holographic polar formatting and realtime optical processing of synthetic aperture radar data

Holographic polar formatting and real-time optical

processing of synthetic aperture radar data

Jack N. Cederquist, Michael T. Eismann, and Anthony M. Tal

A two holographic optical element (HOE) system for polar formatting of spotlight mode synthetic aperture

radar (SAR) data was designed, fabricated, and successfully tested. With the addition of a spatial light

modulator, a third phase-compensating HOE, and a Fourier transform lens, the real-time polar formatting of

SAR data and SAR image formation was experimentally demonstrated.

1. Introduction

Synthetic aperture radar (SAR) image formation isan excellent application for coherent, analog opticalprocessing. However, for spotlight mode SAR, polarformatting of the SAR data is required before theimage can be formed via 2-D Fourier transformation.Previous work has investigated the use of mechanicalrotation and, in one case, spatial light modulators(SLMs) with electron beam writing to perform polarformatting. 1-3 We view mechanical motion as unde-sirable and wish to take advantage of the current inten-sive development of SLMs with rectilinearly placedpixels and electronic addressing. We therefore seekoptical methods for polar formatting. Single holo-graphic optical elements (HOEs) to perform coordi-nate transformations were first designed using the sta-tionary phase approximation by Bryngdahl4 5 in 1974.Unfortunately, a single HOE cannot perform the polarcoordinate transformation for the SAR data process-ing use. However, the use of two coordinate transformHOEs in series has been investigated 6 7 to overcomethe limitations of single HOE systems.

In this paper, the design, fabrication, and testing of athree HOE system for the real-time polar formattingof, and image formation from, spotlight SAR data isdescribed. In Sec. II, background information on SARdata processing is given. The design of the requiredHOEs for polar coordinate transformation is described

The authors are with Environmental Research Institute of Michi-gan, Optical Science Laboratory, Advanced Concepts Division, P.O.

Box 8618, Ann Arbor, Michigan 48107.Received 18 January 1989.0003-6935/89/194182-08$02.00/0.© 1989 Optical Society of America.

in Sec. III. In Sec. IV, the fabrication of the HOEs andthe construction of the optical system is discussed.Experimental results for both polar formatting andreal-time SAR image formation are given in Sec. V.Conclusions and suggestions for further research arepresented in Sec. VI.

II. Synthetic Aperture Radar Data Processing

In synthetic aperture radar an appropriate wave-form is transmitted, and the return signal scatteredfrom a scene is processed to obtain range and Dopplerinformation about the scene.8 Range resolution isachieved in inverse proportion to the frequency band-width of the transmitted signal. The Doppler infor-mation corresponds to azimuthal, or cross-range, spa-tial information. Azimuthal resolution is achieved ininverse proportion to the time interval over whichreturn signals are collected as the SAR flies by thescene (i.e., in inverse proportion to the extent of thesynthetic aperture). Together the range and Dopplerinformation give a 2-D image of the scene. For a SARwhich transmits a sequence of linear FM (chirp)pulses, the required data processing consists of foursteps. The first step is to mix the return signals with areference to demodulate them to an intermediate fre-quency. The return signal from each pulse thus givesa 1-D data set. The second step is to form a 2-D dataset by placing the 1-D data from each of many pulsesside by side. In the important cases where the sceneconsists of a rotating object or the SAR is used in thespotlight mode, Fig. 1(a) shows the resulting data for asingle point scatterer. In these cases, the third stepmakes a polar format of the data as shown in Fig. 1(b)to compensate for the continuously changing viewingangle during data collection. The fourth step is toperform a 2-D Fourier transformation of the data toproduce an image of the scene.

4182 APPLIED OPTICS / Vol. 28, No. 19 / 1 October 1989

Data fromsuccessivepulses (0)

1yCA

Data from each pulse (r)

a) rectilinear format

a1 (x,y) e OXY)

P1

b) polar format

Fig. 1. SAR data for a point scatterer.

I-

L

a2 (U,v)

P2

-b.1f f

Fig. 2. Optical system for coordinate transformation. The lens Lhas focal length f and Fourier transforms plane PI to plane P2 .

