Class Note for ECE 527 at UA 2
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Date Created: 02/06/15
Analysis of Holographic Images If the object field used to record a hologram emanates from a lens the reconstructed image from the resulting hologram will act very similar to the lens used in the recording stage In this manner holographic lenses can be formed and are one class of holographic optical elements Figure X shows the formation of a transmission type holographic lens In this case the hologram is reconstructed with a conjugate reference beam to form a real image of an object Holographic lenses have been used successfully in a number of commercial applications such as headup displays and point of sale optical scanners The performance of the lens can be evaluated using techniques similar to those used for refractive and re ective lenses Paraxial imaging relationships raytracing methods and aberration coefficients can be determined As we will see however the highly dispersive nature of gratings will in general limit the performance of holographic lenses to narrow spectral and angular bandwidths They can also be used in combination with refractive and re ective components to compensate for aberrations and to add some unique properties to the imaging system Recording Holographic Lens Material L ens Conjugate 39 Beam Object Beam Image Refemc e Beam Fig X Formation of a holographic lens left and reconstruction of the holographic lens ri ght Several different methods can be used to evaluate images formed by holograms and holographic optical elements The techniques that we will evaluate include Exact Ray Tracing 7 this technique is based on repeated use of the grating equation Paraxial Imaging Relations7 approximate method for analyzing image formation Analysis of Aberration Coef cients 7 extends the rst order results from paraxial imaging K Vector Closure 7 Useful for highly selective Bragg gratings In general an object can be considered as a collection of source points Each of these points forms a hologram with a reference field Therefore analysis of point source holograms will also provide essential information on the properties of more complex images Hologram Paraxial Imaging Relations The paraxial imaging relations for holographic lenses can be determined from analyzing holograms that are formed with point sources In some sense this can be anticipated from the fact that a lens is a conjugate imaging deVice that produces a 11 relation between an object point and an image point The paraxial relations result from restricting the analysis to regions near the optical zaxis of the optical system This reduces the phase relation for a spherical wave to a quadratic and makes the algebraic analysis more tractable Once the analysis of point sources is complete the results can be used to analyze more complex source point distributions Analysis of the Phase Distribution from a Point Source to a Hologram Plane Consider a hologram formed by a exposing a recording material located at z 0 to a reference point source located at xr yr Zr and object point source at x0 ya 20 The hologram will be reconstructed with a third point source located at xp yp zp The hologram is constructed with light at wavelength X1 and reconstructed at M The eld propagating from the reference point source can be written as jklrr Erhe 11 rhh where k1 27r 21 and rrvh is the distance from xr yr Zr to a point on the hologram xh yh zh For suf ciently large distances from the hologram plane and for small regions around the zaxis the distance rrvh in the exponent can be approximated using the first two terms of the binomial expansion Ref JWG as 2 2 minim gm 2 z 2 2 For sufficiently small regions about the zaxis the term lrrvl1 in Equation Il can be approximated as lzr and for a xed distance from the hologram plane Zr the amplitude A can be taken as a constant Using these approximations the elds from the reference and object source points at the hologram plane can be written as 2 2 z A expj mm Ayelt4 xii z Z r r Similar expressions can also be written for the elds from the object point and the reconstruction point sources The irradiance ofthe exposing beams at the recording plane is 10 Er xhayhEo xhayh lz 2A3 A 421 pp ing x2 yh y2ixh x02 yh y0 A0A eXpj Zixh x2 yh y2zixh x02 yh y0 YE A Aj 4AgeW A0AfeWm Recording Plane Reference Source point xr yr zr HXhyh Local Interference Fringe Object Source point X0 yo 20 After exposure and processing the hologram can be considered as a transmittance function that is linearly proportional to txh yh y xh yh with 7 being a constant related to the processing of the recording material During the reconstruction phase the hologram is illuminated with the field from a point source located at point P jkz ph E p rm pf Using the same assumptions that were used for the reference and object fields the field from the reconstruction point can be written as 2 2 Ep camlp exp jg xh xF yh yp Apew F F The reconstructed field becomes E Eptxhyh and consists of four terms The first two are constants with the same phase as Ep however the third and fourth terms contain a combination of the phase of EF plus that of Er and E0 Ei cl czeJ p c3ej39 p f J p r o E1E2E3E4 c4e Hologram Reconstruction Source Point Xp yp zp As discussed previously the third and fourth terms in this equation will give rise to image formation or reconstruction of the original object phase p0 or conjugate object phase 00 if pp is respectively equal to pr or go When the object is a point source the phase of the image