widehat{\mathbf{E}}}_{ta}$, ${\widehat{\mathbf{E}}}_{tb}$, ${\widehat{\mathbf{E}}}_{rs}$, and ${\widehat{\mathbf{E}}}_{rp}$ can be calculated. The two P matrices for refraction are

$$\begin{array}{l}{\mathbf{P}}_{ta}=\left(\begin{array}{lll}{a}_{inc,s\to ta}{\widehat{\mathbf{E}}}_{ta}& a_{inc,p\to ta}{\widehat{\mathbf{E}}}_{ta}& {\widehat{\mathbf{S}}}_{ta}\end{array}\right)•{\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T\text{and}\\ {\mathbf{P}}_{tb}=\left(\begin{array}{lll}{a}_{inc,s\to tb}{\widehat{\mathbf{E}}}_{tb}& a_{inc,p\to tb}{\widehat{\mathbf{E}}}_{tb}& {\widehat{\mathbf{S}}}_{tb}\end{array}\right)•{\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T,\end{array}$$19.42

19.42

and the two P matrices for reflection are

$$\begin{array}{l}{\mathbf{P}}_{rs}=\left(\begin{array}{lll}{a}_{inc,s\to rs}{\widehat{\mathbf{E}}}_{rs}& a_{inc,p\to rs}{\widehat{\mathbf{E}}}_{rs}& {\widehat{\mathbf{S}}}_{rs}\end{array}\right)•{\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T\text{and}\\ {\mathbf{P}}_{rp}=\left(\begin{array}{lll}{a}_{inc,s\to rp}{\widehat{\mathbf{E}}}_{rp}& a_{inc,p\to rp}{\widehat{\mathbf{E}}}_{rp}& {\widehat{\mathbf{S}}}_{rp}\end{array}\right)•{\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T.\end{array}$$19.43

19.43

The two reflected modes share the same pair of ${\widehat{\mathbf{S}}}_{rs}={\widehat{\mathbf{S}}}_{rp}$, and the couplings between s- and p-states are zero for an uncoated surface (ainc,s→rp = ainc,p→rs = 0). Therefore, Prs and Prp are combined to

$${\mathbf{P}}_r=\left(\begin{array}{lll}{a}_{inc,s\to rs}{\widehat{\mathbf{E}}}_{rs}& a_{inc,p\to rp}{\widehat{\mathbf{E}}}_{rp}& {\widehat{\mathbf{S}}}_r\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,s}& {\widehat{\mathbf{E}}}_{inc,p}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T,$$19.44

19.44

the same as in Equation 19.40. When light refracts or reflects into an isotropic medium, the two modes combine to one P matrix because they have the same S direction. However, when light propagates into a birefringent medium, the incident ray splits into two directions and the exiting modes have two different S. In this case, two P matrices are needed to describe the two modes, which cannot be combined.

19.7.3Case III: Birefringent-to-Isotropic Interface

A ray in a birefringent incident medium with a specified ${\widehat{\mathbf{k}}}_{inc}$ and ${\widehat{\mathbf{S}}}_{inc}$ on an interface is constrained to be one of the two eigenmodes that is calculated from the previous ray intercept by Equations 19.17 through 19.19. The eigenmode is o- or e-modes for uniaxial materials, fast- or slow-modes for biaxial materials, and right- or left-modes for optically active materials. The electric field and magnetic field for this incident ray are ${\widehat{\mathbf{E}}}_m$ and ${\widehat{\mathbf{H}}}_m$with index nm. To construct the P matrix, a pseudo electric field or absent mode${\widehat{\mathbf{E}}}_n={\widehat{E}}_{m\perp }$ conveying no power and orthogonal to ${\widehat{\mathbf{E}}}_m$ is calculated as

$${\widehat{\mathbf{E}}}_n=\frac{{\widehat{\mathbf{S}}}_{inc}\times {\widehat{\mathbf{E}}}_m}{\left|{\widehat{\mathbf{S}}}_{inc}\times {\widehat{\mathbf{E}}}_m\right|}.$$19.45

19.45

This absent mode has no power because the orthogonal polarization has refracted into another direction and is described by the other P matrix. This state needs to be defined, however, to properly set up our 3 × 3 mode matrices.

The exiting modes for ${\widehat{\mathbf{E}}}_m$ are the s- and p-polarized transmitted states and the two bi-reflected rays with ${\widehat{\mathbf{E}}}_v$ and ${\widehat{\mathbf{E}}}_w$. The refraction is depicted in Figure 19.29.

