23Diffractive Optical Elements

23.1Introduction

Diffractive optical elements (DOEs) are optical elements incorporating fine structures to diffract light, as opposed to reflecting or refracting light. In general, DOEs diffract an incident plane wave into a number of reflected and/or transmitted diffraction orders as shown in Figure 23.1. The theory of diffraction grating was first described by James Gregory in 1673.1 David Rittenhouse made the first diffraction grating in 1785 by placing hairs between finely threaded screws.2 In the late nineteenth century, Henry Rowland made major advancements in diffraction grating technology, including concave gratings development, enabling the rapid development of spectroscopic technology.3,4 The diffraction operation is described by a set of amplitude coefficients, like Fresnel coefficients, for each diffraction order. Since each diffraction order diffracts a different amount of transverse electric (TE) and transverse magnetic (TM) modes, each order has different diattenuation and retardance. Often, diffraction orders are quite polarizing.

Figure 23.1A ray diffracts into multiple diffraction orders through a transmission (left) and reflection (right) diffractive grating.

The basic theory of diffractive optics is presented to understand the in-plane and out-of plane propagation of diffracted rays. The calculation of these polarization-dependent amplitude coefficients is generally performed by the rigorous coupled wave analysis (RCWA) algorithm, which is briefly summarized at the end of the chapter.

The oldest application of DOE is reflection diffraction gratings, commonly used in monochromators and spectrometers. Gratings are the source of many polarization problems in monochromators and other optical systems. A typical commercial reflection grating is analyzed in Section 23.3. Another DOE common application, wire grid polarizers, is analyzed by RCWA in Section 23.3.2, where the effect of the depth of the wires on the polarizer performance is considered. Subwavelength phase gratings have retardance, but little diattenuation; hence, the application of DOEs to retarders is considered in Section 23.3.3, where it is shown that the retardance is too small for most applications. Finally, subwavelength gratings used as antireflection coatings on a lens are analyzed in Section 23.3.4 and the resulting polarization aberrations are studied.

The different classes of DOEs are briefly outlined:

• An amplitude grating has alternating opaque and transparent structures imposing amplitude modulation on the reflected and transmitted light. Figure 23.2 shows a periodic amplitude grating.

• A phase grating is a transparent structure with phase variations that produce phase modulation. The phase variations can be created by thickness variations, refractive index variations, and similar means. Figure 23.3 shows a periodic square wave phase grating.

• A surface grating is a thin grating where all diffractions happen at one surface.

• A volume grating is a thick grating, usually thicker than the grating’s period, formed from refractive index variations. Volume gratings can have high diffraction efficiency when operating near the Bragg condition. Because of the thickness, an appreciable interaction can occur, even for small refractive index differences. Volume gratings are not always formed from refractive index variation. For example, a volume hologram can be made on a photographic film, where silver halide particles scatter the incident light and no index variation can be observed.

• A periodic grating has periodic structure, either in amplitude, phase, and/or both. Two example gratings of this type are shown in Figures 23.2 and 23.3.

• A focusing grating uses a grating structure to direct light to certain locations, similar to a lens focusing light. A Fresnel zone plate is an amplitude grating with non-periodic binary circular rings as shown in Figure 23.4a. A Fresnel lens has a structured surface with non-periodic circular blazed structure as shown in Figure 23.4b.

• A non-periodic grating diffracts plane or spherical waves into more complex wavefronts. A hologram is an image recorded as a non-periodic grating; when an appropriate wavefront illuminates the hologram, the resultant diffracted light recreates the image.

• A subwavelength grating (SWG) has a period less than half the illumination wavelength at normal incidence. With a sufficiently small period, all higher diffraction orders are evanescent, leaving only the transmitted and reflected 0th orders. Because light is not lost to the ±1st, ±2nd, and other orders, the efficiency is often high. Later sections consider subwavelength gratings as polarizers, retarders, and antireflection coatings.

Figure 23.2Amplitude grating structure, where the gray area represents the opaque region and the white area in between is the transparent region.

Figure 23.3Phase grating structure, where the (pink) transparent material introduces a spatially alternating phase change.

Figure 23.4Fresnel zone patterns where white has a transmission of 1 and black has 0 transmission.

Two common periodic gratings are the rectangular grating and the blazed grating depicted in Figure 23.5. As shown in Figure 23.5a, a rectangular grating contains a repeating rectangular profile. A blazed grating has a triangular profile. Consider light reflecting from a grating facet. A blazed grating will tend to have high efficiency in orders near the reflected direction. Thus, blazing is used to maximize the diffraction efficiency into a desired order. Arbitrary shape gratings are those with the arbitrary profiles. Several terminologies and notations of these periodic grating structures used in the following sections are highlighted in Table 23.1.

Figure 23.5Structure of a periodic (a) rectangular grating, (b) blazed grating, and (c) multi-step grating.

