Diffractive exit-pupil expander for display applications - CiteSeerX

Diffractive exit-pupil expander for display applications Hakan Urey

Two-dimensional binary diffraction gratings can be used in wearable display applications as exit-pupil expanders 共EPEs兲 共or numerical-aperture expanders兲 to increase the size of the display exit pupil. In retinal scanning displays the EPE is placed at an intermediate image plane between the scanners and the display exit pupil. A focused spot scans across the diffractive EPE and produces multiple diffraction orders at the exit pupil. The overall luminance uniformity across the exit pupil as perceived by the viewer is a function of the uniformity of the diffraction-order intensities, focused-spot size, grating period, scanning-beam profile, and the viewer’s eye-pupil size. The design, the diffraction-order uniformity, and the effects of the grating phase angle on the uniformity for binary diffraction gratings are discussed. Also discussed are the display exit-pupil uniformity and the impact of the diffractive EPE on the point-spread function and the modulation transfer function of the display. Both theoretical and experimental results are presented. © 2001 Optical Society of America OCIS codes: 050.1950, 120.2040, 120.2080, 050.1380, 110.4100.

1. Introduction

Head- and helmet-mounted displays and eye-wear– type microdisplays create a virtual image on the viewer’s retina. These systems require a large exitpupil size to allow eye movements to scan across the field of view 共FOV兲 as well as to provide some extra tolerance in the positioning of the head- or the helmet-mounted display’s exit pupil relative to the viewer’s eye pupil. Typical exit-pupil sizes for large FOV display systems are in the range of 10 to 15 mm.1,2 Retinal scanning display 共RSD兲 systems operate by the scanning of a light beam onto the viewer’s retina in a two-dimensional 共2-D兲 raster format.3–5 The RSD typically creates a small exit pupil that is approximately the size of the eye pupil. To obtain a large exit pupil requires that an exit-pupil expander 共EPE兲 关i.e., a numerical-aperture 共NA兲 expander兴 be placed at an intermediate image plane in the display system. EPE application in display systems provides a novel use for 2-D diffraction gratings. The diffractive EPE employs small-grating-period binary gratings and provides NA expansion by the creation of replicas of the exit-pupil pattern across the display exit-pupil plane. In Section 2 the operation and in Section 3 the design and fabrication of diffractive EPEs are discussed.

The author 共[email protected]兲 is with Microvision, Inc., 19910 North Creek Parkway, Bothwell, Washington 98011. Received 26 January 2001. 0003-6935兾01兾325840-12$15.00兾0 © 2001 Optical Society of America 5840

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The exit-pupil uniformity of a given display system is a measure of the uniformity of the retinal illuminance when the eye pupil moves across the exit pupil, while looking at the same point in the display field. The exit-pupil uniformity is a function of the uniformity of the diffraction-order intensities, diffractedbeam size and profile, and the eye-pupil size. Typically, a less than 30% intensity variation across the exit pupil is desired. Section 4 presents diffraction-order uniformities and tolerances as functions of the phase angle in binary gratings. Analytical solutions for the zeroth-order intensity as a function of the pattern etch depth and the beam incidence angle are developed by use of scalar diffraction theory. In Section 5 the overall exit-pupil uniformity and coherence effects are discussed for the case of scanning a focused spot across the EPE, including display pixel size, binary grating period, and beam-profile effects. Section 6 is dedicated to the effects of a diffractive EPE on the retinal pointspread function 共PSF兲 and the system modulation transfer function 共MTF兲. 2. Exit-Pupil Expander Operation

Figure 1 illustrates RSD optics and EPE operation. The scan-mirror size D and the total optical scan angle ␪TOSA are important performance parameters that are derived from the display resolution 共i.e., the number of pixels N兲 and the MTF requirements of the display system.4 If the EPE is removed from the system the beam angles before and after the EPE become equal 共␪0 ⫽ ␪i 兲. Then the exit-pupil size Pep

Fig. 1. Cross section of the optical layout for a RSD from light source to the viewer’s eye, illustrating the EPE operation. Pep, exit-pupil size of the display; L, intermediate image size; N, number of pixels; p, pixel size; D, scan mirror size; ␪TOSA, total optical scan angle; ␪i , EPE input-beam angle; ␪0, EPE output-beam angle; e, eye-pupil size.

can be computed by use of the optical invariant 共or Lagrange invariant兲 of the system: P ep tan共␪FOV兾2兲 ⫽ D tan共␪TOSA兾2兲.

