Computer Generated Hologram used as an aberration corrector

Transcription

1 Computer Generated Hologram used as an aberration corrector Computer Generated Hologram used as an aberration corrector Oded Arnon, Steve Bloomberg, Dan Ophir Oded Arnon, Steve Bloomberg, Dan Ophir El -Op, Electro- Optics Industries Ltd, P.O.Box 115, Rehovot, 711, Israel El-Op, Electro-Optics Industries Ltd, P.O.Box 115, Rehovot, 711, Israel Abstract Abstract A Computer Generated Hologram (CGH) is used as an aberration corrector in construction optics to record Holographic Optics Elements (HOE's) with preaberrated wave - fronts. In the system described, the CGH provides enough asphericity to record the required HOE, so that other lenses in the construction optics can be selected by paraxial considerations alone. The design process of the recording system is described and a comparison is made between the measured and expected performance of the final lens. A Computer Generated Hologram (CGH) is used as an aberration corrector in construction optics to record Holographic Optics Elements (HOE's) with preaberrated wavefronts. In the system described, the CGH provides enough asphericity to record the required HOE, so that other lenses in the construction optics can be selected by paraxial considerations alone. The design process of the recording system is described and a comparison is made between the measured and expected performance of the final lens. Introduction Introduction Holographic Optical Elements can act as aspheric elements in various optical systems. Such elements we have used in a Helmet Mounted Display System'. The difficulties involved in creating the asphericity in such elements are encountered in the construction optics of the holograms. Conventional optical elements alone are not always able to create the desired aspheric wavefront, even when they are tilted and decentered. A CGH in such a system can help create the aspheric wavefront required to construct the HOE,3. Holographic Optical Elements can act as aspheric elements in various optical systems. Such elements we have used in a Helmet Mounted Display System 1. The difficulties involved in creating the asphericity in such elements are encountered in the construction optics of the holograms. Conventional optical elements alone are not always able to create the desired aspheric wavefront, even when they are tilted and decentered. A CGH in such a system can help create the aspheric wavefront required to construct the HOE ' 3. While examining the suitability of such a solution, the limitations imposed by the CGH are reviewed. Within these limitations, an effort is made to simplify the construction optics for the HOE test case which is presented below. While examining the suitability of such a solution, the limitations imposed by the CGH are reviewed. Within these limitations, an effort is made to simplify the construction optics for the HOE test case which is presented below. Description of desired HOE Description of desired HOE The desired HOE for this test case is a simple f/.5 transmission hologram, operating with unit magnification (m = -1), at 3 off axis angles as shown in 1. The clear aperture is mm and the resonant wavelength nm, i.e. the green He -Ne laser line, to ease laboratory measurements of the optical performance. The lens is characterized basically by two spherical wavefronts, although the hologram itself is sealed between two relatively thick glass plates. The desired HOE for this test case is a simple f/.5 transmission hologram, operating with unit magnification (m=-l), at 3 off axis angles as shown in 1. The clear aperture is mm and the resonant wavelength nm, i.e. the green He-Ne laser line, to ease laboratory measurements of the optical performance. The lens is characterized basically by two spherical wavefronts, although the hologram itself is sealed between two relatively thick glass plates. The recording system The recording system The hologram is recorded with an argon laser operating at nm, a wavelength shorter than the hologram resonance. As a result of this wavelength difference, several geometric modifications are imposed on the recording system. The of axis angles are reduced from 3 to 8.3 and the length of the recording arms are extended from 1 mm to 113 mm. These modifications introduce aberrations into the recording optical system. As shown in, the main components are astigmatism and coma.this aberration must be produced in the construction optics in order to assure good performance in the final holographic element. The hologram is recorded with an argon laser operating at nm, a wavelength shorter than the hologram resonance. As a result of this wavelength difference, several geometric modifications are imposed on the recording system. The off-axis angles are reduced from 3 to 8.3 and the length of the recording arms are extended from 1 mm to 113 mm. These modifications introduce aberrations into the recording optical system. As shown in, the main components are astigmatism and coma.this aberration must be produced in the construction optics in order to assure good performance in the final holographic element. 7 / SPIE Vol. 138 Sixth Meeting in Israel on Optical Engineering (1988) 1 / SPIE Vol. 138 Sixth Meeting in Israel on Optical Engineering (1988) Downloaded From: on /18/1 Terms of Use:

2 DESIRED HOE NM.8 MM DESIRED HOE NM SCrtLE 1. OH -JUN-88 SCALE 1. OR - JUN Layout of desired HOE. 1. Layout of desired HOE. Y-FAN Y-FRN RELflTIVE. RELATIVE FIELD HEIGHT 1. *1 i FIELD HEIGHT (. ) X-FAN X-FRN DESIRED RECORDING RBERRRTION DESIRED RECORDING RBERRRTION RflY RBERRRTIONS (MILLIMETERS) OR -JUN-88 RAY ABERRATIONS (MILLIMETERS) OR - JUN NM. Aberrations caused by wavelength shift.. Aberrations caused by wavelength shift. SPIE Vol. 138 Sixth Meeting in Israel on Optical Engineering (1988) / 13 SPIE Vol. 138 Sixth Meeting in Israel on Optical Engineering (1988) / 13 Downloaded From: on /18/1 Terms of Use:

3 The optical layout of the recording system is shown in 3. The beam splitter splits the laser beam into two. The light reflected to the right forms the first recording beam. This has a simple spherical wavefront diverging towards the photographic plate (PP) from point source Si. The second recording beam illuminates the photographic plate with an aspherical wavefront converging to a spot around point S. The aspherical wavefront is aberrated to suit the aberration shown in. This beam, after expansion and collimation, passes through the CGH and is focussed at the focal plane of the Fourier lens (FL). A mask mounted at this plane isolates the CGH first diffracted order from the other unwanted orders. After passing through an auxiliary lens (AL), the light converges through the photographic plate towards S. The optical layout of the recording system is shown in 3. The beam splitter splits the laser beam into two. The light reflected to the right forms the first recording beam. This has a simple spherical wavefront diverging towards the photographic plate (PP) from point source SI. The second recording beam illuminates the photographic plate with an aspherical wavefront converging to a spot around point S. The aspherical wavefront is aberrated to suit the aberration shown in. This beam, after expansion and collimation, passes through the CGH and is focussed at the focal plane of the Fourier lens (FL). A mask mounted at this plane isolates the CGH first diffracted order from the other unwanted orders. After passing through an auxiliary lens (AL), the light converges through the photographic plate towards S. ccl. ccp FL. e. W J HOE RECORDING SYSTEM 114. MM HOE RECORDING SYSTEM scnle. OR -JON-B8 3. Optical layout of recording system 3. Optical layout of recording system Y-FAN Y-FflN RELATIVE FIELD RETORT. RELATIVE FIELD HEIGHT t. ) 1. '1 X-FAN OVERALL ABERRATIONS OVERFILL flberrrtions RflY flberrfltions (MILLIMETERS 1 RAY ABERRATIONS (MILLIMETERS] $14.5 NM 4. Overall aberrations at Fourier plane to be generated by CGH. 4. Overall aberrations at Fourier plane to be generated by CGH. The Fourier lens and auxiliary lens are chosen to fulfill the power requirements of the system. They are also usually designed to generate aberrations required for the recording of the HOE, as done by Smith4 in interferometric testing of aspheres using a CGH. In this The Fourier lens and auxiliary lens are chosen to fulfill the power requirements of the system. They are also usually designed to generate aberrations required for the recording of the HOE, as done by Smith4 in interferometric testing of aspheres using a CGH. In this 14 / SPIE Vol 138 Sixth Meeting in Israel on Optical Engineering (1988) 14 / SPIE Vol. 138 Sixth Meeting in Israel on Optical Engineering (1988) Downloaded From: on /18/1 Terms of Use:

4 test case however, in order to simplify the system, both lenses were selected off-the-shelf on the basis of paraxial considerations alone. This caused an undesired spherical aberration to be introduced into the system. The CGH was therefore designed to correct this unwanted spherical aberration in addition to producing the aberration caused by the wavelength shift. 4 shows the overall aberrations to be generated by the CGH, as evaluated at the Fourier plane. CGH design considerations test case however, in order to simplify the system, both lenses were selected off-the-shelf on the basis of paraxial considerations alone. This caused an undesired spherical aberration to be introduced into the system. The CGH was therefore designed to correct this unwanted spherical aberration in addition to producing the aberration caused by the wavelength shift. 4 shows the overall aberrations to be generated by the CGH, as evaluated at the Fourier plane. CGH design considerations We have limited the CGH spatial frequency to 5 cycles /mm over a clear aperture of 34 mm. This limit was determined by the resolution of the Versatec plotter used to draw the CGH and its subsequent photo -reduction. The maximum size aberration to be generated can be estimated in the following way. Let ffourier be the effective focal length of the Fourier lens (FL) and a' be the angular aberration of the CGH. The aberration spot size diameter, at the Fourier plane is given by: We have limited the CGH spatial frequency to 5 cycles/mm over a clear aperture of 34 mm. This limit was determined by the resolution of the Versatec plotter used to draw the CGH and its subsequent photo-reduction. The maximum size aberration to be generated can be estimated in the following way. Let Fpourier ^e ^e effective focal length of the Fourier lens (FL) and a' be the angular aberration of the CGH. The aberration spot size diameter, at the Fourier plane is given by: S' = a' ' ffourier (1) a' can be described in terms of the optical path difference OPD across the CGH diameter, DCGH: OPD DCGH The OPD is related to the spatial frequency bandwidth and the recording wavelength by: Therefore S' can be described by: OPD = Ar BW DCGH S' - ffourier Ar BW Furthermore, the spatial frequency bandwidth is related to the maximum frequency vm.ax by: BW c-' v, +. where v, is the carrier spatial frequency of the hologram. $' = <*' /Fourier (!) a 1 can be described in terms of the optical path difference OPD across the CGH diameter, DCG/J- 4= -. () DCGH V ' The OPD is related to the spatial frequency bandwidth and the recording wavelength by: Therefore ' can be described by: OPD = \r -BW- DCGH. (3) «' = /Fourier ' *r ' BW. (4) Furthermore, the spatial frequency bandwidth is related to the maximum frequency vmax by: BW where i/c is the carrier spatial frequency of the hologram. "max ^ Vc + j- (5) As Lees has shown,the carrier frequency must be larger than three times half the bandwidth to ensure proper isolation of the CGH first diffracted order: As Lee 5 has shown,the carrier frequency must be larger than three times half the bandwidth to ensure proper isolation of the CGH first diffracted order: (3) (4) (5) vc > 3 BW (, () Therefore Therefore BW <. (7) - v ' As a result, by substituting this result in Eq. (4), the spot size diameter of the aberration generated by the CGH is limited to: BW < /max As a result, by substituting this result in Eq. (4), the spot size diameter of the aberration generated by the CGH is limited to: b' ffourier Ar ' vmax In our system, ffourier = 4mm, Ar = nm and vm,ax = 5 cy /mm, 8' < ^f Fourier ' ^r ' "max (8) In our system, f ffourier = 4mm, Ar = nm and vm ax 5 cy/mm, (7) (8) SPIE Vol. 138 Sixth Meeting in Israel on Optical Engineering (1988) / 15 SPIE Vol. 138 Sixth Meeting in Israel on Optical Engineering (1988) / 15 Downloaded From: on /18/1 Terms of Use:

5 ' is limited to: ' is limited to: 8'< x 4 x.5145 x 1-3 = 1.7 mm. (9) ' < - x 4 x.5145 x 1~ 3 = 1.7 mm (9) As can be seen in 4, the overall aberration of 1.79 mm exceeds the above limit. However, with slight vignetting of the aperture (5% radial) at the lower edge, the aberration spot size diameter is reduced to an acceptable value of 1.53 mm. This enables the first diffracted order of the CGH to be isolated using a mm slit along the x direction. 5 shows the computed mode structure at the Fourier plane, of the zero, first and second orders. As can be seen in 4, the overall aberration of 1.79 mm exceeds the above limit. However, with slight vignetting of the aperture (5% radial) at the lower edge, the aberration spot size diameter is reduced to an acceptable value of 1.53 mm. This enables the first diffracted order of the CGH to be isolated using a mm slit along the x direction. Fig, 5 shows the computed mode structure at the Fourier plane, of the zero, first and second orders. Ill X Mn 5. Zero, first and second orders as computed at Fourier plane. 5. Zero, first and second orders as computed at Fourier plane. POWER OF X Y POWER OF X Y 4 COEFFICIENT 1.19E E E E E E E E E E F E E E E E E E E E E E E E COEFFICIENT.19E E E E E E E E E E-7.888F E> E E E E E E E E E E E E ' ^ A 1A 1A > A A A A IA 15 1A IA ^ Ó 1 1 TO 19 L IA , IA IB IB IS 17 IF P IS Polynomial defining the OPD which characterizes the CGH.. Polynomial defining the OPD which characterizes the CGH. 7. CGH spatial frequency distribution. 7. CGH spatial frequency distribution. 1 / SPIE Vol 138 Sixth Meeting in Israel on Optical Engineering (1988) 1 / S/V Vol. 138 S/xf/7 Meeting in Israel on Optical Engineering (1988) Downloaded From: on /18/1 Terms of Use:

6 net showing net Moire pattern showing 9. Moiré aberration. correct the aberration. fringes fringes that that correct Obtained by superimposing 88 with cy/rnm cy simple simple grating of /mm representfrequency. carrier frequency. ing the carrier magnified CGH magnified ofcgh partof 8. Central Central part ten times. x COH7: IMPLRNTRT1N COH7: IMPLANTATION I1-ST -ST ORDER ORDER OR. OIFFRRrII"`! IN1ENSITY ''! INTENSITY RINCTION SPREAD SPRERD FUNCTION FLDI -. O-O'OlMRX.l x.1 FLDI.... MM. MM DFFOCUSING --. DEFOCUSING. -JUN-8P -JUN-RR -A WAVELENGTH WOVELENGTH HEIGHT WEIGHT NN NM i O.QJDEG -7E 1/ «H -14 tiw 1. function spread function point spread computed point The computed 1. The of the HOE. HOE, 17 SP /EVol. Vol Sixth Sixth Meeting Meeting in in Israel Israel on on Optical Optical Engineering Engineering(1988} (1988)// 17 SPIE Downloaded From: on /18/1 Terms of Use:

7 A is given given in in the the following followingfour fourfigures. figures. presents A detailed detailed description of the CGH is the polynomial defining the OPD that characterizes 7 depicts the CGH the CGH. 7 depicts the polynomial defining the OPD characterizes the spatial frequency frequency distribution including including the cy cy/mni /mm carrier carrier frequency. frequency. The 88x8 x 8 mm mm of the CGH is is shown Moire pattern shown in in 8, 8, magnified magnifiedten ten times. times. The Moiré pattern in in central part of is obtained by superimposing 88 with with aa simple simple grating grating of of cy cy/mni 9 is /mm representing the carrier frequency. frequency. The pattern pattern shows shows the the net net fringes fringes assigned assigned by by the the CGH to generate generate the aberration The Thecircular circular shape shape of of the the fringes fringes indicates indicates spherical spherical aberration and aberration in in their ellipticity represents astigmatism. Results The computed point spread spread function function of the the HOE to be be produced produced is is shown shown in in The computed 1. The The image image linewidth linewidth is estimated estimated from that figure figure to to be be µm, /nm,which which isisdiffraction diffraction 1. limited. The recording recording system system was was set up on an optical table and and several several holograms made of limited. The dichrornated holograms were dichromated gelatin gelatin were were exposed. exposed. The resulting holograms were obtained obtained with relatively high diffraction efficiency the recording beams has efficiencywhen whenconsidering consideringthat that the CGH in one of the diffraction efficiency efficiency of diffraction efficiency of of slightly slightly below below 1% 1%.The diffraction efficiency of the HOE reaches a peak peak response response of approximately approximately 9% 9% at the the designed designed angle angle of A typical typical angular angular of the the efficiency efficiency of the HOE measured at nm wavelength wavelength is shown in distribution of 11. ino.no an.nn n. no r.o.no an.on ; 8 max. effic.: at X, effic,: 9, at ail 3 deg 34 3 It 543 y: 543 (nn) (m) rlf Fig, 11. Angular Angular distribution of HOE diffraction efficiency as measured measured at 543 nm. efficiency as Comparison Comparison between between image image spot spot size size of of HOE recorded without CGH (left), (left), and with CGH (right). SPIE Vol. Vol Sixth Meeting 18 //SPIE Meeting in in Israel Israel on on Optical Optical Engineering (1988) (19881 Downloaded From: on /18/1 Terms of Use:

8 In order to appreciate the capability of a CGH as an optical corrector, two holograms were compared.the first was recorded through the CGH as described above and the second was recorded through the same system with a simple grating replacing the CGH. This grating had just the carrier spatial frequency in order to keep the recording beam direction. The optical performance of the two holograms was compared using a 1 µm object point source. The results are shown in 1. The first hologram recorded using the aberrated CGH had a resulting spot size of 3 µm, while the uncorrected system with the simple grating had a spot size of approximately 138 x 1 im. This comparison shows that such a CGH is capable of creating a HOE that can correct or generate an angular aberration of over 1 mrad. The frequency limit of the plotter and the maximum size plot that can be photo- reduced are the limiting factors. The corrected hologram has slightly below the expected diffraction limited performance due to the pinhole diameter both in the construction optics and in the evaluation. In order to appreciate the capability of a CGH as an optical corrector, two holograms were compared.the first was recorded through the CGH as described above and the second was recorded through the same system with a simple grating replacing the CGH. This grating had just the carrier spatial frequency in order to keep the recording beam direction. The optical performance of the two holograms was compared using a 1 /mi object point source. The results are shown in 1. The first hologram recorded using the aberrated CGH had a resulting spot size of 3 /mi, while the uncorrected system with the simple grating had a spot size of approximately 138 x 1 /mi. This comparison shows that such a CGH is capable of creating a HOE that can correct or generate an angular aberration of over 1 mrad. The frequency limit of the plotter and the maximum size plot that can be photo-reduced are the limiting factors. The corrected hologram has slightly below the expected diffraction limited performance due to the pinhole diameter both in the construction optics and in the evaluation. Conclusion The experiment described in this paper shows the computer generated hologram as an effective means to correct aberrations. High quality holographic optical elements with high diffraction efficiency can be generated directly with a recording system including a CGH. Such an HOE can be used as a relatively lightweight and inexpensive aspheric optical element in particular optical systems. The recording system can be assembled relatively simply using readily available refractive lenses selected on the basis of their paraxial properties. Undoubtedly the effectiveness of the CGH used as an optical corrector will expand further with improved plotting techniques. Equipment already developed for production of micro -electronic circuits, electron -beam lithography, etc., will certainly improve CGH applicability. Conclusion The experiment described in this paper shows the computer generated hologram as an effective means to correct aberrations. High quality holographic optical elements with high diffraction efficiency can be generated directly with a recording system including a CGH. Such an HOE can be used as a relatively lightweight and inexpensive aspheric optical element in particular optical systems. The recording system can be assembled relatively simply using readily available refractive lenses selected on the basis of their paraxial properties. Undoubtedly the effectiveness of the CGH used as an optical corrector will expand further with improved plotting techniques. Equipment already developed for production of micro-electronic circuits, electron-beam lithography, etc., will certainly improve CGH applicability. Acknowledgements Acknowledgements We would like to thank Mr Gil Mori and Mrs Marina Kaner for their help in the laboratory work. We would also like to thank the management of El -Op for their support of our work. The Israeli Ministry of Trade and Industry is gratefully acknowledged for their support of this work. We would like to thank Mr Gil Mori and Mrs Marina Kaner for their help in the laboratory work. We would also like to thank the management of El-Op for their support of our work. The Israeli Ministry of Trade and Industry is gratefully acknowledged for their support of this work. References References 1. D. Naor, O.Arnon, A.Avnur, "A lightweight innovative helmet airborne display and sight (HADAS)," SPIE vol. 778, 89 (1987). 1. D. Naor, O.Arnon, A.Avnur, "A lightweight innovative helmet airborne display and sight (HADAS)," SPIE vol. 778, 89 (1987).. R.C. Fairchild and R.Fienup, "Computer- Originated aspheric holographic optical elements," Optical Engineering 1, 133, (198). 3. J.M. Tedesco and R.C. Fairchild, "Design and fabrication of aspheric holographic optical elements using computer -generated holograms," SPIE vol. 53, 74 (1985) 4. D.C. Smith, "Testing diamond turned aspheric optics using CGH interferometry," SPIE vol. 3, 11 (1981). 5. W.H. Lee, "Computer generated hologram: Techniques and applications", Progress in Optics, vol. 1, ed. E. Wolf, North -Holland, Amsterdam (1978).. R.C. Fairchild and R.Fienup, "Computer-Originated aspheric holographic optical elements," Optical Engineering 1, 133, (198). 3. J.M. Tedesco and R.C. Fairchild, "Design and fabrication of aspheric holographic optical elements using computer-generated holograms," SPIE vol. 53, 74 (1985) 4. D.C. Smith, "Testing diamond turned aspheric optics using CGH interferometry," SPIE vol. 3, 11 (1981). 5. W.H. Lee, "Computer generated hologram: Techniques and applications", Progress in Optics, vol. 1, ed. E. Wolf, North-Holland, Amsterdam (1978). 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