Table 23.1Grating Parameters and Notation Definitions

Grating Parameters	Definitions	Notation
Grating period		d
Grating height		H
Duty cycle	The positive fraction of the square wave in one period	$D=\frac{\tau }{d}$	(23.1)
Aspect ratio	The height of the grating relative to its period	$A=\frac{H}{d}$	(23.2)
Blaze angle	The base angle of a triangular grating	θB
Amplitude coefficient	A complex coefficient relating the amplitude a and phase ϕ of dth diffractive order to the incident light	$\frac{a_d{e}^{-i{\varphi }_d}}{a_{in}{e}^{-i{\varphi }_{in}}}$	(23.3)
Diffraction efficiency	The ratio between the energy flux in a particular order and the energy flux in the incident beam for incident angle θin and diffractive angle θd	$\frac{{\left|a_d\right|}^2\text{cos}{\theta }_d}{{\left|a_{in}\right|}^2\text{cos}{\theta }_{in}}$	(23.4)

Diffractive structures are manufactured in many different ways. Originally, they were produced as large numbers of parallel wires; this is still done in millimeter and radio frequencies. Conventionally, diffraction gratings have been produced by scribing a diamond tool into a soft metal like aluminum. A replica grating is copied from a master grating into epoxy, allowing many gratings to be produced from the master ruling. Grating copies can also be electroformed. Interference patterns can be generated with interferometers and then exposed in photoresist, dichromated gelatin, photographic film, and other materials to create DOEs. Microlithography is a flexible tool for producing very fine gratings in photoresist, which can then be processed into many materials through etching, coating, and so on. The computer-controlled electron beam etching allows patterning gratings with a resolution of a few nanometers, with a trade-off of long processing time. The diffraction pattern can be fabricated on a thin sheet, flat or curved surface.

23.2The Grating Equation

To ray trace a DOE, the grating period and orientation must be specified along with the direction of the incident light. A grating is operated in-plane when the plane perpendicular to the rulings contains the incident light’s k vector. All the diffracted orders will lie in the same plane, and the mathematics is easier than the general case of out-of-plane diffraction, where the light has an arbitrary direction. The directions of diffracted light are given by the grating equation in one of these two forms, Equation 23.6 or 23.9. Either equation describes both transmission and reflection gratings. These equations specify the diffracted light’s direction for all diffracted orders, since this only depends on the grating period d, but say nothing about the amount of energy carried by each order, which depends on the grating profile. This can be calculated using the RCWA algorithm.

For a grating in the x–y plane with a surface normal along the +z-direction and the grating’s grooves along the y-direction, as shown in Figure 23.6, an in-plane incident ray at an angle of incidence α diffracts into the x–z plane with diffraction angles βm for each diffraction order m as

Figure 23.6An in-plane incident light ray is diffracting from a refractive grating (left) and a reflective grating (right). The colors of the arrows distinguish the incident and diffractive directions for one wavelength. The incident ray shown in blue results in multiple diffraction orders shown in different colors. The projections from the unit sphere are equally spaced on the x–y plane. Red indicates the zero order, with orange, yellow, and green arrows indicating negative orders and red–purple is the only positive order present.

$$n_{m}d\text{sin}{\beta }_m-n_{i}d\text{sin}\alpha =m\lambda .$$23.5

23.5

With media refractive index ni = nm = 1,1

$$\text{sin}{\beta }_m=\text{sin}\alpha +\frac{m\lambda }{d},$$23.6

23.6

where λ is the light’s wavelength.

In the 0th order (m = 0), all wavelengths are diffracted in the same direction; this corresponds to the ordinary reflected and transmitted beams. In this 0th order, white light diffracts into white light. The other orders disperse the light, with shorter wavelengths diffracting nearer to the 0th order and longer wavelengths diffracting further from the 0th order,2 as shown in Figure 23.7. From the incident beam, the positive orders diffract beyond the 0th order; negative orders lie on the same side of the 0th order as the incident light. The normal always lies in the region of the negative orders.

Figure 23.7An incident beam diffracts into multiple orders. The color of the arrows represents different wavelengths. Because of dispersion, the diffraction angles are changed with wavelength except for the 0th diffraction order, which diffracts white light. The shorter wavelength (blue) diffracts closer to the 0th order, while the longer wavelength (red) diffracts further from the 0th order.

The angular dispersion is the rate of change of the diffracted angle with wavelength,

$$\frac{d\beta }{d\lambda }=\frac{m}{d\text{cos}\beta }.$$23.7

23.7

The more general out-of-plane diffraction grating equation is derived from the three-dimensional grating equation ki − km = Kg depicted in Figure 23.8 where $\left|{\mathbf{k}}_m\right|=\frac{2\pi n_m}{\lambda }$ and $\left|{\mathbf{K}}_g\right|=\frac{2\pi }{d}$.

Figure 23.8Grating vector Kg describes the period and orientation of the grating grooves.

Consider a plane grating surface in the x–y plane with an incident medium refractive index of nI, exiting medium index nII, and incident wavevector

$${\mathbf{k}}_i=\left(\begin{array}{l}{k}_{xo}\\ k_{yo}\\ k_{I,zo}\end{array}\right)=\left(\begin{array}{l}{k}_o{n}_I\text{sin}\theta \text{cos}\varphi \\ k_o{n}_I\text{sin}\theta \text{sin}\varphi \\ k_o{n}_I\text{co`s}\theta \end{array}\right),$$23.8

23.8

with $k_o=\frac{2\pi }{\lambda }$. The diffracted wavevectors are

$${\mathbf{k}}_{I,m}=\left(\begin{array}{l}{k}_{xm}\\ k_{yo}\\ -k_{I,zm}\end{array}\right)\text{and}{\mathbf{k}}_{II,m}=\left(\begin{array}{l}{k}_{xm}\\ k_{yo}\\ k_{II,zm}\end{array}\right)$$23.9

23.9

for mth reflected kI,m and transmitted kII,m diffraction order wavevectors, respectively,5–7 where

$$k_{xm}=k_o(n_I\text{sin}\theta \text{cos}\varphi +m\lambda /d),$$23.10

23.10

$$k_{I,zm}=\{\begin{array}{lll}\sqrt{{(k_o{n}_I)}^2-\left(k_{xm}^2+k_{yo}^2\right)}\hfill & \text{for}\hfill & \sqrt{k_{xm}^2+k_{yo}^2}\le k_o{n}_I,\hfill \\ -i\sqrt{{\left(k_{xm}^2+k_{yo}^2\right)}^2-{(k_o{n}_I)}^2}\hfill & \text{for}\hfill & \sqrt{k_{xm}^2+k_{yo}^2}>k_o{n}_I,\hfill \end{array}$$23.11