(1)

In typical RSDs the exit-pupil size without an EPE is of the order of 1 to 3 mm. For enlarging the NA of the incoming beam to the required pupil size the EPE is placed at the intermediate image plane between the scanner and the exit pupil. Note that the optical invariant before and after the EPE plane does not remain constant in the presence of an EPE: D tan共␪TOSA兾2兲 ⫽ L tan共␪ i 兲, L tan共␪ 0兲 ⫽ P ep tan共␪FOV兾2兲.

(2)

A number of technological devices, such as controlled angle diffusers that are used for screens, appear to be good EPE candidates. However, for reducing system size, while maintaining resolution, the display pixel size p and the size of the focused scanned spot that is incident upon the EPE are designed to be smaller than 30 ␮m. The small spot size prohibits the use of diffuser materials, as they are dependent on the beam geometry’s being much larger than the features of the diffuser material. Binary diffractive EPEs for monochrome green and monochrome red display systems were developed in collaboration with Rochester Photonics Corporation and previously reported in Ref. 6. Figure 2 illustrates the circular array of uniform-intensity beamlets 共i.e., diffraction orders兲 at the exit pupil for largecollimated-beam and small-focused-spot incidence cases. The beamlet size is inversely proportional to the incident-beam size at the diffraction grating, as is discussed in detail in Section 5. The eye pupil collects more than one beamlet at any instant and at any position within the exit pupil. Each beamlet contains all the information of the original beam and, when focused by the eye’s lens, produces overlapping images on the retina. If the diffractive EPE is removed all the beam power goes into the central beamlet. The required NA 共⫽sin ␪0兲 of the EPE and the required number of beamlets can be computed with the help of Eqs. 共2兲 and by use of the FOV and the Pep requirements of the display system.

Fig. 2. 共a兲 Far-field pattern created by a diffractive EPE when illuminated with a collimated laser beam that is much larger than the EPE grating period. 共b兲 Typical exit-pupil pattern created by the same diffractive EPE used for 共a兲 in a RSD system when illuminated with a focused beam scanning across the element. 10 November 2001 兾 Vol. 40, No. 32 兾 APPLIED OPTICS

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Fig. 3. 共a兲 Desired 11 ⫻ 11 circular diffraction-order intensity pattern. 共b兲 The resultant diffraction-order intensities for the design in 共c兲. The binary phase-grating designs in 共c兲 and 共d兲 were obtained by use of the Halomaster II diffractive optical element design software,12 and both designs result in nearly identical diffraction-order intensity patterns.

3. Binary Diffractive Exit-Pupil Expander Design

The literature discusses diffraction-grating designs in detail.7–9 Binary, multilevel,10 or continuoussurface-profile11 共or gray-scale兲 technologies can aid in the design and fabrication of EPEs. Rigorous electromagnetic solutions for the diffraction-order intensities require heavy computations. If the grating period and the smallest feature size are much larger than the wavelength ␭, one can use the thin-element approximation and the scalar diffraction theory. For the NA-expander application, the maximum NA required is typically smaller than 0.25, and the cell size 共i.e., the grating period兲 is larger than 10 ␮m. Even though the feature sizes approach ␭, the binary grating designs based on scalar theory produce quite accurate results. Small feature sizes make it very difficult to align multiple masks to produce multilevel gratings. The discussion in this paper is limited to binary diffraction gratings designed by use of the scalar diffraction theory. The main performance parameters for a diffractive EPE design are diffraction efficiency, uniformity, and manufacturability. The diffraction efficiency ␩ is defined as the ratio of the total beam intensity that goes into the intended diffraction orders to the input5842

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beam intensity. The diffraction efficiency does not take optical losses into account. The uniformity U is a measure of the deviation from uniformity and can be calculated by use of the maximum IMax and the minimum IMin diffraction-order intensities within the intended NA: U⫽

