System Design

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 EPEs or numerical-aperture expanders 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 pro le, 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 exper-

imental results are presented. 2001 Optical Society of America

OCIS codes: 050.1950, 120.2040, 120.2080, 050.1380, 110.4100.

1. Introduction The exit-pupil uniformity of a given display system

is a measure of the uniformity of the retinal illumi-

Head- and helmet-mounted displays and eye-wear

nance when the eye pupil moves across the exit pupil,

type microdisplays create a virtual image on the

while looking at the same point in the display eld.

viewer s retina. These systems require a large exit-

The exit-pupil uniformity is a function of the unifor-

pupil size to allow eye movements to scan across the

mity of the diffraction-order intensities, diffracted-

eld of view FOV as well as to provide some extra

beam size and pro le, and the eye-pupil size.

tolerance in the positioning of the head- or the

Typically, a less than 30% intensity variation across

helmet-mounted display s exit pupil relative to the

the exit pupil is desired. Section 4 presents

viewer s eye pupil. Typical exit-pupil sizes for large

diffraction-order uniformities and tolerances as func-

FOV display systems are in the range of 10 to 15

tions of the phase angle in binary gratings. Analyt-

mm.1,2 Retinal scanning display RSD systems op-

ical solutions for the zeroth-order intensity as a

erate by the scanning of a light beam onto the view-

function of the pattern etch depth and the beam in-

er s retina in a two-dimensional 2-D raster

cidence angle are developed by use of scalar diffrac-

format.3 5 The RSD typically creates a small exit

tion theory. In Section 5 the overall exit-pupil

pupil that is approximately the size of the eye pupil.

uniformity and coherence effects are discussed for the

To obtain a large exit pupil requires that an exit-pupil

case of scanning a focused spot across the EPE, in-

expander EPE i.e., a numerical-aperture NA ex-

cluding display pixel size, binary grating period, and

pander be placed at an intermediate image plane in

beam-pro le effects. Section 6 is dedicated to the

the display system. EPE application in display sys-

effects of a diffractive EPE on the retinal point-

tems provides a novel use for 2-D diffraction gratings.

spread function PSF and the system modulation

The diffractive EPE employs small-grating-period bi-

transfer function MTF .

nary gratings and provides NA expansion by the cre-

ation of replicas of the exit-pupil pattern across the

display exit-pupil plane. In Section 2 the operation

2. Exit-Pupil Expander Operation

and in Section 3 the design and fabrication of diffrac-

Figure 1 illustrates RSD optics and EPE operation.

tive EPEs are discussed.

The scan-mirror size D and the total optical scan

angle TOSA are important performance parameters

that are derived from the display resolution i.e., the

The author abqmff@r.postjobfree.com is with Microvision, Inc.,

number of pixels N and the MTF requirements of the

19910 North Creek Parkway, Bothwell, Washington 98011.

display system.4 If the EPE is removed from the

Received 26 January 2001.

system the beam angles before and after the EPE

000*-****-** 325840-12$15.00 0

become equal 0 i. Then the exit-pupil size Pep

2001 Optical Society of America

5840 APPLIED OPTICS Vol. 40, No. 32 10 November 2001

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 or

Lagrange invariant of the system:

P ep tan 2 D tan 2. (1)

FOV TOSA

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 2 L tan,

TOSA i

L tan P ep tan 2. (2)

0 FOV

A number of technological devices, such as con-

trolled angle diffusers that are used for screens, ap-

pear 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 de-

signed 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 illus-

trates the circular array of uniform-intensity beam-

lets i.e., diffraction orders at the exit pupil for large-

collimated-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 col-

lects more than one beamlet at any instant and at any

position within the exit pupil. Each beamlet con-

tains 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 re-

moved all the beam power goes into the central beam- Fig. 2. a Far- eld pattern created by a diffractive EPE when

let. The required NA sin 0 of the EPE and the illuminated with a collimated laser beam that is much larger than

required number of beamlets can be computed with the EPE grating period. b Typical exit-pupil pattern created by

the help of Eqs. 2 and by use of the FOV and the Pep the same diffractive EPE used for a in a RSD system when

requirements of the display system. illuminated with a focused beam scanning across the element.