The necessary polar formatting operation is a diffi-cult data processing step. In any sampled data proces-sor (such as an electronic digital computer), polar for-matting requires interpolation which often introduceserror and is computationally intensive-about twicethat of the FFT operation. Since a spatial light modu-lator (SLM) must be used to input the data to anoptical processor, one possibility that has been investi-gated is to polar format the data by writing it onto arotating SLM with an electron beam.' An opticallyaddressed SLM could also be used, but it must have astorage time as long as the time taken to write all the 2-D data coupled with a short erasure time to allow real-time input of the next 2-D data set. Other possibilitiessuch as rotating the output detector of the data proces-sor and processing each 1-D data set separately havealso been investigated.2 3

The approach investigated and demonstrated in thispaper uses a rectilinearly addressed 2-D SLM to inputthe data in the form shown in Fig. 1(a) and a two HOEsystem to optically polar format the data as shown inFig. 1(b). It does not suffer from the frame rate limita-tions of optically or electron beam addressed SLMsand does not involve mechanical motion. By using arectilinearly addressed SLM, this approach will bene-fit from the current intensive development of electron-ically addressed SLMs, such as pixel addressed liquidcrystal light valves9 and the Si-PLZT device.10 Thesedevices are compact, have controllable storage anderasure rates, and are relatively fast. Together withoptical polar formatting and optical processing, theyhave great potential for application to real-time pro-cessing of SAR data.

Ill. Design of Polar Coordinate Transformation HOEs

As an introduction to the design method, considerthe optical system shown in Fig. 2 where the input fieldat Pi in the front focal plane of a lens L is uniphase andhas an amplitude distribution a(xy). A HOE withphase (x,y) is placed in plane P1. The complex am-plitude distribution a2(uv) found in the back focalplane P2 of the lens is given approximately by theFourier transform:

a2(u,v) -- J J a1 (x,y)2 f _E, _

X exp[io(x~y) - ir (xu + YV)]dxdy, (1)

where X is the wavelength and f is the lens focal length.

Following Bryngdahl,4 5 this integral can be approxi-mated by the method of stationary phase 1 to showthat a point (x,y) in the input is mapped to a point (u,v)in the output where

F Xf ao(xy)[xl [u] = 2r oxLYJ Lv If a(xy)

L27r ?y ]If a desired coordinate transformation is

[Y] l- [v h(X"y)J]

(2)

(3)

then the required HOE phase k(x,y) is found by solv-ing the set of partial differential equations

Xf ao = g(xXY)

(4)

2 y h(xy).

Equation (4) has a solution if and only if12

(5)(xy)= ahxay ax

Unfortunately for many coordinate transformationsof interest, Eq. (5) is not satisfied. Specifically forpolar formatting, the desired coordinate transforma-tion is

(6)[X] -U [X xCosy]

and the transformation functions are

g(x,y) = x cosy

h(x,y) = x siny.

The partial derivatives

ag(xy) = -x smy,

(7)

(8)bh(x,y) =

a sy,do not satisfy Eq. (5) and, hence, the transformation isnot realizable with a single coordinate transform HOE.

However the polar formatting transformation can beseparated into two cascaded transformations, each ofwhich does satisfy Eq. (5).13,14 The transformationsare

1 October 1989 / Vol. 28, No. 19 / APPLIED OPTICS 4183

I _ * I 4

I

r 4 1

/0

I

y=Oj , x=r

Input ' f

v=0i~~~~~1 Xu

I _,- I

=ln r

0

H1 lens H1'and H2 lens H2'

Fig. 3. Optical system for polar formatting.

E x] [u] = [lnx],

t]l = [ expu cosVIv i _ Lexpu sinv J

(9) 0

(10)

where s and t are Cartesian coordinates. Equation (5)is satisfied for both transformations since

b(lnx) -

by (11)

by 0,ax

o(expu cos) = -expu sinv,

bV (12)

6(-expu sinv) = _expu sinv.au

Therefore, both transformations are realizable optical-ly. Note that the minus sign in Eq. (10) is necessary tomake the second transformation realizable. It corre-sponds to an inversion in one of the output coordi-nates. Similar use of a minus sign in Eq. (6) does notmake that transformation realizable, however.