23.11

and

$$k_{II,zm}=\{\begin{array}{lll}\sqrt{{(k_o{n}_{II})}^2-\left(k_{xm}^2+k_{yo}^2\right)}\hfill & \text{for}\hfill & \sqrt{k_{xm}^2+k_{yo}^2}\le k_o{n}_{II}\hfill \\ -i\sqrt{{\left(k_{xm}^2+k_{yo}^2\right)}^2-{(k_o{n}_{II})}^2}\hfill & \text{for}\hfill & \sqrt{k_{xm}^2+k_{yo}^2}>k_o{n}_{II}.\hfill \end{array}$$23.12

23.12

Equations 23.6 and 23.9 describe the directions of the diffracted orders for gratings in the x–y plane. For other grating orientations, these results must be rotated. This out-of-plane grating equation is not so widely presented, but is essential for ray tracing (Figure 23.9).

Figure 23.9A graphic depiction of the out-of-plane grating equation. The blue vectors pointing into the origin are the incident light. In the six figures, the incident vector is rotated about the normal, starting in the x–z plane and ending in the y–z plane. The upper left figure shows the in-plane case, where multiple diffraction orders (red, orange, and yellow vectors), extending to a unit hemisphere, have equally spaced projections onto the x-axis. As the incident vector rotates toward the y-axis, the diffraction orders lie on the surface of a cone, but the projection onto the x–y plane remains equally spaced.

23.3Ray Tracing DOEs

Geometrical ray tracing through a DOE calculates the propagation vectors of the diffraction orders, but not, the amplitude coefficients, and the state of the electric fields resulting from the light–DOE interaction. The wavevectors are obtained from Equation 23.9 to Equation 23.12 for all diffracted rays. The derivation of the amplitude coefficients for all the diffraction orders is calculated from Maxwell’s equations for periodic boundary conditions. This is most often performed by the RCWA calculation summarized in Section 23.4. The eigenpolarizations of DOEs are not necessarily the s- and p-polarizations. But the changes to the light’s amplitude and phase can be expressed with Jones matrices or P matrices for each reflected and transmitted diffraction order. This section includes four examples of DOEs showing their polarization characteristics simulated by RCWA.

23.3.1Reflection Diffractive Gratings

Diffraction gratings are essential elements in spectrometers and monochromators. However, they are also the root cause of many serious polarization problems in these systems. A reflection grating example is presented to assist in understanding how to interpret the manufacturer’s specifications and measurement data as well as using the outputs from RCWA grating simulations in polarization ray tracing.

Reflection gratings are in widespread use in spectroscopy to disperse polychromatic light for the purpose of measuring spectra or producing monochromatic light.3 For example, a Czerny–Turner monochromator, shown in Figure 23.10 (left), uses a mirror to collimate light from the incident slit onto the grating. The dispersed light is focused by a second mirror and forms an image, the spectrum. An exit slit selects and passes a small spectral band. By rotating the grating, the monochromator scans the exiting wavelength. Another common monochromator is the Littrow monochromator, Figure 23.10 (right), which uses one focusing mirror twice.

Figure 23.10(Left) The Czerny–Turner monochromator and (right) the Littrow monochromator.

Reflective diffraction gratings commonly have triangular profiles, where one facet receives most of the illumination. Aluminum grating surfaces are common for visible applications while gold surfaces are common for infrared applications. Light is distributed among all the diffraction orders in a complex fashion and can be calculated by RCWA. Despite the diffraction, light has a natural tendency to reflect from the facet. The diffraction orders near this reflection direction are typically the brightest. The term blazing means choosing the reflective facet angle to reflect one wavelength, the blaze wavelength λB, into the direction of a desired diffraction order. Thus, blazing will provide high diffraction efficiency for wavelengths near the blaze wavelength. By convention, the blaze wavelength and blaze angle θB are defined as the wavelength and facet angle where, in a Littrow monochromator, the incident light and the −1st order lie along the same line, as depicted in Figure 23.11. The blaze wavelength satisfies the equation

Figure 23.11(Left) The Littrow configuration where the −1st diffraction order shown is optimized for maximum efficiency by setting the incident angle and the blaze angle equal to θB. Gratings can be blazed for other negative orders. (Right) The Littrow blaze angle as a function of wavelength for a grating period of 1.67 μm.

$${\lambda }_B=2d\text{sin}{\theta }_B.$$23.13

23.13

Example 23.1Reflection Diffractive Grating Example

A reflection diffractive grating is simulated, made of aluminum, with 600 grooves per millimeter, a blaze wavelength λB = 500 nm, and a blaze angle θB = 8.6°. The triangular blaze structure is simulated by 30 layers of rectangular steps as shown in Figure 23.12 to mimic the triangular shape.

Figure 23.12The profile of a blazed diffraction grating simulated with 30 rectangular steps.

The diffraction efficiency of this grating is calculated by RCWA5,6 for in-plane illumination near the Littrow configuration for both TE and TM modes, where TE field is parallel to the grating grooves and TM field is perpendicular to the grating grooves. In our convention, TE mode is s-polarization and TM mode is p-polarization. Figure 23.13 shows measured and simulated grating efficiencies for the −1st diffraction order. Note that the diffraction efficiencies peak around 500 nm at the blaze wavelength.