Figure 19.29Mode coupling in refraction through a birefringent-to-isotropic interface. The incident ray is polarized along Em (blue) and has a zero amplitude component En (red, dashed arrow). The incident polarization couples to the s- and p-states in the isotropic medium, which are Ev (pink) and Ew (green) and propagate in the same direction S′.

The four exiting electric fields in transmission and reflection are ${\widehat{\mathbf{E}}}_{ts}$, ${\widehat{\mathbf{E}}}_{tp}$, ${\widehat{\mathbf{E}}}_{rc}$, and ${\widehat{\mathbf{E}}}_{rd}$. Inside the incident birefringent medium, only ${\widehat{\mathbf{E}}}_m$ carries non-zero amplitude,

$$\left(\begin{array}{l}{a}_{inc,m\to ts}\\ a_{inc,m\to tp}\\ a_{inc,m\to rc}\\ a_{inc,m\to rd}\end{array}\right)={\mathbf{F}}^{-1}\cdot \left(\begin{array}{l}{\overrightarrow{\mathbf{s}}}_1\cdot {\widehat{\mathbf{E}}}_m\\ {\overrightarrow{\mathbf{s}}}_2\cdot {\widehat{\mathbf{E}}}_m\\ {\overrightarrow{\mathbf{s}}}_1\cdot {\overrightarrow{\mathbf{H}}}_m\\ {\overrightarrow{\mathbf{s}}}_2\cdot {\overrightarrow{\mathbf{H}}}_m\end{array}\right),$$19.46

19.46

and the amplitude coefficients from ${\widehat{\mathbf{E}}}_n$, (ainc,n→ts, ainc,n→tp, ainc,n→ts, ainc,n→ts) are zeros. As depicted in Figure 19.29, the three pairs of conditions for each transmitted P matrix are

$$\begin{array}{lll}\{\begin{array}{l}{\mathbf{P}}_{ts}{\widehat{\mathbf{E}}}_m=a_{inc,ms\to ts}{\widehat{\mathbf{E}}}_{ts},\\ {\mathbf{P}}_{ts}{\widehat{\mathbf{E}}}_n=a_{inc,ns\to ts}{\widehat{\mathbf{E}}}_{ts}=\mathbf{0},\\ {\mathbf{P}}_{ts}{\widehat{\mathbf{S}}}_{inc}={\widehat{\mathbf{S}}}_{ts},\end{array}& \text{and}& \{\begin{array}{l}{\mathbf{P}}_{tp}{\widehat{\mathbf{E}}}_m=a_{inc,ms\to tp}{\widehat{\mathbf{E}}}_{tp},\\ {\mathbf{P}}_{tp}{\widehat{\mathbf{E}}}_n=a_{inc,ns\to tp}{\widehat{\mathbf{E}}}_{tp}=\mathbf{0},\\ {\mathbf{P}}_{tp}{\widehat{\mathbf{S}}}_{inc}={\widehat{\mathbf{S}}}_{tp},\end{array}\end{array}$$19.47

19.47

where 0 is a 3 × 1 zero vector. Since the transmitted medium is isotropic, ${\widehat{\mathbf{S}}}_{ts}={\widehat{\mathbf{S}}}_{tp}={\widehat{\mathbf{S}}}_t$, and the refracted P matrix is

$${\mathbf{P}}_t=\left(\begin{array}{lll}{a}_{inc,m\to ts}{\widehat{\mathbf{E}}}_{ts}+a_{inc,m\to tp}{\widehat{\mathbf{E}}}_{tp}& \mathbf{0}& {\widehat{\mathbf{S}}}_t\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_m& {\widehat{\mathbf{E}}}_n& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T.$$19.48

19.48

The two reflected P matrices for the two rays reflecting back to the birefringent medium are

$$\begin{array}{l}{\mathbf{P}}_{rc}=\left(\begin{array}{lll}{a}_{inc,m\to rc}{\widehat{\mathbf{E}}}_{rc}& \mathbf{0}& {\widehat{\mathbf{S}}}_{rc}\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,m}& {\widehat{\mathbf{E}}}_{inc,n}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T\text{and}\\ {\mathbf{P}}_{rd}=\left(\begin{array}{lll}{a}_{inc,m\to rd}{\widehat{\mathbf{E}}}_{rd}& \mathbf{0}& {\widehat{\mathbf{S}}}_{rd}\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,m}& {\widehat{\mathbf{E}}}_{inc,n}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T.\end{array}$$19.49

19.49

The calculation shown in this section is for one incident mode only. In general, each incident mode has its own associated calculations of P matrices for transmission and reflection.