Polar formatting can therefore be performed opti-cally by the successive use of two holographic coordi-nate transformations as shown in Fig. 3. HOEs HIand H2 perform coordinate transformations which,when cascaded, produce the polar coordinate transfor-mation. HOEs Hi' and H2' correct the phase at theoutput of each transformation so that the output phaseis uniform for a uniphase input. It should be notedthat the phase correction for the output of the firstcoordinate transformation and the phase for the sec-ond transformation can be combined and implement-ed as a single HOE, so the complete system, includingthe final phase correction, requires only three HOEs.

The next step is to solve for the phase functions ofthe two coordinate transformation HOEs. To do this,we need to define coordinate systems in each of thethree optical planes in Fig. 3 and to establish the rela-tionships of these systems to coordinate systems forthe SAR data. As explained in Sec. II, after the secondstep in the data processing, the input SAR data lie in arectangular region from r, to r2 and -01 to 01 in acoordinate system in which r and 0 are orthogonal [seeFig. 4(a)]. This SAR data will be loaded into an elec-tronically addressed SLM with rectilinear pixel place-ment such that the return signal data from each pulsewill reside in one row of SLM pixels. In the opticalplane located at the HOE HI, this data can be placed in

V

V1

In r

In r1 In r2

r

001

-01

r2

(b)

717

t

ti

-1

U

S

(C)

Fig. 4. SAR data and optical coordinate systems. (a) Input, (b)intermediate, and (c) output planes of polar formatting system.

a region defined to be from -xl to xi and -yj to Yi[again, see Fig. 4(a)]. The relationship between thesetwo coordinate systems (one optical and one for theinput SAR data) is

2x, / r2+r,x= r - ,2 (13)

Y10

x /r2 -rl\ r2+rl1=-V 2y / 2 (14)

0= 01Y.Y1

The first HOE Hi will transform [see Eq. (9)] theSAR data into a region from lnrl to lnr2 and -01 to 01 ina coordinate system in which lnr and 0 are orthogonal.In the optical system, this data can be placed in aregion from-ul to ul and-vi to vl [see Fig. 4(b)]. Therelationship between these data and optical coordinatesystems at the output plane of the first coordinatetransformation (at the plane of the HOE incorporatingHi' and H2) is


t ris DATA

0

Yi

-Yi

OPTICAL

X

-x1 I Xir1 r2

(a)

* l Z b l - b l - -

f f f

Si

2u ln ln(r2 r,)U ln(r 2 /r1) - 2

v1

(u ln(r2/rl)

2u, )I

01V0=-.V1

Combining Eqs. (14) and (15) gives the requiredform of the first optical coordinate transformation

[y a (17a)

where

n(r2/r) {n[x ( 2 - r) + 2 r- ln(r2r)} bVly

=Yb

Note that Eq. (17) is simply a more general form of Eq.(9) which still satisfies Eq. (5). Solving the partialdifferential Eq. (4) by using the expressions in Eq. (17)and integrating gives the required holographic phasefunction for the HOE H1

(15)

2s, (u ln(r2/rl) /0 1V\ r2 + r11I = r2r1 exp ICos- -

r 2 -r 1 L k 2u, / \ V ) 2 1(21b)

-2s, pu ln(r2/r,) .in(lV\r2-r exp 2u sin 1 /

(16) Note that Eq. (21) is simply a more general form of Eq.(10) which still satisfies Eq. (5) (when written in termsof u-v coordinates) provided that

u1 ln(r2/rl)

V1 20,(22)

Equation (22) expresses the requirement that the SARdata be scaled identically in both coordinates in the u-vplane. In these experiments, ul was chosen equal to vland r2 /rl = 5/3, so 01 = 14.60. Solving the partialdifferential Eq. (4) (in u-v coordinates) by using theexpressions given in Eqs. (21) and (22) and integratinggives the required holographic phase function for theHOE H2

-2- / 2s1 \ 2u /1 u ln(r2/r)\02(U'V) =r 2-reL lnpr2/ p 1 /Xf 2 - r ln(r2/rl) 2u,

x cos(v ln(r2/r)) (r2 )+ r)] (23)

27r 2u1 /r2 - r 1 ' F- x r 2 - r r2 + r1k1 (xy) = - -I 1Xf \ln(r2/rj) \ 2x1 / [x \ 2 / 2 J