10 November 2001 Vol. 40, No. 32 APPLIED OPTICS 5841

Fig. 3. a Desired 11 11 circular diffraction-order intensity pattern. b The resultant diffraction-order intensities for the design in

c . The binary phase-grating designs in c and d 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 beam intensity. The diffraction ef ciency does not

take optical losses into account. The uniformity U is

The literature discusses diffraction-grating designs

a measure of the deviation from uniformity and can

in detail.7 9 Binary, multilevel,10 or continuous-

be calculated by use of the maximum IMax and the

surface-pro le11 or gray-scale technologies can aid

minimum IMin diffraction-order intensities within the

in the design and fabrication of EPEs. Rigorous

intended NA:

electromagnetic solutions for the diffraction-order in-

tensities require heavy computations. If the grating

I Max I Min

period and the smallest feature size are much larger

U 100. (3)

than the wavelength, one can use the thin-element I Max I Min

approximation and the scalar diffraction theory.

For the NA-expander application, the maximum NA Figure 3 a shows the desired diffraction pattern

required is typically smaller than 0.25, and the cell for an exemplary NA expander. There are 89 dif-

size i.e., the grating period is larger than 10 m. fraction orders all with uniform intensity within a

Even though the feature sizes approach, the binary circular window, and the diffraction orders outside

grating designs based on scalar theory produce quite the window have zero intensity. The binary diffrac-

accurate results. Small feature sizes make it very tion gratings shown in Figs. 3 c and 3 d are de-

dif cult to align multiple masks to produce multilevel signed by use of the commercial design program

gratings. The discussion in this paper is limited to Holomaster II.12 In the gures, the basic unit-cell

binary diffraction gratings designed by use of the design is tiled to form a 2 2 array of cells, and there

scalar diffraction theory. is a phase difference between the black and the

The main performance parameters for a diffractive white levels.

EPE design are diffraction ef ciency, uniformity, and Holomaster II employs a simulated-annealing al-

manufacturability. The diffraction ef ciency is gorithm.13 The program starts with a random de-

de ned as the ratio of the total beam intensity that sign and can converge to a different solution after

goes into the intended diffraction orders to the input- each run. The best design should be selected based

5842 APPLIED OPTICS Vol. 40, No. 32 10 November 2001

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

on ef ciency, uniformity, and manufacturability e.g., In the scalar diffraction theory the optical eld

the critical dimension CD size . Both designs have immediately after the diffractive optical element can

an approximately 75% theoretical diffraction ef - be modeled as

ciency and an approximately 3% uniformity. Figure

U out x, y U in x, y exp j x, y, (4)

3 b shows the resultant diffraction-order intensities

for Fig. 3 c . The design shown in Fig. 3 d has a

where Uin x, y is the incident optical eld and x, y

nearly identical far- eld intensity pattern.

is the phase of the periodic diffractive element. Fig-

The designs shown in Figs. 3 c and 3 d have a

ure 4 a shows one period of a 2-D binary phase grat-

64 64 pixel digital representation for each EPE cell.

ing with a period d, and Fig. 4 b shows the chief ray

Increasing the number of pixels in the design can

angles for the incident eld and the transmitted dif-

further improve the ef ciency and the uniformity and

fraction orders. If we assume that the incident eld

reduce the uniformity errors caused by the digitiza-

is a unit-amplitude plane wave Uin x, y 1 the

tion of the 2-D grating pattern. The CD, which can

diffracted-order intensities can be calculated with the

be de ned as the size of the smallest channel in the

Fourier integral

binary pattern, is an important parameter for the

manufacturability of the design. A binary grating d d

that meets the CD requirements can be produced by S k, m exp j x, y

use of various mask-writing technologies, followed by 0 0

replication onto glass or plastic substrates. Fabri-

kx my

cation tolerances add up and make it hard to produce exp j2 d x dy, (5)

gratings that achieve the theoretical uniformity val- d d

ues for small grating periods with large output NAs.