19.7.4Case IV: Birefringent-to-Birefringent Interface

Similar to Section 19.7.3, the basis for the incident fields is chosen as ${\widehat{\mathbf{E}}}_m$ and ${\widehat{\mathbf{E}}}_n$, where ${\widehat{\mathbf{E}}}_n$ is an absent mode with zero energy constructed orthogonal to ${\widehat{\mathbf{E}}}_m$. The four exiting modes all propagate in different directions. The refraction is depicted in Figure 19.30.

Figure 19.30Mode coupling in refraction through a birefringent-to-birefringent interface. The incident ray propagating in S is polarized along Em (blue) and has a zero amplitude component En (red, dashed arrow). It splits into two exiting modes as v (green) and w (pink) in two directions. Their orthogonal states Ev⊥ and Ew⊥ (dashed arrows) both have zero amplitude.

The amplitude coefficients associated with ${\widehat{\mathbf{E}}}_m$ are calculated by Equation 19.45, and the amplitude coefficients associated with ${\widehat{\mathbf{E}}}_n$ are zeros. The P matrices for the two transmitted rays are

$$\begin{array}{l}{\mathbf{P}}_{ta}=\left(\begin{array}{lll}{a}_{inc,m\to ta}{\widehat{\mathbf{E}}}_{ta}& \mathbf{0}& {\widehat{\mathbf{S}}}_{ta}\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,m}& {\widehat{\mathbf{E}}}_{inc,n}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T\text{and}\\ {\mathbf{P}}_{tb}=\left(\begin{array}{lll}{a}_{inc,m\to tb}{\widehat{\mathbf{E}}}_{tb}& \mathbf{0}& {\widehat{\mathbf{S}}}_{tb}\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,m}& {\widehat{\mathbf{E}}}_{inc,n}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T.\end{array}$$19.50

19.50

Similarly, the P matrices for the two reflected rays are

$$\begin{array}{l}{\mathbf{P}}_{rc}=\left(\begin{array}{lll}{a}_{inc,m\to rc}{\widehat{\mathbf{E}}}_{rc}& \mathbf{0}& {\widehat{\mathbf{S}}}_{rc}\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,m}& {\widehat{\mathbf{E}}}_{inc,n}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T\text{and}\\ {\mathbf{P}}_{rd}=\left(\begin{array}{lll}{a}_{inc,m\to rd}{\widehat{\mathbf{E}}}_{rd}& \mathbf{0}& {\widehat{\mathbf{S}}}_{rd}\end{array}\right)\cdot {\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_{inc,m}& {\widehat{\mathbf{E}}}_{inc,n}& {\widehat{\mathbf{S}}}_{inc}\end{array}\right)}^T.\end{array}$$19.51

19.51

Example 19.3Construct the P Matrix for a Uniaxial-to-Isotropic Interface

Consider the ray doublings shown in Figure 19.31. The o–i and e–i couplings at the second surface are described by two P matrices.

Figure 19.31An incident ray refracts through two uniaxial interfaces to two rays. A ray starts from an isotropic medium, splits into o- and e-modes, and refracts into oi- and ei-modes.

Following the one ray path from o to oi, the Poynting vector changes from ${\widehat{\mathbf{S}}}_o$ to ${\widehat{\mathbf{S}}}_i$; the o⊥-mode (orthogonal states of o-mode) along ${\widehat{\mathbf{S}}}_o$ has zero energy, so the coupling from o⊥ to o⊥–i is zero. Then, applying Equation 19.30,

$$\{\begin{array}{l}{\mathbf{P}}_{oi}{\widehat{\mathbf{E}}}_o=a_{o\to ts}{\widehat{\mathbf{E}}}_{ts}+t_{o\to tp}{\widehat{\mathbf{E}}}_{tp},\hfill \\ {\mathbf{P}}_{oi}{\widehat{\mathbf{E}}}_{o\perp }=\mathbf{0},\text{and}\hfill \\ {\mathbf{P}}_{oi}{\widehat{\mathbf{S}}}_o={\widehat{\mathbf{S}}}_i.\hfill \end{array}$$