X I (r2- r)+ r2+r, ln(r2r,) + vjy2 )xi 2 / 2 2 2y1)

(18)

The second transformation [see Eq. (10)] will putthe SAR data into a region from r to r2 and -01 to 01 ina coordinate system in which r and 0 are polar coordi-nates. In the optical system, the data will be placed ina region from -sl to s on the s-axis and -tj to t on thearc corresponding to r = 1/2(r1 to r2) [see Fig. 4(c)]. Therelationship between these data and optical coordinatesystems is:

2sl / r+ls = s, ir cos0- rir2 - r 2 (19)

-2t(r sin0t=(r2 + rl)sin0l

It is convenient to require that

t - (r2 + r)sin0(s1 r 2 - r(

so that SAR image produced is scaled identically inboth coordinates.

Combining Eqs. (16) and (19) gives the requiredsecond optical coordinate transformation:

L i -] L I' (21a)where

IV. HOE Fabrication and Optical Setup

The phase functions of Eqs. (18) and (23) were im-plemented as computer generated holograms (CGHs)by encoding them as real, nonnegative amplitudetransmittance functions t(x,y) of the form:

t(x,y) = b + a cos[o(xy) + ax + fly], (24)

where a and b are the amplitude and bias of the cosinu-soidal fringe pattern. For these experiments, a = 0.4and b = 0.5. The term of the form ax + y was addedto each phase to give a spatial carrier frequency. Theconstants a and fi were chosen so that, in a Fourierplane with respect to the CGH, the desired first dif-fraction order (the coordinate transform region) didnot overlap with the zero order. The amplitude trans-mittance functions were recorded on high resolutionfilm using a rotating drum laser beam recorder. ACGH size of (1 cm)2 was used with a 5-Am pixel size toproduce a CGH space-bandwidth product of 2048 X2048. The film was mounted between /4-in. microflatplates using a UV-curing optical cement to eliminateany phase errors due to thickness variations in the film.

These CGHs are thin absorption holograms and aretherefore too low in diffraction efficiency (1%) to beused in a multiple HOE system. To improve theirdiffraction efficiency, the CGHs were copied onto avolume phase material and placed on a high carrierfrequency. The result is a computer originated holo-graphic optical element (COHOE).15 ,16 The opticalsetup for producing a COHOE is shown in Fig. 5. Alaser beam is split into two parts. One part is colli-mated and used to read out the CGH. The desireddiffracted order of the CGH is passed by a Fourierplane spatial filter and imaged to the COHOE record-


BS M Collimator

\ "I CGH

M _

Collimator SpatialFilter

Fig. 5. COHOE recording method.

ing plane. There it interferes with a reference planewave at an offset angle suitable for forming a volumehologram. When illuminated by the reference planewave, the phase distribution encoded in the CGH isreconstructed with high efficiency by the COHOE.

As shown in Fig. 3, two coordinate transform HOEsand a third phase-compensating HOE are used in thepolar formatting system. The first HOE is a COHOEof the phase function for the first coordinate transfor-mation [Eq. (18)]. The second HOE corrects the out-put phase of the first coordinate transformation andproduces the phase for the second transformation [Eq.(23)]. This second HOE was fabricated by interferingthe output wavefront of the first HOE (for a planewave input) with the desired phase produced by thesecond CGH. It is therefore a COHOE made with anonplanar reference wavefront. The third HOE cor-rects the output phase of the second coordinate trans-formation. It was made by interfering the outputwavefront from the second HOE with a uniform refer-ence beam. All three HOEs were made on spectro-scopic plates (Kodak 649F) using a HeNe laser (X =632.8 nm). The HOEs were processed by the silverhalide (sensitized) gelatin process'7 and then coveredwith microflats to avoid degradation of diffraction effi-ciency when exposed to humidity. The resulting dif-fraction efficiencies for the three HOEs were 37%,14%,and 36%, respectively. Because the second and thirdHOEs are made by interfering beams of nonuniformintensity, the recording must be confined to the linearexposure region to avoid intermodulation noise. This,of course, decreases the maximum diffraction efficien-cy that can be achieved.