Figure 23.13Data for a 600 line/mm aluminum grating in the −1st order. The reflectance of aluminum at normal incidence and the measured diffraction efficiency data are shown in dashed lines. The blue line is the reflectance dispersion of the aluminum used in the simulation. Red and green lines are the diffraction efficiency for s- and p-polarizations (TE and TM modes), respectively. Woods anomalies can be seen at 1.05 and 1.21 μm. Very good agreement was found between the measured and the simulated grating efficiencies for the −1st order.

The abrupt variations of the TM mode’s diffraction efficiency at 1.02 and 1.22 μm (Figure 23.13) are examples of Woods anomalies.8,9 Woods discovered in 1902 that abrupt changes in diffraction efficiency from metal gratings are observed at certain wavelengths and angles. In 1907, Rayleigh10,11 explained that the anomalies occur when the diffracted light in one order is diffracting at βm = 90° (i.e., tangential to the grating’s surface) and is transitioning from a real into an evanescent wave. In this situation, the energy of the diffracted light is redistributed to lower diffraction orders, which causes abrupt changes in the spectrum. In 1941, Fano12 and others13,14 described the anomalies as the results of surface plasmon resonance effects, and the characteristics of the materials close to the interface also affect the sharp changes to the spectrum. The anomalies are generally observed for both s-polarization and p-polarization, with the p-polarization showing larger anomalies.15–18

The diffraction efficiencies of the TE and TM modes are very different and change rapidly with wavelength. These lead to large grating polarization effects. The diattenuation of the example grating, in Figure 23.14 (right), is significant, rapidly changing, and not described by any simple linear or quadratic function. As a result, when a polarized source is measured by a spectrometer using these gratings, very different spectra are obtained if the incident light is in the TE or TM mode. Gratings also have retardance and the retardance of the −1st diffraction order for this example grating is plotted in Figure 23.14 (left) with an interesting shape.

Figure 23.14The simulated retardance in radian (left) and diattenuation (right) of −1st diffraction order.

Figure 23.13 showed the grating efficiency as measured in a “Littrow” monochromator. But a Littrow can never operate at exactly the Littrow condition, the entrance and exit slit would be on top of each other. In this case the angle between the slits is not specified, making matching calculations with measurements difficult. To obtain a close match between the simulation and the measurement, the grating was simulated for a series of incident and diffracted angles about the Littrow condition. The best match, shown in Figure 23.15, was found when the incident beam was about 0.15 radians from Littrow, and the diffracted beam is collected at about 0.15 radians on the other side. In particular, the locations and the shapes of the two Woods anomalies at 1.02 and 1.22 μm were very sensitive to the incident angle and provided an accurate method to estimate the angle from Littrow in the monochromator configuration used for the measurements. The measured and simulated diffraction efficiency agree well, particularly since real gratings have many defects, including not perfectly rectangular or triangular profiles and small variability in the grating’s spacing.

Figure 23.15TE and TM’s diffraction efficiencies calculated with RCWA for the −1st order at several angles of incidence. The incident angles are specified in radians from the Littrow condition: 0.05 offset is light green, 0.1 and 0.15 in darker greens, and 0.2 offset is deep green. The best fit to the diffraction efficiencies occurred for an offset from the Littrow condition of 0.15 radians.

23.3.2Wire Grid Polarizers

A wire grid polarizer is formed as a series of fine parallel metal strips on a transparent substrate. To be an effective polarizer, the wire spacing should be less than half of the wavelength of the light. This group of wires forms the diffraction grating as shown in Figure 23.2. When an electromagnetic field is incident onto the metal wires, it easily moves the free electrons along the wires, and this current generates a strong reflected beam polarized along the wires. The orthogonal components of the field, oscillating perpendicular to the metal wires, have a much smaller interaction and thus nearly all the light passes through the polarizer. Therefore, a wire grid polarizer with wires oriented in the y-direction transmits mostly x-polarized light and reflects mostly y-polarized light.

Example 23.2An Aluminum Wire Grid Polarizer

An aluminum wire grid polarizer with 0.2 μm period, 30% duty cycle, and 0.225 μm wire height is simulated at 550 nm wavelength. A circularly polarized converging wavefront passes through the vertically oriented wire grid shown in Figure 23.16a. The horizontal component transmits and forms the wavefront shown in Figure 23.16b, while the vertical component reflects as shown in Figure 23.16c. Thus, this wire grid acts as a horizontal polarizer in transmission.

Figure 23.16(a) A circularly polarized converging incident ray grid (green) focuses on a wire grid polarizer with vertically oriented wires. This has a horizontal transmission axis. (b) A horizontally polarized wavefront (blue) transmits through the polarizer. (c) A vertically polarized wavefront (red) reflects from the polarizer for incident circularly polarized light.

The wires are oriented vertically in a 2D plane on the polarizer’s surface, while the incident light rays are converging. For the on-axis ray, its transverse plane is in the plane of the polarizer’s surface, and thus transmitted ray has horizontal linear polarization. For the other off-axis rays, their tranverse planes do not coincide with the polarizer’s surface. The higher the incident angle, the further its transverse plane is tilted from the polarizer’s surface, and the more light in vertical polarization leaks through, rotating the planes of polarization, and yielding elliptically polarized transmitted rays.

To view the polarization states of the incident, transmitted, and reflected beams, the dipole local coordinate system4 with dipole axis along the wires was selected, since this closely matches the physics. The polarization of the incident, reflected, and transmitted beams is plotted in Figure 23.17. The exiting states are predominantly linear with a small amount of ellipiticity visible at some off-axis rays.

Figure 23.17The local polarization ellipses of the (a) incident, (b) transmitted, and (c) reflected rays in Figure 23.16 over a 45° spherical wavefront.