Then,

$${\mathbf{P}}_{oi}=\left(\begin{array}{lll}{a}_{o\to ts}{\widehat{\mathbf{E}}}_{ts}+a_{o\to tp}{\widehat{\mathbf{E}}}_{tp}& \mathbf{0}& {\widehat{\mathbf{S}}}_i\end{array}\right)•{\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_o& {\widehat{\mathbf{E}}}_{o\perp }& {\widehat{\mathbf{S}}}_o\end{array}\right)}^T.$$

Similarly,

$${\mathbf{P}}_{ei}=\left(\begin{array}{lll}{a}_{e\to ts}{\widehat{\mathbf{E}}}_{ts}+a_{e\to tp}{\widehat{\mathbf{E}}}_{tp}& \mathbf{0}& {\widehat{\mathbf{S}}}_i\end{array}\right)•{\left(\begin{array}{lll}{\widehat{\mathbf{E}}}_e& {\widehat{\mathbf{E}}}_{e\perp }& {\widehat{\mathbf{S}}}_e\end{array}\right)}^T.$$

Example 19.4Calculate the P Matrix for an Isotropic-to-Biaxial Interface

This example uses the calculation results (E, S, and a’s) from Example 19.2 to construct reflection and transmission P matrices. With Equations 19.42 and 19.44,

$$\begin{array}{l}{\mathbf{P}}_{tf}=\left(\begin{array}{lll}0.653& 0& 0\\ 0& 0& 0.321\\ 0& 0& 0.947\end{array}\right){\left(\begin{array}{lll}1& 0& 0\\ 0& 0.819& 0.574\\ 0& -0.574& 0.819\end{array}\right)}^{-1}=\left(\begin{array}{lll}0.653& 0& 0\\ 0& 0.184& 0.263\\ 0& 0.543& 0.776\end{array}\right),\\ {\mathbf{P}}_{ts}=\left(\begin{array}{lll}0& 0& 0\\ 0& 0.644& 0.286\\ 0& -0.192& 0.958\end{array}\right){\left(\begin{array}{lll}1& 0& 0\\ 0& 0.819& 0.574\\ 0& -0.574& 0.819\end{array}\right)}^{-1}=\left(\begin{array}{lll}0& 0& 0\\ 0& 0.692& -0.135\\ 0& 0.392& 0.895\end{array}\right),\text{and}\\ {\mathbf{P}}_r=\left(\begin{array}{lll}-0.347& 0& 0\\ 0& -0.175& 0.574\\ 0& -0.123& -0.819\end{array}\right){\left(\begin{array}{lll}1& 0& 0\\ 0& 0.819& 0.574\\ 0& -0.574& 0.819\end{array}\right)}^{-1}=\left(\begin{array}{lll}-0.347& 0& 0\\ 0& 0.185& 0.570\\ 0& -0.570& -0.601\end{array}\right).\end{array}$$

The singular value decomposition of each exiting P matrix gives the incident and exiting E fields and S vectors. The singular value of 1 corresponds to the S vector. The other two singular values represent the magnitude of the amplitude coefficients of its two exiting modes.

The P matrix for the transmitting slow-mode,

$${\mathbf{P}}_{ts}=\left(\begin{array}{lll}0& 0& 0\\ 0& 0.692& -0.135\\ 0& 0.392& 0.895\end{array}\right)=\left(\begin{array}{lll}0& 0& 1\\ 0.286& 0.958& 0\\ 0.958& -0.286& 0\end{array}\right)\left(\begin{array}{lll}1& 0& 0\\ 0& 0.672& 0\\ 0& 0& 0\end{array}\right){\left(\begin{array}{lll}0& 0& 1\\ 0.574& 0.819& 0\\ 0.819& -0.574& 0\end{array}\right)}^{†},$$

shows that the incident Sinc maps to Sts and Einc,p maps to Ets with 0.672 attenuation.

The reflection matrix,

$$\begin{array}{l}{\mathbf{P}}_r=\left(\begin{array}{lll}-0.347& 0& 0\\ 0& 0.185& 0.570\\ 0& -0.570& -0.601\end{array}\right)=\left(\begin{array}{lll}0& -1& 0\\ -0.574& 0& 0.819\\ 0.819& 0& 0.574\end{array}\right)\left(\begin{array}{lll}1& 0& 0\\ 0& 0.347& 0\\ 0& 0& 0.214\end{array}\right){\left(\begin{array}{lll}0& 1& 0\\ -0.574& 0& -0.819\\ -0.819& 0& 0.574\end{array}\right)}^{†},\\ \end{array}$$

shows that the incident Sinc maps to Sr, Einc,s maps to Ers with −0.347 amplitude coefficient, and Einc,p maps to Erp with 0.214 attenuation.