I Max ⫺ I Min 100. I Max ⫹ I Min

(3)

Figure 3共a兲 shows the desired diffraction pattern for an exemplary NA expander. There are 89 diffraction orders all with uniform intensity within a circular window, and the diffraction orders outside the window have zero intensity. The binary diffraction gratings shown in Figs. 3共c兲 and 3共d兲 are designed by use of the commercial design program Holomaster II.12 In the figures, the basic unit-cell design is tiled to form a 2 ⫻ 2 array of cells, and there is a ␲ phase difference between the black and the white levels. Holomaster II employs a simulated-annealing algorithm.13 The program starts with a random design and can converge to a different solution after each run. The best design should be selected based

Fig. 4. 共a兲 Repeated binary grating-cell design for an exemplary EPE and 共b兲 a cross-sectional view of the cell.

on efficiency, uniformity, and manufacturability 关e.g., the critical dimension 共CD兲 size兴. Both designs have an approximately 75% theoretical diffraction efficiency and an approximately 3% uniformity. Figure 3共b兲 shows the resultant diffraction-order intensities for Fig. 3共c兲. The design shown in Fig. 3共d兲 has a nearly identical far-field intensity pattern. The designs shown in Figs. 3共c兲 and 3共d兲 have a 64 ⫻ 64 pixel digital representation for each EPE cell. Increasing the number of pixels in the design can further improve the efficiency and the uniformity and reduce the uniformity errors caused by the digitization of the 2-D grating pattern. The CD, which can be defined as the size of the smallest channel in the binary pattern, is an important parameter for the manufacturability of the design. A binary grating that meets the CD requirements can be produced by use of various mask-writing technologies, followed by replication onto glass or plastic substrates. Fabrication tolerances add up and make it hard to produce gratings that achieve the theoretical uniformity values for small grating periods with large output NAs. For instance, the widening and the narrowing of the features 共e.g., CD errors兲 and the pattern-depth errors that occur during fabrication cause uniformity variations. In particular, the zeroth diffractionorder intensity is highly sensitive to the depth errors of the fabricated patterns. In Section 4, we discuss the pattern-depth error and other parameters that affect beamlet uniformity by way of changing the grating phase angle. 4. Diffraction-Order Uniformity: Collimated Large-Beam Incidence

In this section, we first give a mathematical description of the grating operation and state an important property of binary gratings. We then discuss the zeroth-order and the off-axis–order intensity variations with etch depth and incidence angle.

In the scalar diffraction theory the optical field immediately after the diffractive optical element can be modeled as U out共 x, y兲 ⫽ U in共 x, y兲exp关 j␾共 x, y兲兴,

(4)

where Uin共x, y兲 is the incident optical field and ␾共x, y兲 is the phase of the periodic diffractive element. Figure 4共a兲 shows one period of a 2-D binary phase grating with a period d, and Fig. 4共b兲 shows the chief ray angles for the incident field and the transmitted diffraction orders. If we assume that the incident field is a unit-amplitude plane wave 关Uin共x, y兲 ⫽ 1兴 the diffracted-order intensities can be calculated with the Fourier integral S关k, m兴 ⫽

兰兰 d

0

d

0

exp关 j␾共 x, y兲兴

冋 冉

⫻ exp ⫺j2␲

kx my ⫹ d d

冊册

dxdy,

(5)

where the 2-D phase function ␾共x, y兲 is defined by ␾共 x, y兲 ⫽

再

␾0 0

共 x, y兲 僆 A 1 , 共 x, y兲 僆 A 0

(6)

where ␾0 is the phase angle of the grating and A1 and A0 are nonoverlapping 共complementary兲 regions within the cell area 兵i.e., A0 艛 A1 ⫽ 共关0, d兴, 关0, d兴兲 and A0 艚 A1 ⫽ 0其. The A1 and A0 representations are also used to denote the areas of the regions A1 and A0 throughout the paper. The sum of the areas of A1 and A0 is equal to the area of the cell d2. The diffraction efficiency of the grating can be defined as the sum of diffraction-order intensities within the window of interest W: ␩⫽

兺

兩S关k, m兴兩 2.