Figure 23.18 plots the diattenuation of the reflected and transmitted light; both are above 90% diattenuation, but are never 100%, such as achieved by crystal polarizers. The reflected light has a maximum diattenuation on-axis, while the transmitted diattenuation is not at maximum on-axis. The extinction ratio shown in Figure 23.19 for this wire grid polarizer in transmission has a maximum of more than 1000 for the on-axis beam and decreasing to around 20 at the corner of the field (~56°).

Figure 23.18The transmitted (left) and reflected (right) diattenuation magnitude of the polarizer.

Figure 23.19Angle of incidence variation of the extinction ratio for the simulated wire grid polarizer.

When designing a wire grid polarizer, the design parameters include the grating’s metal layer thickness, cross-sectional shape, period, duty cycle, and the choice of metal. In most cases, the transmission and/or extinction ratio are optimized. As a rule of thumb, a metal with a large real index and an absorption index in the range of 1 to 2 works well. A smaller duty cycle with a thicker metal layer is preferred to a larger duty cycle and thinner metal layer since it can yield better extinction with smaller absorption. The wire grids produce substantial retardance but since the flux of orthogonal polarization is generally very small, the effect is barely noticeable. Equation 23.14 is an example merit function for wire grid polarizer optimization,

$$\text{Merit}=\sqrt{c_1{(1-T_p)}^2+c_2{T}_s^2+c_3{(1-R_s)}^2+c_4{R}_p^2},$$23.14

23.14

where (c1, c2, c3, c4) are weighted to trade off the transmission and reflection coefficients for s- and p-polarizations (Ts, Tp, Rs, Rp). For high contrast ratio, c1 should be large. For better transmitted flux, c2 should increase. The ideal wire grid polarizer would have a merit function of zero. Wire grid polarizers are somewhat angle sensitive, with more variation for the polarization orthogonal to the wire direction or along the transmission axis, than along the wire direction.

Example 23.3Performance as a Function of Aluminum Wire Thickness

To understand these trade-offs in Equation 23.14, consider the following simulation (Figure 23.20) showing the TE and TM mode transmissions as a function of aluminum layer thickness at various wavelengths. The TE transmission, parallel to the wires, is not well extinguished until the thickness is greater than 0.2 μm. The TM transmission has a rather small oscillation with thickness.

Figure 23.20Wire grid polarizer transmission of TM polarization (red) and TE polarization (blue) versus aluminum wire height at six wavelengths.

When selecting commercial wire grid polarizers, it is helpful to understand that these polarizers can be optimized for high extinction, transmission efficiency, or a balance between the two, as shown in Figure 23.21. These graphs make clear the design trade-off between contrast and transmission; higher contrast polarizers have a lower transmission.

Figure 23.21Performance comparison of three wire grid polarizers: (left) transmission of p-state and (right) contrast ratio. Red for ultra-high transmission polarizer, black for high transmission polarizer, and blue for high contrast polarizer.

23.3.3Diffractive Retarders

Diffractive polarizers, wire grid polarizers, are very common optical components, so one might wonder why diffractive retarders are not as common. Light transmitted through grating structures experiences retardance; thus, it is possible to fabricate retarders from DOEs. However, it is difficult to obtain substantial retardance from such DOEs. This section explain why that is the case.

An SWG only reflects and transmits the 0th order; the angle of the first diffraction order would be greater than 90°, if that was possible. In our terminology, all of the orders for a subwavelength grating except the 0th order are evanescent. Thus, SWGs are more efficient than gratings that are not subwavelength, which must divide their flux between orders. Example 23.4 explains the difficulty with diffractive retarders.

Example 23.4A Family of Fused Silica Diffractive Retarders

An etched fused silica grating is simulated with a period of 170 nm and various aspect ratios correspond to grating heights of 42.5, 85, 170, and 340 nm. Figure 23.22 shows the retardance spectra for different aspect ratios. Higher retardance is obtained with a larger aspect ratio, though these SWGs are difficult to make, especially when the aspect ratio increases! Even for an aspect ratio of 2, these diffractive retarders have such weak retardance that it is hardly practical to make a quarter wave retarder, much less for half wave retarders.

Figure 23.22(Left) The retardance of subwavelength fused silica gratings with a 170-nm period and aspect ratios of 0.25 (blue), 0.5 (red), 1 (green), and 2 (orange). (Right) A comparison of the retardance of the aspect ratio of 1 SWG (green) with comparable birefringent waveplates with thicknesses chosen to match the 600-nm retardance: sapphire (purple), calcite (pink), and quartz (cyan).

The retardance dispersions of the SWG with an aspect ratio of 1 from Example 23.4 is compared in Figure 23.22 (right) with three waveplates made of different crystals: a 2.198-μm-thick sapphire A-plate, a 0.103-μm-thick calcite A-plate, and a 1.95-μm-thick quartz A-plate. The thicknesses are chosen for a 0.186-rad retardance at 600 nm. Note the similarity of the retardance spectra! If the shape of the diffractive retardance dispersion was different from the crystal’s dispersion, the diffractive retarders could play a valuable role in achromatizing retarders even with small retardance, but this is not the case. Thus, diffractive retarders appear to be difficult to fabricate due to the need for very large aspect ratios and do not significantly enlarge the design space for controlling the dispersion of retardance.

23.3.4Diffractive Subwavelength Antireflection Coatings

The SWG can be used to replace thin film coatings, such as anitireflection coatings and polarizing beam splitters. An ideal antireflection SWG can be structured to create an effective medium layer that has a gradient index, from the substrate index to the index of the surrounding medium.19 An antireflection SWG20,21 shown in Figure 23.23 is simulated with the RCWA, and the associated wavefront aberrations and polarization aberrations are studied. These aberrations depend on the direction of the incident light, the polarization state, and the orientation of the grating’s grooves.