19.8Example: Ray Splitting through Three Biaxial Crystal Blocks

The ray tracing example of a normal incident ray propagating through a sequence of anisotropic materials in Figure 19.2 will help explain the calculation of the P matrix. The principal refractive indices and orientations of the three biaxial plane parallel plates, KTP, aragonite, and mica, are given in Table 19.5 for λ = 500 nm.

Table 19.5Principal Refractive Indices and Orientation for the KTP, Aragonite, and Mica Example

Biaxial Materials	Principal Refractive Indices(nF, nM, nS) at Wavelength 500 nm	Principal Axes OrientationsUnit Vector of nF Axis, Unit Vector of nS Axis
KTP	(1.786, 1.797, 1.902)	(0.00, 0.64, 0.77), (0.00, −0.77, 0.64)
Aragonite	(1.530, 1.681, 1.685)	(0.38, 0.64, 0.66), (0.32, −0.77, 0.56)
Mica	(1.563, 1.596, 1.601)	(−0.12, 0.74, 0.66), (0.74, −0.38, 0.56)

The three biaxial blocks produce 23 = 8 exiting modes as shown in the ray tree in Figure 19.32. The directions of ray doubling at each of the three interfaces are different depending on the principal indices and orientations relative to the ray’s electric field and propagation direction. The ray locations at the exit surface of each biaxial block are shown in Figure 19.33. The first crystal (KTP) splits the two rays up and down. The second crystal (aragonite) splits the rays diagonally. Looking straight onto the interface, these four rays form a parallelogram. The last crystal (mica) splits the rays up and down with a slight shift and the resulting rays form a double parallelogram.

Figure 19.32The ray tree shows that one incident ray results in eight exiting modes after propagating through three blocks of biaxial materials.

Figure 19.33The ray locations at the end of each biaxial block.

For the normal incident ray down the z-axis, all the intermediate k vectors and the final k vectors remains the same as the incident k, while the orientation of S vectors changes along each ray segment. Each exiting mode has a unique P matrix that tracks the polarization and electric field amplitude. The P matrix for each ray intercept for the fast–slow–fast-mode is shown as an example in Table 19.6.

Table 19.6Resultant Ray Trace Parameter for Fast–Slow–Fast-Mode

Mode	n for the Mode	P	Normalized E of the Mode
f	1.797	$\left(\begin{array}{ccc}0.715& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right)$	$\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)$
fs	1.683	$\left(\begin{array}{ccc}0.750& 0& 0.003\\ -0.461& 0& -0.002\\ -0.003& 0& 1\end{array}\right)$	$\left(\begin{array}{c}0.852\\ -0.523\\ -0.003\end{array}\right)$
fsf	1.578	$\left(\begin{array}{ccc}0.041& -0.025& 0.002\\ -0.517& 0.317& -0.020\\ -0.009& 0.005& 1\end{array}\right)$	$\left(\begin{array}{c}0.080\\ -0.997\\ -0.022\end{array}\right)$
fsfi exiting the surface	1	$\left(\begin{array}{ccc}0.008& -0.097& -0.002\\ -0.097& 1.216& 0.027\\ 0.002& -0.022& 1\end{array}\right)$	$\left(\begin{array}{c}0.080\\ -0.997\\ 0\end{array}\right)$

The cumulative P matrix from the entrance to beyond the exit surface is

$${\mathbf{P}}_{fsfi,\mathrm{total}}={\mathbf{P}}_{fsfi}{\mathbf{P}}_{fsf}{\mathbf{P}}_{fs}{\mathbf{P}}_f=\left(\begin{array}{lll}0.037& 0& 0\\ -0.467& 0& 0\\ 0& 0& 1\end{array}\right),$$

which is calculated by matrix multiplication. The resultant E is along the unit direction (0.080, −0.997, 0) and its amplitude depends on the incident polarization state. For an incident E = (1, 0, 0), the exiting E is (0.037, −0.467, 0). For an incident E of (0, 1, 0), the exiting E is (0, 0, 0). Therefore, this mode sequence acts as a linear polarizer with a transmission axis along the fsf-mode. In fact, all the exiting modes are linearly polarized and all the associated P matrices have the form of linear polarizers.