where the 2-D phase function x, y is de ned by

For instance, the widening and the narrowing of the

features e.g., CD errors and the pattern-depth er-

x, y A1

0

rors that occur during fabrication cause uniformity x, y, (6)

0 x, y A0

variations. In particular, the zeroth diffraction-

order intensity is highly sensitive to the depth errors where 0 is the phase angle of the grating and A1 and

of the fabricated patterns. In Section 4, we discuss A0 are nonoverlapping complementary regions

the pattern-depth error and other parameters that within the cell area i.e., A0 A1 0, d, 0, d and

affect beamlet uniformity by way of changing the A0 A1 0 . The A1 and A0 representations are

grating phase angle. also used to denote the areas of the regions A1 and A0

throughout the paper. The sum of the areas of A1

4. Diffraction-Order Uniformity: Collimated

and A0 is equal to the area of the cell d2.

Large-Beam Incidence

The diffraction ef ciency of the grating can be de-

In this section, we rst give a mathematical descrip- ned as the sum of diffraction-order intensities

tion of the grating operation and state an important within the window of interest W:

property of binary gratings. We then discuss the

zeroth-order and the off-axis order intensity varia- 2

S k, m . (7)

tions with etch depth and incidence angle. k,m W

10 November 2001 Vol. 40, No. 32 APPLIED OPTICS 5843

The propagation angles of the diffracted beams are

given by

sin cos sin k d,

km km inc

sin sin m d, (8)

km km

where is the wavelength of light, k and m are the

indices describing the diffraction order k, m, 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 prop-

agation direction of the diffraction order k, m in the Fig. 5. Ratio of the zeroth-order intensity to the average

horizontal direction. These angles are valid for both diffraction-order intensity I0 Iavg plotted as a function of the per-

cent etch-depth error . Assumptions are that there are Nepe

scalar and rigorous design.

121 diffraction orders within the intended circular NA, the diffrac-

For the case illustrated in Fig. 4 b, 0 can be com-

tion ef ciency is 75%, the refractive index is n 1.46, A1

puted by use of the phase angle of the rays that pass

0.53d2, and A0 0.47d2, producing I0 Iavg 1 for 0.

through two different levels of the binary grating

2t n

n,, t,

0 inc 1

cos sin sin n Note that, for normal incidence inc 0 and no

inc

etch-depth error 0, t t, we nd that 0,

1

A0 A1 2, and I0 goes to zero for A0 A1.

I0, (9)

cos The average diffraction-order intensity Iavg within

inc

the intended NA can be expressed as

where n is the refractive index of the material and t

is the thickness of the binary pattern on the sub-

I avg, (14)

strate. For inc 0, the value of t that is needed to

N epe

produce a phase difference at the design wave-

length is given by 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 inten-

t . (10)

2n 1 sity Iavg has the opposite sign from that of the

change in the zeroth-order intensity I0, and, be-

Except for the zeroth-order intensity the ratio of the

cause of the large values for Nepe for the EPE appli-

intensities of all the diffraction orders relative to each

cation, Iavg is a small fraction of I0 and can be

other is independent of the phase angle 0. From

ignored.

the conservation-of-energy principle the amount of

Figure 5 shows I0 Iavg plotted as a function of the

change in the energy of the zeroth diffraction order is

etch-depth error for the EPE used to obtain the

counteracted by the total amount of change in all the

far- eld patterns shown in Fig. 2. Figure 5 suggests

off-axis diffraction orders.14,15 In mathematical

that the etch depth should be controlled to within

terms, for a binary diffraction-grating function x, y

2% 3% to keep I0 smaller than 30%. Such a small

that has a constant amplitude of 0, if the diffraction-

tolerance makes the fabrication of these parts dif -

order amplitudes are S k, m, where k, m 0, 1,

cult. Equation 14 can also be used to compute I0 as

2, 3, . . ., then S k1, m1 S k2, m2 is independent

a function of wavelength by one s treating the wave-

of 0 if k1, m1 0 and k2, m2 0.