Figure 23.23(Left) An antireflection grating’s groove orientation. (Right) The subwavelength structure21 of the grating acts like a gradient material since the duty cycle decreases moving from the substrate.

Figure 23.24 compares the transmission of the SWG with the transmission of a standard quarter wave antireflection MgF2 coating. The SWG shows very high transmission up to 99.5%. In the case where the plane of incidence is parallel to the grooves, it has less polarization dependence than when the plane of incidence is perpendicular to the grooves. For light propagating in the plane of incidence perpendicular to the grooves, as the incident angle increases, the apparent grating period shrinks, and resonant interactions between the grating and the incident field occurs. As the angle of incidence increases to 28°, a 1st diffraction order comes into existence (the first grating order ceases to be evanescent) and a rapid reduction in transmission occurs as the 1st order gains energy. The corresponding diattenuation for the 0th order rapidly changes. This sets a limit on the practical angle of incidence range of the antireflection SWG.

Figure 23.24The intensity transmission of the antireflection SWG compared with a standard quarter wave MgF2 antireflection coating. The SWG data correspond to the grating’s grooves parallel (a) and perpendicular (b) to the plane of incidence.

The SWG introduces retardance even at normal incidence, unlike a typical antireflection coating. Figure 23.25 shows the phase delay in transmitted light for TE and TM polarized light in two orthogonal planes. As the plane of incidence switches from being parallel to perpendicular to the grating’s groves, the resultant retardance switches sign, because the polarization orientation along the grating’s groove has switched. As the angle of incidence increases, the two retardance functions also diverge. The phase shifts and retardance at 24° incident angle for each polarization and grating orientation are given in Table 23.2. The phases are given in terms of the s- and p-components, but which of these is parallel to the grating lines depends on the plane of incidence. Therefore, the s-component parallel to the grating should be compared against the p-component perpendicular to the grating. This shows the phase in the relative orientation of the plane of incidence and the gratings produce a 0.005 wave change in the electric field component perpendicular to the gratings and a 0.012 wave change in the component parallel to the gratings. This causes a 0.007 wave difference in the retardance magnitude between the two grating orientations at 24°.

Figure 23.25The transmitted phase and retardance of the SWG for the plane of incidence parallel (a) and perpendicular (b) to the grating’s groves.

Table 23.2Phase Shift of SWG at 24° Incident Angle for Plane of Incidence (POI) Parallel and Perpendicular to the Grating’s Grooves

	s-Phase (Wave)	p-Phase (Wave)	Retardance (Wave)
POI // to the groove	0.3037	0.1785	0.1252
POI ⊥ to the groove	0.1662	0.2986	−0.1324

The polarization aberration function of the example SWG is analyzed on an ideal spherical surface using the Polaris-M polarization ray tracing program (Airy Optics, Inc.). Since the plane of incidence is radially oriented on a spherical surface, the groove’s orientation plays a role in the resultant polarization aberration as well. The polarization aberration functions are shown in Figures 23.26 through 23.28 for an on-axis beam with the grating lines oriented horizontally. Both co-polarized terms with subscripts xx and yy have a small quadratic apodization due to incident angle changes across the field. The cross-coupled terms xy and yx have the Maltese cross pattern with a 3% maximum amplitude leakage. The abrupt amplitude and phase change toward the top and bottom of the field indicates where the subwavelength condition fails and the 1st order begins robbing energy. The resultant retardance pattern contains elements of piston, astigmatism, and defocus. The retardance orientation has a very small 3° variation toward the edge of the field. Overall, the resulting wavefront aberrations are dominated by astigmatism, with a magnitude less than 0.1 waves.

Figure 23.26The amplitude ρ of the Jones pupil for the example lens with two SWG antireflection-coated surfaces. Dark bands across the tops and bottoms of the xx and yy amplitudes are due to increasing amounts of light lost into the first diffraction order.

Figure 23.27The wavefront aberration, the phase ϕ of the Jones pupil, for the example lens with two SWG antireflection-coated surfaces.

Figure 23.28Retardance magnitude in waves (a) and retardance orientation in degrees (b).

The polarization-dependent aberrations from SWG antireflection coatings give an additional complexity to aberration compensation compared to the traditional geometrical wavefront optimization. In the example described here, using two identical gratings in orthogonal orientation on two similar surfaces can compensate for the on-axis retardance. The higher-order aberrations remain, but with very small magnitudes. Using the RCWA simulation capability to fine-tune the shape of an SWG in a point-by-point basis can potentially result in high levels of polarization correction.

23.4Summary of the RCWA Algorithm

The RCWA algorithm is widely used for analyzing periodic diffractive structures, such as diffraction gratings, wire grid polarizers, holograms, and all the examples presented in this chapter. RCWA calculates exact solutions for the reflection from and transmission through periodic structures. It is a relatively straightforward, non-iterative, and deterministic technique that calculates the amplitude coefficients for each diffraction order. The accuracy depends solely on the number of Fourier terms retained in the Floquet–Fourier expansions of the grating structure. The algorithm was initially developed for modeling volume holographic gratings and then extended to surface-relief and multilevel grating structures. RCWA has been successfully applied to transmission and reflection planar dielectric and/or absorption holographic gratings, arbitrarily profiled dielectric and/or metallic surface-relief gratings, multiplexed holographic gratings, two-dimensional surface-relief gratings, and anisotropic gratings, for both planar and conical diffraction.5,6,22–24 This section will provide a summary of the RCWA algorithm.