(7)

共k,m兲僆W

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The propagation angles of the diffracted beams are given by sin ␪ km cos ␾ km ⫽ sin ␪ inc ⫹ k␭兾d, sin ␪ km sin ␾ km ⫽ m␭兾d,

(8)

where ␭ is the wavelength of light, k and m are the indices describing the diffraction order 共k, m兲, d is the cell period, ␪inc is the angle between the incident beam and the normal of the EPE, ␪km is the angle that the diffracted order makes with the normal of the EPE plane, and ␾km is the component of the propagation direction of the diffraction order 共k, m兲 in the horizontal direction. These angles are valid for both scalar and rigorous design. For the case illustrated in Fig. 4共b兲, ␾0 can be computed by use of the phase angle of the rays that pass through two different levels of the binary grating

再

2␲t n ⫺1 ␭ cos关sin 共sin ␪inc兾n兲兴

␾ 0共n, ␭, t, ␪ inc兲 ⫽

⫺

冎

1 , cos ␪ inc

(9)

where n is the refractive index of the material and t is the thickness of the binary pattern on the substrate. For ␪inc ⫽ 0, the value of t that is needed to produce a ␲ phase difference at the design wavelength ␭ is given by ␭ t␲ ⫽ . 2共n ⫺ 1兲

(10)

Except for the zeroth-order intensity the ratio of the intensities of all the diffraction orders relative to each other is independent of the phase angle ␾0. From the conservation-of-energy principle the amount of change in the energy of the zeroth diffraction order is counteracted by the total amount of change in all the off-axis diffraction orders.14,15 In mathematical terms, for a binary diffraction-grating function ␾共x, y兲 that has a constant amplitude of ␾0, if the diffractionorder amplitudes are S关k, m兴, where k, m ⫽ 0, ⫾1, ⫾2, ⫾3, . . . , then S关k1, m1兴兾S关k2, m2兴 is independent of ␾0 if 共k1, m1兲 ⫽ 0 and 共k2, m2兲 ⫽ 0. The zeroth diffraction-order intensity I0 can be expressed as I 0 ⫽ 兩S关0, 0兴兩 ⫽ 2

冏兰 兰 d

0

冏

d

2

exp关 j␾共 x, y兲兴dxdy ,

0

(11)

I 0 ⫽ 兩 A 0 ⫹ A 1 exp共 j␾ 0兲兩 2 ⫽ A 02 ⫹ A 12 ⫹ 2 A 0 A 1 cos ␾ 0. (12) Equation 共9兲 can now be rewritten in terms of the fractional etch-depth error ε ⫽ t兾t␲ ⫺ 1: ␾0 ⫽

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再

冎

␲共1 ⫹ ε兲 1 n ⫺ . 共n ⫺ 1兲 cos关sin⫺1共sin ␪inc兾n兲兴 cos ␪inc (13) APPLIED OPTICS 兾 Vol. 40, No. 32 兾 10 November 2001

Fig. 5. Ratio of the zeroth-order intensity to the average diffraction-order intensity I0兾Iavg plotted as a function of the percent etch-depth error ε. Assumptions are that there are Nepe ⫽ 121 diffraction orders within the intended circular NA, the diffraction efficiency is ␩ ⫽ 75%, the refractive index is n ⫽ 1.46, A1 ⫽ 0.53d2, and A0 ⫽ 0.47d2, producing I0兾Iavg ⫽ 1 for ε ⫽ 0.

Note that, for normal incidence 共␪inc ⫽ 0兲 and no etch-depth error 共ε ⫽ 0, t ⫽ t␲兲, we find that ␾0 ⫽ ␲, I0 ⫽ 共 A0 ⫺ A1兲2, and I0 goes to zero for A0 ⫽ A1. The average diffraction-order intensity Iavg within the intended NA can be expressed as I avg ⫽