The experimental multiple HOE system for polarformatting and SAR image formation is shown in Fig.6. The SAR data are collected either by recording iton film and placing it in a liquid gate or by an electroni-cally addressed, liquid crystal television spatial lightmodulator. In either case, the data are coherentlyilluminated and imaged onto the first HOE (Hi).Fourier transformation with a lens then produces the

log transformed data in the first diffracted order whereit illuminates the second HOE (Hi' + H2). The sec-ond HOE and a second lens then perform the remain-der of the polar formatting operation. The third HOE(H2') corrects the output phase, and a final lens is usedto perform a Fourier transformation and produce thedesired SAR image.

V. Experimental Results

The multiple HOE system was used experimentallyto polar format and form images from SAR data ofmultiple point scatterers. Both film and a liquid crys-tal television (LCTV) are used as input devices. SARdata are written on the input devices by computergeneration techniques and the output image is magni-fied and detected by a COHU CCD camera. Theoverall efficiency of the system is -1.6%; however, theoutput intensity using a 50-mW laser source is morethan sufficient for the CCD detector.

Figure 7 shows the operation of the system at eachstage of the formatting and processing of SAR data fora single point scatterer. For this experiment, in whichthe input is recorded on film and placed in a liquidgate, the ability of the multiple HOE system to proper-ly format the data and provide both azimuth and rangecompression can clearly be seen.

To measure system performance quantitatively,SAR data representing eight point scatterers in a sin-gle line was generated and recorded on film. Becausethe data were recorded on a bias [as in Eq. (24)], anintense DC reference point and high sidelobes alongthe x' and y' axes are present in the output imageplane. The eight point test image was therefore inten-tionally placed along a line at 450 to these axes toreduce the effect of these sidelobes. In Fig. 8(a), theimage is shown for an experiment in which the data areprocessed without polar formatting (i.e., optical Fouri-er transformation alone). The output of the multipleHOE system is shown in Fig. 8(b) with a line scanthrough the points shown in Fig. 8(c). Detector noiseis reduced by temporal averaging. As expected, theimage obtained with polar formatting shows far supe-rior azimuth compression.

Several performance characteristics of the multipleHOE system can be determined from the line scan inFig. 8(c). First, the space-bandwidth product(SBWP) can be measured. Theoretically, the SBWPis given approximately by the square root of the holo-gram SBWP.6 For the (1 cm)2 hologram used with2048 X 2048 pixels, a SBWP of approximately 45 X 45

Hl"+ H2 H-12

Output

Fig. 6. Experimental multiple HOE system for polar formatting and SAR image formation.


Input H1

4Y

InputData

Data After FirstTransformation

PolarFormatted

Data

SARImage

Fig. 7. Experimental results showing polar formatting and SARimage formation.

would be predicted. The SAR test data are computedso that the image points are spaced by 10 diffraction-limited impulse response widths. The system modu-lation transfer function (MTF), specified by the enve-lope of the line scan, falls to -3 dB of the peak value at62 diffraction-limited impulse response widths alongthe diagonal from the DC reference point, giving anexperimental SBWP of 44 X 44.

Second, to evaluate the system phase errors, theimpulse response widths of the images of the eightpoints are measured and found to be -1.25 X thediffraction-limited width predicted for an ideal sys-tem. Not all of this increase in width is due to phase

errors, however. Aperture weighting due to the MTFof the CGH recorder is partially responsible for thisincrease. Another part of the increase is due to thecoordinate transformation itself. This apertureweighting is the result of the change in the differentialelement of area from drd6 (rectilinear r-O coordinates)to rdrdO (polar coordinates)' 8 and can also be predict-ed by evaluating Eq. (1) using the method of stationaryphase." Both of these effects can be corrected byappropriate inverse weighting in the input plane of thepolar formatting system, but this correction was notimplemented in these experiments.

The final system performance characteristic whichcan be computed is the signal-to-noise ratio (SNR).The system SNR is signal dependent for two reasons.First, as the number of points in the image increases,the available system dynamic range is shared by morepoints and the SNR decreases. Second, SNR varia-tions over the image roughly follow the system MTFand therefore the SNR decreases as the location of thepoint moves farther from the DC reference point (i.e.,SNR decreases as carrier frequency increases). Forthe point of the eight point image which was nearest tothe DC point, a SNR of 34 dB was measured.