19.9Example: Reflections Inside a Biaxial Cube

An example of a non-sequential ray trace in a biaxial crystal involving evanescent waves is considered in this section. Aragonite is a natural form of calcium carbonate CaCO3, different from calcite, which is biaxial with principal indices (1.530, 1.681, 1.685) at 500 nm. Figure 19.34 shows a cube of aragonite with its crystal axes aligned along the edge of the cube. When a laser shines into this cube at a certain range of angles, part of the light will refract into the crystal, then reflect within the crystal, and eventually refract out through the entrance surface. The example ray enters the aragonite block, reflects three times within the crystal, and exits through the front surface. There are four birefringent interfaces along this ray path; thus, a maximum of 24 = 16 modes can potentially exit the front surface. For some incident directions, due to total internal reflections and inhibited reflections, the number of modes decreases. Since the number of exiting modes depends on the crystal axis orientations, ray tracing results will be compared with two sets of crystal axis orientations for the same set of incident rays.

Figure 19.34Ray paths through a cube of aragonite arising from one incident ray (top left corner of block) at an angle of incidence in the y–z plane of 39.5°. Multiple rays are generated at each internal interface and six modes exit back out the entrance surface. The crystal axis orientations (shown below the CA label) are aligned with the edges of the block; (vF, vM, vS) = (y, z, x). Note that not all reflected modes carry significant energy. The white bar shows the location of the exiting rays. The shade of the color red shows the amplitude associated with each exiting mode; four modes have negligible flux. The red arrows to the left of the white bar show the polarization ellipse, all linear, for each exiting mode.

When the crystal axes are aligned with the edge of the cube, rays incident in the y–z plane remain propagating within the y–z plane of Figure 19.34, and each mode couples entirely into only one mode at the next surface. In general, during refraction, the energy distributes between fast- and slow-modes. In this case, all the energy from the fast-mode couples to the fast–fast-mode and all the energy from the slow-mode couples to the slow–slow-mode; hence, the fast–slow-mode and the slow–fast-mode carry no energy. This behavior applies to all of the reflections within the crystal. Eventually, only two exiting rays carry energy, the purely fast-mode and the purely slow-mode as shown in Figure 19.35. The exiting intensities from these two modes are shown in Figure 19.36.

Figure 19.35Rays are traces through a block of aragonite with crystal axes aligned with the edges of the block for incident angles of 10°, 30°, 50°, and 70°. The figures show only the exiting rays with non-zero amplitude, which are the purely slow-mode and the purely fast-mode. The amplitude coefficient distribution at the exiting surface is shown in the white bar, where red denotes high amplitude and white denotes zero amplitude. The red arrows on the left of the white bar represent the polarization ellipses of the exiting modes.

Figure 19.36Exiting intensity from the pure fast-mode and pure slow-mode versus angle of incidence in air for unit incident intensity.

When the incidence angle is small, the two modes reflect twice inside the cube and most energy is lost due to inhibited reflection at the second reflection since one mode is evanescent at this steep incident angle. As the angle of incidence increases, the top surface inhibited reflection ceases, and total internal reflection occurs after the first reflection. For incident angles greater than 56° for the pure fast-mode and 63° for the pure slow-mode, there are three reflections instead of two reflections inside the cube. The ray path changes rapidly with incident angle and the crystal axis orientation, so non-sequential ray tracing is used. The ray tree for the 39.5° ray is shown in Figure 19.37, which has inhibited reflection in the slow-mode paths and total internal reflection in the end of the fast-mode paths.

Figure 19.37The ray tree showing mode splitting for 39.5°. The entrance surface is surface 1, the top surface is surface 2, the right side surface is surface 3, and the bottom surface is surface 4. IR denotes inhibited refraction; TIR denotes total internal reflection.

Next, consider the case with crystal axis orientations rotated from xyz (alignment with the cube edges). Now the slow- and fast-modes couple at each interface, modes that previously had zero amplitude acquire flux, and propagation within the crystal is no longer confined to a plane. As shown in Figure 19.38, many exiting rays have amplitudes near zero. The exiting linear polarization and amplitude distribution also change with the incident angle. By such methods, tolerance analysis can be performed on the crystal axis orientations, fabrication angles, thicknesses, and other parameters of birefringent devices.

Figure 19.38A block of aragonite with crystal axes not aligned to the block edges is simulated with various incident angles. The fast crystal axis orientation is (−0.36, 0.39, 0.85), and the slow crystal axis orientation is (0.15, 0.92, −0.36).