␩ , N epe

(14)

where Nepe is the number of diffraction orders within the intended NA. As stated above, if ␾0 changes, the resultant change in the average off-axis–order intensity ⌬Iavg has the opposite sign from that of the change in the zeroth-order intensity ⌬I0, and, because of the large values for Nepe for the EPE application, ⌬Iavg is a small fraction of ⌬I0 and can be ignored. Figure 5 shows I0兾Iavg plotted as a function of the etch-depth error ε for the EPE used to obtain the far-field patterns shown in Fig. 2. Figure 5 suggests that the etch depth should be controlled to within 2%–3% to keep ⌬I0 smaller than 30%. Such a small tolerance makes the fabrication of these parts difficult. Equation 共14兲 can also be used to compute I0 as a function of wavelength by one’s treating the wavelength difference between the operation and the design wavelengths as a phase error. For instance, if an EPE that is designed for a green 共532-nm兲 monochrome display system is used in a red 共635-nm兲 monochrome system, the fractional phase error becomes ε ⫽ 532兾635 ⫺ 1 ⫽ ⫺0.16, and I0 becomes approximately 15Iavg, whereas the uniformity of the off-axis orders 共excluding the zeroth order兲 does not change with wavelength. Because the zeroth order is normally a part of the display exit-pupil pattern, the rapid growth of I0 with the wavelength makes the binary diffractive EPE unsuitable for full-color display systems. Other NA-expander technologies need to be explored for full-color displays. Figure 6 shows I0 plotted as a function of the beam’s incidence angle, which is essentially the scan angle illustrated in Fig. 1. For a negative etch-

5. Exit-Pupil Uniformity: Focused-Spot Incidence

Fig. 6. Theoretical and measurement results for I0 and measurement results for one of the off-axis–order intensities plotted as functions of the beam’s incidence angle at the EPE. All values are relative to the average diffraction-order intensity of 1.0. Assumptions for Fig. 6 are the same as those for Fig. 5.

So far, we discussed the uniformity of diffraction orders. In this section, we discuss the impact of the spot-size–to–EPE-cell-size ratio 共s兾d ratio兲, the beam shape, and the viewer eye-pupil size on the display exit-pupil uniformity. As is illustrated in Fig. 7, the diffractive exit pupil with a period d produces replicas of the non-EPE system exit-pupil pattern. The angle between the diffracted orders or the beamlets is given by ␪d ⫽ ␭兾d, whereas the angular width of the beamlets, 2␪s, is determined by the focusing geometry of the scanner. The focusing optics typically has a small ␪s and produces a diffraction-limited spot at the EPE plane. The spot size s is defined as the FWHM intensity and is inversely proportional to ␪s s ⫽ K T␭f #,

depth error 共ε ⬍ 0兲, I0 starts to increase rapidly with the incidence angle. If the beamlet-uniformity requirement is 30% the incidence angle at the EPE becomes limited to less than 10° even for a zero etchdepth error 共ε ⫽ 0兲. Figure 6 reveals that slightly overetching the pattern 共by approximately 1%–2%兲 can help to reduce the effects of the incidence angle across the EPE and allow for incidence angles larger than 15°. Experimental results for I0兾Iavg and Ioff-axis for one of the off-axis orders are also plotted in the figure. The measurements were obtained by the rotation of the EPE and the monitoring of the beamlet intensity. As predicted, Ioff-axis changes by only a very small amount for ␪inc up to 20°. The measured value of I0 increases rapidly for ␪inc ⬎ 15°, and, based on the shape of the measured I0 curve, the etch-depth error for the EPE used in the experiments is predicted to be approximately ⫹1%. For larger values of ␪inc the measured value of I0 does not increase as fast as the theoretical predictions. The difference can be attributed to the decrease in diffraction efficiency, the increase in Fresnel losses at the tilted surfaces, and the increase in the backreflectedorder intensities with increasing ␪inc.

(15)

where KT is a function of the Gaussian beamtruncation ratio T and f# ⫽ z兾D is the f-number of the focusing geometry. For an incident Gaussian beam the beam-truncation ratio T is defined as the ratio of the Gaussian beam width at 1兾e2 intensity 共2wm兲 to the system’s aperture diameter D. As T increases, the spot profile changes from a Gaussian 共T ⬍ 0.4兲 to an Airy disc 共as T 3 ⬁兲. For small T 共i.e., negligible clipping at the aperture, a Gaussian spot兲 an analytical expression can be found for KT. For large T empirical formulas for K can be obtained by the numerical solution of the Fresnel diffraction integral16: T ⫽ 2w m兾D, KT ⫽