In the RCWA algorithm, the grating is divided into a large number of sufficiently thin planar slabs, as shown in Figure 23.29, to approximate the grating profile to an arbitrary degree of accuracy. The information required is listed in Table 23.3 and Figure 23.29.

Figure 23.29(Left) Coordinate system for the incident light onto a grating structure. (Right) The specification for a grating or diffractive optical element in RCWA divides the grating into many layers, and each layer into rectangles of uniform refractive index or dielectric tensor. Only one period of the grating is specified as input.

Table 23.3Rigorous Coupled Wave Analysis Input and Output Parameters

Incident Light Descriptions	Incident Light Parameters
Incident wavelength	λ
Angle of incidence	θ
Azimuthal angle	ϕ
Incident polarization	E
Grating Descriptions	Grating Parameters
Dielectric constant or tensor of incident mediuma	ε1
Dielectric constant or tensor of grating substratea	ε2
Layers of dielectric constant or tensor of grating structurea	{{ε11, ε12,…, ε1M},…{εM1, εM2, …, εMN}}
Subperiod dimensions for each layer	{{x11, x12,…, x1M},…{xN1, xN2, …, xNM}}
Layer thicknesses	{h1, …, hN}
Grating period	d
RCWA Parameter Descriptions	Algorithm’s Parameters
Number of Fourier orders retained in calculations	(2f + 1) orders: −fth, −(f − 1)th, …, −2nd, −1st, 0, +1st, +2nd, …, (f − 1)th, fth
Diffracted Light Parameters
km	mth diffraction order propagation vector
Sm	mth diffraction order Poynting vector
Dm	mth diffraction order D-field
Em	mth diffraction order polarization state
Hm	mth diffraction order H-field
am	mth diffraction order complex amplitude coefficients
Rm and Tm	mth diffraction order diffraction efficiencies

a Similarly for gyrotropic tensor, when optical activity is present.

The basic algorithm for RCWA is similar to the algorithms for Fresnel equations and thin film–coated interfaces (Figure 23.30). The principal difference is the superposition of many diffracted orders of reflected, transmitted, forward, and backward waves in the grating region because of the periodic grating structure. The explicit form of the incident, reflected, and transmitted waves given by the superposition of individual modes in the grating region is obtained from Floquet’s condition, described in Figure 23.8, which relates the wavevectors of the diffracted waves with the grating vector and the dispersion relationships. The fields propagating in each grating layer are calculated via a coupled wave approach. In the lth grating layer, the fields can be expanded to

Figure 23.30The basic algorithm of RCWA solves the boundary value problem for forward and backward propagating waves for all diffraction orders. Each wave interacts with all the components of a three-dimensional refractive structure described by its Fourier series. The incident, reflected, and transmitted waves are obtained by Floquet’s condition and the dispersion relations.

$$\begin{array}{lll}{\mathbf{E}}_{g,l}={\displaystyle \sum _n{\mathbf{S}}_{l,n}(z)e^{-i\left(k_{xn}x+k_{yn}y\right)}}\hfill & \text{and}\hfill & {\mathbf{H}}_{g,l}=\sqrt{{\epsilon }_o/{\mu }_o}{\displaystyle \sum _n{\mathbf{U}}_{l,n}(z)e^{-i\left(k_{xn}x+k_{yn}y\right)}},\hfill \end{array}$$23.15

23.15

where Sn and Un are the nth components of the vector Fourier coefficients function. The coupled wave equations are obtained by substituting the constitutive relations and electromagnetic fields in their Fourier series forms into Maxwell’s equations.

After the substitutions, Sz and Uz are eliminated and the coupled wave equations are obtained:

$$\frac{1}{k_o}\frac{\partial }{\partial z}\left(\begin{array}{l}{\mathbf{S}}_{l,x}(z)\\ {\mathbf{S}}_{l,y}(z)\\ {\mathbf{U}}_{l,x}(z)\\ {\mathbf{U}}_{l,y}(z)\end{array}\right)=i\mathbf{\Gamma }\left(\begin{array}{l}{\mathbf{S}}_{l,x}(z)\\ {\mathbf{S}}_{l,y}(z)\\ {\mathbf{U}}_{l,x}(z)\\ {\mathbf{U}}_{l,y}(z)\end{array}\right),$$23.16

23.16

where Γ is a block matrix described in Refs. [22] and [25]. The coupled wave equation is a 1st-order ordinary differential equation, and its solution for the field in the z-component (orthogonal to interface) is analytically obtained as

$$\begin{array}{lll}{\mathbf{V}}_l(z)={\displaystyle \sum _m{c}_{l,m}{\mathbf{w}}_{l,m}{e}^{i{k}_o{\lambda }_{l,m}\left(z-z_{l-1}\right)}}\hfill & \text{and}\hfill & {\mathbf{V}}_l=\left(\begin{array}{l}{\mathbf{S}}_{l,x}\\ {\mathbf{S}}_{l,y}\\ {\mathbf{U}}_{l,x}\\ {\mathbf{U}}_{l,y}\end{array}\right).\hfill \end{array}$$23.17

23.17

λl,m are the set of eigenvalues and wl,m are the eigenvectors of Γl for the lth layer. This yields the amplitude coefficients cl,m, determined by the boundary conditions. By applying the electromagnetic boundary conditions—continuity of the tangential electric and magnetic field components—to the interfaces at the output region, the individual grating layers, and finally the input region, the Fresnel coefficients of the reflected and transmitted diffractive fields are obtained for each order. More detailed description of the RCWA algorithm and its solution for isotropic material is provided in Glytsis and Gaylord,26 section 2, Appendix C. For anisotropic and optically active gratings, the dielectric and gyrotropic tensors are needed in the grating description, and the scattering matrix method is employed to solve the boundary condition equations for numerical stability.27,28 The diffraction efficiencies, complex amplitude reflectance and transmittance coefficients, wavevectors and the electromagnetic field vectors from RCWA, Jones matrices, Mueller matrices, or polarization ray tracing matrices can be constructed for each order for polarization analysis.