19.10Conclusion

The definition of the 3D polarization ray tracing matrix has been extended to incorporate birefringent ray tracing. The calculations of the P matrix shown in this chapter provide the basis to perform systematic ray tracing through complex sequences of birefringent interfaces. Propagation through systems with birefringent materials generates multiple exiting rays from each incident ray. Tracing a single ray samples a single point on the incident wavefront. This wavefront splits into multiple wavefronts propagating through a birefringent assembly with a potential of 2N resultant wavefronts after N birefringent interface interactions. With birefringent components, accounting for ray doubling is only the first step for accurate analysis. Further calculations may be required to manipulate the ray tracing results of all the bifurcated modes exiting the system. Each of these P matrices represents polarization coupling between a single pair of eigenmodes in the incident and exiting materials. Optical system performance is usually evaluated through recreating the exiting wavefront, after tracing a grid of rays from the incident wavefront. The multiple exiting wavefronts require algorithms such as discussed in Chapter 20 (Beam Combination with Polarization Ray Tracing Matrices) to combine them appropriately in the image space. When the exiting rays are propagating in the same direction, the $\stackrel{⌣}{\mathbf{P}}$(defined in Section 9.6) form of the P matrices can be added. When the exiting rays have different S, the ${\stackrel{⌣}{\mathbf{P}}}^{\prime }\text{s}$ don’t add and the resultant E’s must be added instead. Although the ray parameter amplitude calculations in Section 19.5 are for uncoated birefringent interfaces, for coated birefringent interfaces, the amplitude coefficients calculated for these coated interfaces22–26 can be substituted into the P matrix calculations in Section 19.7.

Often, assumptions can simplify the analysis of the multiple exiting modes, such as cases with small shear, small ray separation, or parallel exiting rays. Simple systems such as plane parallel waveplates and birefringent crystal polarizers are designed for a small range of incident angles; hence, the shear will often be small. In the case of a quarter waveplate, a normally incident beam produces two orthogonally polarized modes propagating in the same direction with a quarter wave phase delay, and a circularly polarized incident beam yields a linearly polarized exiting beam. With non-normal incident beams, the exiting modes have a slight displacement; the optical path lengths of the two modes may increase or decrease, and the result is an elliptically polarized exiting beam. The larger the incident angle is, the more elliptical the exiting polarization state becomes. Because of the different angles and ray paths, there are two crescent-shaped regions of light around an exiting circular beam area with only one mode present; thus, the majority of the beam may be circularly polarized with thin strips of horizontally polarized light around one side, and vertically polarized light around the other side, as shown in Figure 19.39. Further analysis on waveplates is included in Chapter 21. A detailed analysis of the Glan–Taylor crystal polarizer is included in Chapter 22.

Figure 19.39The modes exiting a birefringent plate such as a waveplate with off-axis illumination are slightly shifted (shear). Thus, for a quarter waveplate illuminated with 45° light, the majority of the exiting beam is circularly polarized, but two crescent-shaped areas are illuminated by only one mode, in this case horizontally and vertically polarized light.

The simple meaning of optical path length and optical path difference becomes complicated when more than two exiting rays emerge from each incident ray. Many modes might propagate close to each other in the same direction; thus, the concept of optical path length from conventional optical design must be generalized in polarization optical design. To simulate a measurement, all the partial waves need to be added correctly, and the resultant amplitude, phase, and polarization state of the exiting wave need to be calculated at the exit pupil or terminal surface of the optical system. After this beam combination, the phase of the light remains well defined; it is the optical path length that becomes multivalued. For example, in a multibeam interferometer with monochromatic light, the phase of the light can always be measured despite a large number of overlapping beams. For birefringent systems illuminated by short laser pulses, the addition of partial waves needs to account for multiple pulses that can arrive in the exit pupil at different times.

In some components such as lenses with stress birefringence or electro-optic devices, the magnitude of the birefringence is small enough that the ray splitting angle is small and safely ignored because the deviation between the two ray paths is negligible. However, the ray’s polarization changes may still be significant if the accumulated retardance is large enough. These close ray paths are typically handled as follows in polarization ray tracing. Instead of tracing two rays through the remainder of the system, a retardance matrix is associated with the ray segment, and then the ray segment can be handled as a single ray. Stress birefringence is further discussed in Chapter 25. Another example of close ray paths is liquid crystal cells. The liquid crystal interfaces are parallel, so all the modes exit in the same direction. Because the cell is so thin, 1 to 7 μm, the ray paths do not separate by a significant distance; the shear is small. Depending on the objective of the calculation, a retardance matrix can usually be used to describe each ray segment through the cells, and the light propagation can be handled as a single ray. The simulation of liquid crystal is described in Chapter 24.