再

1.036 ⫺ 0.058兾T ⫹ 0.156兾T 2 0.75兾T

(16) if T ⱖ 0.4 . if T ⱕ 0.4 (17)

Figure 8 shows the exit-pupil patterns generated by means of changing the incident-beam focusing NA and thereby changing the diffraction-limited spot size8 s. Note that the EPE used in the experiments had poor diffraction-order uniformity. The objective

Fig. 7. RSD focusing geometry and EPE operation illustrating Gaussian beam truncation at the limiting system aperture. wm, the 1兾e2-intensity beam radius; d, the grating period. 10 November 2001 兾 Vol. 40, No. 32 兾 APPLIED OPTICS

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Fig. 8. Experimental results for the exit-pupil pattern as a function of the spot-size–to– cell-size ratio s兾d. focusing geometry varies the s兾d ratio.

in the experiment was to illustrate how the exit-pupil pattern and the exit-pupil uniformity changed by the reduction of the 2␪s兾␪d ratio 共or the increase of the s兾d ratio兲. The system aperture was illuminated with a converging uniform beam, and the 2␪s兾␪d ratio was adjusted by the reduction of the aperture stop size D. When 2␪s ⬎⬎ ␪d 共s ⬍⬍ d, the top row in Fig. 8兲, the spot

Changing the NA of the

focused on the grating does not illuminate a sufficiently large portion of the grating, and thus distinct diffraction orders do not form properly. Furthermore, the exit-pupil pattern changes dramatically by one’s moving the spot across the grating, resulting in poor exit-pupil uniformity. When 2␪s ⬎ ␪d 共s ⬍ d, the middle row in Fig. 8兲 overlapping beamlets start

Fig. 9. Exit-pupil patterns produced by two different beam profiles at the system aperture 共the aperture size is adjusted to yield the same FWHM spot size in each case兲: 共a兲 small truncation 共Gaussian beam and Gaussian spot兲 and 共b兲 large truncation 共uniform beam and Airy disc spot兲. 5846

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Fig. 10. 共a兲 Randomly generated beamlet intensities for a uniformity of 30% 共U ⫽ 0.3兲 and a zeroth-order diffraction ratio of I0 ⫽ 1. 共b兲 Computed exit-pupil pattern obtained by the convolution of the beam profile at the aperture by the beamlet intensities shown in 共a兲.

to form; however, the random phase variations of the beamlets cause interference at the overlap areas and reduce the exit-pupil uniformity. Moving the spot across the grating causes variations on the diffractedorder intensities and reduces the exit-pupil uniformity further. Small spot sizes also cause some fringelike intensity modulation across the displayed image, reducing the image quality. The left-most picture in the bottom row in Fig. 8 shows the case of 2␪s ⫽ ␪d 共s ⬃ d, as discussed below兲. The beamlets fill the space without causing any interference, the diffraction-order intensities do not change significantly by the movement of the spot across the grating, and the intensity modulation disappears. This situation represents the optimal choice of spot size relative to the EPE cell size for the best exit-pupil uniformity. The other images in the bottom row il-

Fig. 11. Exit-pupil uniformity as perceived by the viewer for 共a兲 a 2-mm and 共b兲 a 5-mm eye-pupil size as obtained by use of the computed exit-pupil pattern of Fig. 10共b兲.

lustrate the case of ␪s ⬍ ␪d 共s ⬎ d兲, where the beamlets are getting smaller and start to reduce the exitpupil uniformity by causing gaps between the beamlets. Another important parameter that affects the exitpupil uniformity is the beam profile across the limiting system aperture. One can obtain the same size spot at the EPE plane by use of different focusing geometry. Figure 9 shows two exit-pupil patterns obtained by use of the same size spot but with the first one’s having a small truncation ratio and the second one, a large truncation ratio. Beamlets interfere heavily for the case of the small truncation ratio and cause variation of the interference patterns as the spot moves across the EPE. The large truncation ratio provides better uniformity across the exit-pupil pattern. The exit-pupil plane for a scanning display system is located at a plane conjugate with the scanner. 10 November 2001 兾 Vol. 40, No. 32 兾 APPLIED OPTICS

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Fig. 12. System exit-pupil uniformity plotted as a function of the s兾d ratio and T. It is assumed that Pep ⫽ 15 mm, there is a 13 ⫻ 13 array of beamlets, U ⫽ 0.3, I0 ⫽ 1, and the eye-pupil size is 2 mm. The crosses show to which contour line the contour labels belong.