For an accurate calculation from RCWA, not just the reflected and diffracted orders are calculated, but also a large number of evanescent orders. This depends on the number of Fourier terms needed to accurately express the discontinuous piecewise electric permittivity ε, electric field E, and magnetic field B at the edge of the grating’s grooves.29–32 In general, more Fourier terms yield greater accuracy, but with a longer simulation time. As the number of Fourier terms is increased, the diffraction efficiency oscillates and converges. Usually, the RCWA calculation for metal gratings converges slower than that for dielectric gratings. Also, RCWA calculation for binary gratings requires more Fourier terms and takes longer to compute than that for sinusoidal and/or continuous gradient index gratings due to the convergence of the Fourier series of the grating index function. For conventional one-dimensional dielectric gratings, 10 to 20 Fourier terms are usually sufficient for accurate RCWT calculation. As an example, the convergence of the RCWA algorithm for the wire grid polarizer example of Section 23.3.2 is shown in Figure 23.31. The resultant 0th-order diffraction efficiency converges nicely after 100 terms while the p-polarization efficiency fluctuates more and converges slowly.

Figure 23.31The convergence of the 0th-order transmitted and reflected diffraction efficiencies for the wire grid polarizer as the number of Fourier terms increases.

23.5Problem Sets

1. Derive the angular dispersion equation (Equation 23.7) from Equation 23.6 assuming α > 0.

2. How many transmitted diffraction orders are presented from a fused silica grating at normal incident with wavelength 0.5 m for periods d = 800 nm, 1600 nm, 2000 nm, and 2800 nm?

3. How many reflected diffraction orders are presented from an aluminum grating for periods d = 950 nm, 1600 nm, and 2400 nm with wavelength 0.5 μm for out-of-plane angle of incidence θ = 40° and ϕ being (a) 10°, (b) 30°, (c) 60°, and (d) 80°?

4. Calculate the Littrow angle for blaze grating with 0.5 μm period.

   1. Diffraction order m = 1 and wavelength λ = 0.5, 0.6, and 0.7 μm.

      The Littrow condition is not limited to the −1st order. What is the Littrow blaze angle for the following configurations?

   2. Diffraction order m = 2 and wavelength λ = 0.5, 0.6, and 0.7 μm.

5. Calculate the Woods anomaly locations for a 600 line/mm diffraction grating in the Littrow condition and compare the result to Figure 23.13.

6. In Figure 23.22, what would be the aspect ratio needed for a half wave and quarter wave retarder?

7. At what wavelength is the diattenuation of the example grating in Figure 23.13 zero? Approximate how fast is the diattenuation varying at nearby wavelengths, that is, estimate $\frac{dD}{d\lambda }$.

8. A diffraction grating has the following diffraction efficiencies (output flux/input flux) as a function of wavelength for its −1st order. Green is for light polarized parallel to the rulings, and red is for light polarized perpendicular to the rulings (Figure 23.32).

   1. For an unpolarized incident light, at what wavelengths would the output light have a degree of linear polarization (DoLP) of about 1/4?

   2. Estimate in which spectral regions is the variation of DoP with wavelength the smallest?

Figure 23.32Diffraction efficiencies of diffracted light with polarization parallel (green) and perpendicular (red) to the grating grooves.

Acknowledgments

This chapter incorporates the work of several colleagues including Karlton Crabtree, who performed the analysis of the SWG antireflection-coated lens, and Michihisa Onishi, who wrote the RCWA code used for many of these examples.

References

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1 A sign convention can be defined to have positive diffraction orders refract/reflect away from surface normal while negative orders refract/reflect toward the surface normal. In this convention, Equation 23.6 becomes $\text{sin}{\beta }_m=\text{sin}\alpha +\frac{\text{Sign}[\alpha ]\times m\lambda }{d}$, where Sign[α] = +1 for positive α and Sign[α] = −1 for negative α.

2 Each photon that diffracts from a diffraction grating transfers mh/λ of momentum to the grating in the x–y plane where h is Plank’s constant.

3 The linear dispersion of a monochromator or spectrometer is the rate of change of length per wavelength at the exit slit, $\frac{dx}{d\lambda }=\frac{fm}{d\text{cos}\beta }$, where f is the focal length of the focusing mirror. The grating resolving power is the ability to discern closely spaced spectral lines. It is proportional to the number of rulings N being illuminated. The grating free spectral range is the maximum wavelength range detected without overlapping diffraction orders. When a monochromator is illuminated with monochromatic light, the flux through the exit slit as a function of wavelength is the line spread function, the spectral profile of a monochromatic input. The line spread function is calculated as the convolution of the entrance slit (a rectangle function) with the point spread function (PSF) of the imaging optics in the monochromator, which is then convolved with the exit slit,

The convolution operation is indicated by *. The spectrum measured by a monochromator is the convolution of the input light’s spectrum with the line spread function. The spectral resolution describes the minimum separation in wavelength, which can be identified as two separate spectral lines. The spectral resolution can be estimated as the width of the line spread function in millimeters multiplied by the linear dispersion in nanometers per millimeter.

4 See Chapter 11 (The Jones Pupil and Local Coordinate Systems).