19.11Problem Sets

1. Explain the difference between ray tracing the combinations of isotropic, uniaxial, biaxial, and optically active interfaces.

2. Calculate the dielectric tensor in global coordinates for a material with principal refractive indices (1.3, 1.4, 1.5) and principal axes oriented at $\left(\begin{array}{c}0.66\\ -0.24\\ 0.71\end{array}\right)$, $\left(\begin{array}{c}0.34\\ 0.94\\ 0\end{array}\right)$, and $\left(\begin{array}{c}-0.66\\ 0.24\\ 0.71\end{array}\right)$ in global coordinates.

3. The refractive indices for the biaxial material sulfoiodide SbSI are (2.7, 3.2, 3.8). What is the principal dielectric tensor? What is the largest angle obtainable between D and E? What is the corresponding direction of propagation?

4. A sample is measured to have the dielectric tensor $\mathbf{\epsilon }=\left(\begin{array}{ccc}1.906& -0.076& 0.199\\ -0.076& 1.971& -0.023\\ 0.199& -0.023& 2.355\end{array}\right)$. Find the unitary transformation that diagonalizes this matrix. How are the crystal axes oriented with respect to x, y, and z? What are nx, ny, nz?

5. Given a compound retarder formed from two materials with fast axes 45° apart, fast mode optical path lengths OPL1 and OPL2, and retardances δ1 and δ2, find the four optical path lengths associated with the four resultant modes. Combine the four modes into a Jones matrix.

6. Consider a collimated beam incident on a tilted plane parallel plate of a uniaxial material. Is it possible to separate two polarizations by total internal reflecting one mode at the back face and transmitting the orthogonal mode at Brewster’s angle?

7. What is the number of potential modes from the aragonite block example in Figure 19.34, including all the modes with zero energy? What is the polarization of the two modes from Figure 19.35 relative to each other? According to Figure 19.36, what incident angle for the aragonite block gives the highest diattenuation?

8. Given a 10 mm thick plane parallel anisotropic plate oriented along the z-axis (0,0,1), what is the OPL for a normal incident ray, where the two modes have indices ns = 1.85124 and nf = 1.79718, propagation vectors ${\mathbf{k}}_{\mathbf{s}}={\mathbf{k}}_{\mathbf{f}}=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)$, and Poynting vectors ${\mathbf{S}}_{\mathbf{s}}=\left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)$ and ${\mathbf{S}}_{\mathbf{f}}=\left(\begin{array}{c}0\\ -0.0627\\ 0.998\end{array}\right)$?

9. Consider a biaxial material whose refractive indices are described by its index ellipsoid: $\frac{x^2}{n_x^2}+\frac{y^2}{n_y^2}+\frac{z^2}{n_z^2}=1$. The only two circular cross sections through this ellipsoid are shown in Figure 19.20, associated with propagation directions k1 and k2. These directions have no birefringence, because the electric field in any transverse direction has the same refractive index. Show the angle, θ, of these special directions known as two optic axes of a biaxial material, from the nz axis given by $\text{tan}(\theta )=\sqrt{\frac{\left(\frac{1}{n_y^2}-\frac{1}{n_x^2}\right)}{\left(\frac{1}{n_z^2}-\frac{1}{n_y^2}\right)}}$, where ny < nx < nz. How will the polarization evolve for propagation along the optic axis?

10. Build a ray tree for an off-axis ray refracting into a biaxial plate with two internal reflections. Each ray intercept produces multiple reflected and refracted rays. Figure 19.40 shows the ray splitting at the first internal reflection, which generates six rays.

Figure 19.40An off-axis ray refracts into a biaxial plate.

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1 See Preface xxvii for Polaris-M.

2 Polarization vector is described in Chapter 2.

3 Uniaxial waveplates are described in Chapter 21 (Uniaxial Materials and Components).

4 Convention in Chapter 10 (Optical Ray Tracing).

5 A combination of the P matrix is explained in Section 9.6 (The Addition Form of Polarization Ray Tracing Matrices), where ${\widehat{\mathbf{S}}}^{\prime }=\mathbf{P}{\widehat{\mathbf{S}}}_{inc}$ should not be double counted.