Thus for a system without an EPE the exit-pupil profile is the same as the beam profile at the scanner

共assuming the focusing optics and the ocular are diffraction limited and sufficiently large compared with the beam size兲. Because the function of the diffractive EPE is to create replicas of the exit-pupil pattern, one can easily compute the system exit-pupil uniformity for different values of eye-pupil size after T, the diffracted-order intensities, and the s兾d ratio are known. Figure 10共a兲 shows the relative diffracted-order intensities for an exemplary 13 ⫻ 13 circular array of diffracted orders that form a 15 mm ⫻ 15 mm exit pupil. Figure 10共b兲 shows the simulated exit-pupil pattern, assuming that the grating design has a uniformity of U ⫽ 30% and I0 ⫽ 1. Figures 11共a兲 and 11共b兲 show the resultant displaysystem exit-pupil uniformity after the convolution of Fig. 10共b兲 with a 2-mm and a 5-mm circular eye pupil, respectively. The 5-mm eye pupil is fairly large, and the exit-pupil uniformity is good, independently of the choice of system parameters. For the 2-mm pupil case, however, the choice of system parameters can make a large difference in the resultant system exit-pupil uniformity. Figure 12 shows how the choice of system parameters T and the s兾d ratio affect the system exit-pupil

Fig. 13. PSF measurements obtained by use of the EPE of Fig. 2共b兲 that has a cell period of 16 ␮m. The spot in the EPE is imaged onto a CCD by use of a microscope objective and an aperture that mimic a 2-mm eye pupil. Top row, left-most image: the PSF without an EPE. Remaining images: same measurement with the EPE in place and obtained by the movement of the EPE horizontally relative to the focused spot in 2-␮m steps. 5848

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Fig. 14. Results of the repetition of the experiment of Fig. 13 but with an aperture that mimics a 5-mm eye pupil.

uniformity for an EPE design that produces U ⫽ 30% and I0 ⫽ 1.0, assuming a 2-mm eye pupil. The contour curves show the percent exit-pupil uniformity. The lower left-hand region of the figure

shows an undesired area, where beamlets interfere. Depending on the s兾d ratio and T, the exit-pupil uniformity changes from approximately 30% to ⬎50%.

Fig. 15. Horizontal component of the MTF for the PSFs shown in 共a兲 Fig. 13 共2-mm eye-pupil case兲 and 共b兲 Fig. 14 共5-mm eye-pupil case兲. The vertical MTF remains constant when the grating is moved horizontally relative to the spot. The curve that represents no use of the EPE is marked with open circles. The other curves each correspond to a different spot position across the EPE. 共a兲 Eight and 共b兲 16 different spot positions were used to compute the MTFs. 10 November 2001 兾 Vol. 40, No. 32 兾 APPLIED OPTICS

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Fig. 16. Extremum values of the horizontal 共Hor.兲 and the vertical 共Ver.兲 MTFs at the display cutoff frequency plotted as a function of the eye-pupil size. The variation of the MTF is obtained by the movement of the EPE diagonally relative to the focused static spot.