length difference between the operation and the de-

The zeroth diffraction-order intensity I0 can be ex-

sign wavelengths as a phase error. For instance, if

pressed as

an EPE that is designed for a green 532-nm mono-

2 chrome display system is used in a red 635-nm

d d

2

monochrome system, the fractional phase error be-

I0 S 0, 0 exp j x, y d x dy, (11)

comes 532 635 1 0.16, and I0 becomes

0 0

approximately 15Iavg, whereas the uniformity of the

2

A 02 A 12

I0 A0 A 1 exp j 2 A 0 A 1 cos . off-axis orders excluding the zeroth order does not

0 0

change with wavelength. Because the zeroth order

(12) is normally a part of the display exit-pupil pattern,

the rapid growth of I0 with the wavelength makes the

Equation 9 can now be rewritten in terms of the

binary diffractive EPE unsuitable for full-color dis-

fractional etch-depth error t t 1:

play systems. Other NA-expander technologies

need to be explored for full-color displays.

1 n 1

Figure 6 shows I0 plotted as a function of the

.

0 1

n1 cos sin sin n cos beam s incidence angle, which is essentially the scan

inc inc

(13) angle illustrated in Fig. 1. For a negative etch-

5844 APPLIED OPTICS Vol. 40, No. 32 10 November 2001

5. Exit-Pupil Uniformity: Focused-Spot Incidence

So far, we discussed the uniformity of diffraction or-

ders. In this section, we discuss the impact of the

spot-size to EPE-cell-size ratio s d ratio, 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 d,

whereas the angular width of the beamlets, 2 s, is

determined by the focusing geometry of the scanner.

Fig. 6. Theoretical and measurement results for I0 and measure-

The focusing optics typically has a small s and pro-

ment results for one of the off-axis order intensities plotted as

duces a diffraction-limited spot at the EPE plane.

functions of the beam s incidence angle at the EPE. All values are

The spot size s is de ned as the FWHM intensity and

relative to the average diffraction-order intensity of 1.0. Assump-

tions for Fig. 6 are the same as those for Fig. 5. is inversely proportional to s

s K T f #, (15)

where KT is a function of the Gaussian beam-

depth error 0, I0 starts to increase rapidly with truncation ratio T and f# z D is the f-number of the

the incidence angle. If the beamlet-uniformity re- focusing geometry. For an incident Gaussian beam

quirement is 30% the incidence angle at the EPE the beam-truncation ratio T is de ned as the ratio of

becomes limited to less than 10 even for a zero etch- the Gaussian beam width at 1 e2 intensity 2wm to

depth error 0 . Figure 6 reveals that slightly the system s aperture diameter D. As T increases,

overetching the pattern by approximately 1% 2% the spot pro le changes from a Gaussian T 0.4 to

can help to reduce the effects of the incidence angle an Airy disc as T 3 . For small T i.e., negligible

across the EPE and allow for incidence angles clipping at the aperture, a Gaussian spot an analyt-

larger than 15 . Experimental results for I0 Iavg ical expression can be found for KT. For large T

and Ioff-axis for one of the off-axis orders are also plot- empirical formulas for K can be obtained by the nu-

ted in the gure. The measurements were obtained merical solution of the Fresnel diffraction integral16:

by the rotation of the EPE and the monitoring of the

beamlet intensity. As predicted, Ioff-axis changes by T 2wm D, (16)

only a very small amount for inc up to 20 . The

measured value of I0 increases rapidly for inc 15, 0 .156 T 2

1 .036 0 .058 T if T 0 .4

KT .

and, based on the shape of the measured I0 curve, the 0.75 T if T 0 .4

etch-depth error for the EPE used in the experiments

is predicted to be approximately 1%. For larger (17)

values of inc the measured value of I0 does not in-

crease as fast as the theoretical predictions. The Figure 8 shows the exit-pupil patterns generated

difference can be attributed to the decrease in diffrac- by means of changing the incident-beam focusing NA

tion ef ciency, the increase in Fresnel losses at the and thereby changing the diffraction-limited spot

tilted surfaces, and the increase in the backre ected- size8 s. Note that the EPE used in the experiments

order intensities with increasing inc. 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 e2-intensity beam radius; d, the grating period.