Real-time operation of the SAR processor was dem-onstrated by using a low cost input spatial light modu-lator (Radio Shack Model 16-151 LCD PocketvisionTV). The LCTV was modified at ERIM for this appli-cation by removing the polarizers and cementing thedevice between optical flats to obtain higher opticalquality.'9 In order to fully utilize the SBWP of theLCTV, the input imaging optics were altered to pro-vide 3X demagnification of the LCTV onto the firstHOE H1. The LCTV was driven by the video outputof a frame grabber (Data Translation DT2861) in acomputer (IBM PC-AT). This allowed SAR test datato be placed on a standard video signal and sent to theLCTV. A polarizer oriented at 90° to the input laserpolarization was placed just before the detector.

The output of this real-time, multiple HOE, SARpolar formatter and processor for an input represent-ing SAR data from three point scatterers is shown inFig. 9. The input data are shown in Fig. 9(a) and theoutput image in Fig. 9(b). Note that Fig. 9(b) is mag-nified with respect to Fig. 8(b). Because the SBWPand dynamic range of the LCTV are less than that ofthe film input, the processor performance is reducedsomewhat. The smaller SBWP of the LCTV can beseen by comparing the MTF of the real-time processorto that of the film input processor (see Fig. 10). Thereal-time processor MTF falls off much more rapidly,limiting the processor SBWP to approximately 20 X20. The limited dynamic range reduces the signal-to-bias ratio and lowers the SNR. For the image of thetest point nearest the DC reference point, a SNR of-18 dB was measured.

VI. Conclusion

A multiple HOE system was designed, constructed,and shown experimentally to perform polar formatting


t

Y I

y

I X

DCreference

point

4

(a)Y'

DCreference

point

-- X'

scan direction

I--- 0- X'

(b)

NormalizedIntensity

Position

(C)

Fig. 8. Experimental results for SAR data for eight point scat-terers. 2-D image produced (a) without polar formatting and (b)with polar formatting. (c) Line scan through point images in (b).

for the real-time optical processing of SAR data. Anelectronically addressed SLM was used for data input.The experimental system demonstrated high SNR,low aberrations, acceptable efficiency, and a SBWP atthe output image of 44 X 44. In order to increase theprocessor SBWP, the SBWP of the CGHs used mustbe increased by reducing the pixel size and increasingthe CGH size. Using a (6.25-cm)2 CGH with a 1-,qmspot size, a 250 X 250 coordinate transform SBWPcould be achieved which would also match the perfor-mance of current real-time input and output devices.

The promising results obtained in this research pro-vide a useful foundation for future development ofcompact, real-time spotlight SAR processors. Fur-ther work in this area should first be aimed at improv-

(a)YI

4'

X'

(b)

Fig. 9. Experimental results with LCTV input. (a) Data for threepoint scatterers which was input to LCTV. (b) Output SAR image.

1.0

.F_

C:a)

I

0.8

0.6

0.4

0.2

0.00.0 2.0 4.0 6.0

Spatial frequency (p/mm)

Fig. 10. Relative output image intensity versus spatial frequency ofinput data for film and LCTV input devices.

ing the system performance by increasing the SBWPand reducing the scattering and intermodulation noisein the COHOEs. The next step would be the design ofa more compact system with the Fourier transformlenses incorporated into the coordinate transformHOEs.

This research was supported by the U.S. Army Re-search Office. Portions of this research were present-ed at the Optical Society of America Annual Meetingin Seattle, Washington, 20-24 Oct. 198613 and at the


tI

Society of Photo-Optical Instrumentation EngineersMeeting on Holographic Optics: Design and Applica-tions in Los Angeles, California, 13-14 Jan. 1988.14

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7. W. J. Hossack, A. M. Darling, and A. Dahdouh, "CoordinateTransformations with Multiple Computer-Generated OpticalElements," J. Modern Optics 34, 1235-1250 (1987).

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10. S. H. Lee, S. C. Esener, M. A. Title, and T. J. Drabik, "Two-Dimensional Silicon/PLZT Spatial Light Modulators: DesignConsiderations and Technology," Opt. Eng. 25, 250-260 (1986).

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