6. Modulation Transfer Function of Numerical-Aperture Expanders

In the previous sections, we discussed the far-field effects of diffractive EPEs. In this section, we focus on the near-field effects and discuss the PSF and the MTF implications of the EPE. The beamlets that pass through the eye pupil are focused by the eye lens onto the retina and form overlapping images on the retina. The random phase of the beamlets results in dynamic interference patterns on the retina. The phase of the beamlets and the interference pattern change as the scanned light beam moves across the retina. Figures 13 and 14 show CCD pictures of the PSF as a function of the spot position across the cell for the 2-mm and the 5-mm eye-pupil cases, respectively. The focused spot at the EPE is imaged onto a CCD camera by use of a high-NA microscope objective that is followed by a circular aperture that mimics the eye pupil. The top left-most image of Fig. 13 shows the PSF when the EPE is removed from the optical train. The EPE grating period is 16 ␮m, and it is moved in increments of 2 ␮m horizontally between the different images shown in the figures. Note that, for the 2-mm pupil case, only an approximately 2 ⫻ 2 array of beamlets can pass through the pupil. Therefore, depending on the phases of the beamlets, the interference pattern has either one lobe or two lobes across each dimension. On the other hand, for the 5-mm pupil case, there is an approximately 5 ⫻ 5 array of beamlets interfering at the retina, causing a more complicated interference pattern. Figures 15共a兲 and 15共b兲 show the corresponding horizontal MTF variations. Moving the grating horizontally does not vary the vertical MTF. The MTF images reveal several interesting points. First, the diffractive EPE causes high-spatial-frequency intensity variations. The display’s cutoff spatial frequency was set to approximately 30 cycles兾mm in the experiments. Display systems are often designed so that single pixels are not seen. Therefore viewers may not perceive the high-spatial-frequency features beyond the display cutoff frequency. Note that, at spatial frequencies below the display’s cutoff fre5850

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quency, the 5-mm pupil causes smaller MTF variations with spot position across the cell compared with the 2-mm pupil case. Figure 16 shows the range of MTF values at the display cutoff frequency 共30 cycles兾mm兲 obtained by the movement of a focused spot across the EPE in both the horizontal and the vertical directions. Repeating the experiment for pupil sizes from 1 to 10 mm 共2 to 8 mm is a more realistic range of human eye-pupil diameter兲 shows that the MTF values fluctuate over a wide range for small pupil sizes. For pupil sizes larger than 6 mm the MTF variation with spot position across the EPE is negligible.

7. Conclusions

Binary diffraction gratings can be used as EPEs in scanning display systems. The uniformity of diffraction-order intensities is sensitive to the fabrication tolerances and the grating phase. The grating phase is a function of pattern depth, beam incidence angle, and wavelength. Although off-axis diffraction-order intensities relative to each other remain the same, the zeroth-order intensity changes rapidly with the phase angle. Good exit-pupil uniformity can be obtained by one’s controlling the uniformity of the diffraction-order intensities and also by the adjustment of the system parameters, such as the beam-truncation ratio and the spot-size–to–EPE-cellsize ratio. The diffractive EPE causes PSF and MTF variations as a result of the interference of the beamlets at the viewer’s retina. The spot-positiondependent MTF variation is more pronounced for small eye-pupil sizes, which occur in high-luminance systems. I am thankful to Peggy Lopez for her help with the experiments. Financial support from the U.S. Army for parts of this research is gratefully acknowledged. References 1. H. L. Task, “HMD image source, optics, and the visual interface,” in Head Mounted Displays: Designing for the User, J. E. Melzer and K. Moffitt, eds. 共McGraw-Hill, New York, 1997兲, Chap. 3, pp. 55– 82. 2. C. E. Rash, ed., Helmet-Mounted Displays: Design Issues for Rotary-Wing Aircraft, Vol. PM93 of SPIE Monographs and Handbooks Series 共SPIE Press, Bellingham, Wash., 2001兲. 3. H. Urey, “Retinal scanning displays,” in Encyclopedia of Optical Engineering, R. B. Johnson and R. G. Diggers, eds. 共Marcel Dekker, New York, 2001兲. 4. H. Urey, N. Nestorovic, B. Ng, and A. Gross, “Optics designs and system MTF for laser scanning displays,” in Helmet and Head-Mounted Displays IV, R. J. Lewandowski, L. A. Haworth, and H. J. Girolamo, eds., Proc. SPIE 3689, 238 –248 共1999兲. 5. H. Urey, D. W. Wine, and T. D. Osborn, “Optical performance requirements for MEMS-scanner based microdisplays,” in MOEMS and Miniaturized Systems, M. E. Motamedi and R. Go¨ ring, eds., Proc. SPIE 4178, 176 –185 共2000兲. 6. A. Stevens, H. Urey, P. Lopez, T. R. M. Sales, R. McGuire, and D. H. Raguin, “Diffractive optical elements for numerical aperture expansion in retinal scanning displays,” in Diffractive

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