10 November 2001 Vol. 40, No. 32 APPLIED OPTICS 5845

Fig. 8. Experimental results for the exit-pupil pattern as a function of the spot-size to cell-size ratio s d. Changing the NA of the

focusing geometry varies the s d ratio.

in the experiment was to illustrate how the exit-pupil focused on the grating does not illuminate a suf -

pattern and the exit-pupil uniformity changed by the ciently large portion of the grating, and thus distinct

reduction of the 2 s d ratio or the increase of the s d diffraction orders do not form properly. Further-

ratio . The system aperture was illuminated with a more, the exit-pupil pattern changes dramatically by

converging uniform beam, and the 2 s d ratio was one s moving the spot across the grating, resulting in

adjusted by the reduction of the aperture stop size D. poor exit-pupil uniformity. When 2 s ds d,

When 2 s ds d, the top row in Fig. 8, the spot the middle row in Fig. 8 overlapping beamlets start

Fig. 9. Exit-pupil patterns produced by two different beam pro les at the system aperture the aperture size is adjusted to yield the same

FWHM spot size in each case : a small truncation Gaussian beam and Gaussian spot and b large truncation uniform beam and Airy

disc spot .

5846 APPLIED OPTICS Vol. 40, No. 32 10 November 2001

Fig. 11. Exit-pupil uniformity as perceived by the viewer for a a

2-mm and b a 5-mm eye-pupil size as obtained by use of the

Fig. 10. a Randomly generated beamlet intensities for a unifor-

computed exit-pupil pattern of Fig. 10 b .

mity of 30% U 0.3 and a zeroth-order diffraction ratio of I0

1. b Computed exit-pupil pattern obtained by the convolution of

the beam pro le at the aperture by the beamlet intensities shown

in a .

lustrate the case of s ds d, where the beam-

lets are getting smaller and start to reduce the exit-

to form; however, the random phase variations of the pupil uniformity by causing gaps between the

beamlets cause interference at the overlap areas and beamlets.

reduce the exit-pupil uniformity. Moving the spot Another important parameter that affects the exit-

across the grating causes variations on the diffracted- pupil uniformity is the beam pro le across the limit-

order intensities and reduces the exit-pupil unifor- ing system aperture. One can obtain the same size

mity further. Small spot sizes also cause some spot at the EPE plane by use of different focusing

fringelike intensity modulation across the displayed geometry. Figure 9 shows two exit-pupil patterns

image, reducing the image quality. The left-most obtained by use of the same size spot but with the

picture in the bottom row in Fig. 8 shows the case of rst one s having a small truncation ratio and the

2s ds d, as discussed below . The beamlets second one, a large truncation ratio. Beamlets in-

ll the space without causing any interference, the terfere heavily for the case of the small truncation

diffraction-order intensities do not change signi - ratio and cause variation of the interference patterns

cantly by the movement of the spot across the grat- as the spot moves across the EPE. The large trun-

ing, and the intensity modulation disappears. This cation ratio provides better uniformity across the

situation represents the optimal choice of spot size exit-pupil pattern.

relative to the EPE cell size for the best exit-pupil The exit-pupil plane for a scanning display system

uniformity. The other images in the bottom row il- is located at a plane conjugate with the scanner.

10 November 2001 Vol. 40, No. 32 APPLIED OPTICS 5847

assuming the focusing optics and the ocular are dif-

fraction limited and suf ciently large compared with

the beam size . Because the function of the diffrac-

tive EPE is to create replicas of the exit-pupil pattern,

one can easily compute the system exit-pupil unifor-

mity for different values of eye-pupil size after T, the

diffracted-order intensities, and the s d ratio are

known.

Figure 10 a shows the relative diffracted-order in-

tensities for an exemplary 13 13 circular array of

diffracted orders that form a 15 mm 15 mm exit

pupil. Figure 10 b shows the simulated exit-pupil

pattern, assuming that the grating design has a uni-

formity of U 30% and I0 1.

Figures 11 a and 11 b show the resultant display-

system exit-pupil uniformity after the convolution of

Fig. 10 b with a 2-mm and a 5-mm circular eye pupil,

respectively. The 5-mm eye pupil is fairly large, and

Fig. 12. System exit-pupil uniformity plotted as a function of the

the exit-pupil uniformity is good, independently of

s d ratio and T. It is assumed that Pep 15 mm, there is a 13

the choice of system parameters. For the 2-mm pu-

13 array of beamlets, U 0.3, I0 1, and the eye-pupil size is 2 mm.

pil case, however, the choice of system parameters

The crosses show to which contour line the contour labels belong.

can make a large difference in the resultant system

exit-pupil uniformity.

Figure 12 shows how the choice of system param-

Thus for a system without an EPE the exit-pupil

pro le is the same as the beam pro le at the scanner eters T and the s d ratio affect the system exit-pupil

Fig. 13. PSF measurements obtained by use of the EPE of Fig. 2 b 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 APPLIED OPTICS Vol. 40, No. 32 10 November 2001

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 shows an undesired area, where beamlets interfere.

30% and I0 1.0, assuming a 2-mm eye pupil. The Depending on the s d ratio and T, the exit-pupil

contour curves show the percent exit-pupil unifor- uniformity changes from approximately 30% to

mity. The lower left-hand region of the gure 50%.

Fig. 15. Horizontal component of the MTF for the PSFs shown in a Fig. 13 2-mm eye-pupil case and b Fig. 14 5-mm eye-pupil case .

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. a Eight and b 16

different spot positions were used to compute the MTFs.

10 November 2001 Vol. 40, No. 32 APPLIED OPTICS 5849

quency, the 5-mm pupil causes smaller MTF varia-

tions 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 30 cycles mm obtained by

the movement of a focused spot across the EPE in

both the horizontal and the vertical directions. Re-

peating the experiment for pupil sizes from 1 to 10

mm 2 to 8 mm is a more realistic range of human

eye-pupil diameter shows that the MTF values uc-

tuate 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.

Fig. 16. Extremum values of the horizontal Hor. and the verti-

cal Ver. 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.

7. Conclusions

Binary diffraction gratings can be used as EPEs in

scanning display systems. The uniformity of

6. Modulation Transfer Function of

diffraction-order intensities is sensitive to the fabri-

Numerical-Aperture Expanders

cation tolerances and the grating phase. The grat-

In the previous sections, we discussed the far- eld

ing phase is a function of pattern depth, beam

effects of diffractive EPEs. In this section, we focus

incidence angle, and wavelength. Although off-axis

on the near- eld effects and discuss the PSF and the

diffraction-order intensities relative to each other re-

MTF implications of the EPE. The beamlets that

main the same, the zeroth-order intensity changes

pass through the eye pupil are focused by the eye lens

rapidly with the phase angle. Good exit-pupil uni-

onto the retina and form overlapping images on the

formity can be obtained by one s controlling the uni-

retina. The random phase of the beamlets results in

formity of the diffraction-order intensities and also by

dynamic interference patterns on the retina. The

the adjustment of the system parameters, such as the

phase of the beamlets and the interference pattern

beam-truncation ratio and the spot-size to EPE-cell-

change as the scanned light beam moves across the

size ratio. The diffractive EPE causes PSF and MTF

retina.

variations as a result of the interference of the beam-

Figures 13 and 14 show CCD pictures of the PSF as

lets at the viewer s retina. The spot-position-

a function of the spot position across the cell for the

dependent MTF variation is more pronounced for

2-mm and the 5-mm eye-pupil cases, respectively.

small eye-pupil sizes, which occur in high-luminance

The focused spot at the EPE is imaged onto a CCD

systems.

camera by use of a high-NA microscope objective that

is followed by a circular aperture that mimics the eye

I am thankful to Peggy Lopez for her help with the

pupil. The top left-most image of Fig. 13 shows the

experiments. Financial support from the U.S. Army

PSF when the EPE is removed from the optical train.

for parts of this research is gratefully acknowledged.

The EPE grating period is 16 m, and it is moved in

increments of 2 m horizontally between the differ-

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