eld will also resemble the phase of a spherical wave therefore the reconstructed phase must look like a field converging to or diverging from a point With this constraint the phase of the third and fourth terms of the reconstructed field can be expressed as 1 U3 036 03 exp cop lt0r we 1 U4 96 quot C4 exp cop or 00 The superscript l in the p3 and p4 terms indicate that they are first order in 12 of the expansion for lri These terms are quadratic approximations to the spherical wavefront It can be seen that two images U3 and U4 will be formed for each reconstruction point In general one image will be much better than the other ie better resemble the phase of a spherical wave The approximate locations of the two point images can be found by matching the first order terms in the expansions for the spherical wavefronts lt1gt2 lx2y2 2xx3 2yy3 3 22 2 z3 zzlxzyz 2xxP 2yyp27rlxzy2 2xxo 2yyo 12 2 2P 21 2 20 27 l x2 y2 2xxr 2yyr 11 2 Zr If the phase of the reconstruction eld pp is equal to or close to the phase of the reference eld pr used during the exposure an image point will form at the location X3 y3 23 where 44m fwnag x 3 44yag yag y 3 44yag yag 222 23 p 0 739 44yag yag and u xii 21 Similarly matching the phase in the expression for U4 results in 1 22 11 y2 2xx3 2yy3 4 12 2 z 3 anxf W ng ng2zleW 2xg 2yn 22 2 2P 11 2 20 2 7rlx2y2 2xxr 2yyr 11 2 Zr This term results in the formation of an image at a point X4 y4 24 where 44 m wnag x 4 44yaqyaa y 4 44yaqyaa 222 24 p 0 739 44yagyag The link between the reconstruction point coordinates Xpypzp and the image point coordinates X3y323 provide a set of paraXial imaging relations for a holographic lens in the same manner as the paraXial imaging relations for a refractive lens or a focusing mirror In many cases image point X3y323 will result in a virtual image point in object space with rays diverging away from it on the image side of a hologram and X4y4Z4 a real image However in general this depends on the geometry of the hologram and the reconstruction conditions Image Magni cation Effects The paraXial imaging relations for a holographic lens can be used to determine a set of magni cation features for these elements The lateral image magnification can be specified as the change in lateral image height Hologram A X 3 X3 AXO i l with the change in the object point height The figure illustrates this relation Notice that unlike a conventional lens the image height is not necessarily inverted as for a refractive lens The expressions for lateral magnification can be found by differentiating X34 and y374 with respect to X0 Mmz34 34 or Mmz34 f l l M or M 1123 1 1 Lat4 lzt7 l zt7 Lzzp z Lzzp z The magnification of a change in angle in object space can be found by differentiating the ratio of lateral image height to the distance from the hologram plane d x 2 it MAng3 3 3gt Ang3zlu2 2 39 d x0 20 39 11 Similarly the longitudinal magnification can be found by differentiating the paraXial relation for Z3 with respect to Z0 ld z M 0 Lang dzg u dzg 1zg1uzp lz l l l M I lz7 lyzp lzr 1 1 2 MLong4 MLat4 u Hologram 063 AX3 AXQ oco X0 X3 47 ZO 74lt Z3 gt Hologram A 4 AZOIlt7 4 A23 7 Figure XX Illustrations of paraXial angular magni cation upper gure and longitudinal magni cation lower gure of point source holograms Effect of Spectral Bandwidth on Hologram Resolution The effect of spectral bandwidth on the image quality can also be estimated using the paraXial imaging relations For this analysis it is assumed that the hologram is formed with a narrow bandwidth optical source at wavelength X1 and reconstructed with a source having a nominal wavelength of M and a bandwidth of All The object point is located at 0x0 ya zt7 and both the reference and reconstruction points are located at zp z 00 The source points at 00 can be described as angles with tan 9 Zr x tan 6p p ZF and it is assumed that yr yp 0 Recording Material Hologram R Reconstruction Beam er 9p k1 M A7b2 OXOYOZO Figure XX Schematic diagram of the construction and reconstruction conditions for analyzing the effects of spectral bandwidth of the reconstruction source With these construction and reconstruction conditions the reconstructed paraXial image position reduces to x x Z x 9 p 0 420 Zp I Zr 23 zt7 u with u 22 21 The change in image position with the change in M can be determined taking the derivative of X3 with respect to M dxg 2 g M 2p y A 6 23 inji M If the nominal construction and reconstruction wavelengths are nearly equal 22 xii and u l A reconstruction source with a bandwidth of All will result in a variation in the image location of x M 20 ml 12 I IAZSI 20 12 Notice that if zt7 0 then A23 0 This is the condition for an image plane hologram Consider an example where a hologram is formed with an argon laser at 5145 nm and reconstructed with a mercury Hg arc lamp with M 546 nm and A1 571m The construction and reconstruction beams are plane waves at an angle of 30 It is desired to have the resultant blur in the lateral x direction no larger than 05 mm At what distance should an object point be located to satisfy this requirement Using the above relation we can write 2 20 ii A22 xp Snm tan30 7 Therefore the object point used during the recording should not be any closer than 95 mm from the hologram plane The corresponding spread in the longitudinal position of the J 0924mm image is IAZSI 95mm 522m nm Third order aberration coefficients and related imaging properties Exact ray tracing Discussion of Zemax for analyzing holograms Preaberration of wavefronts to control aberrations OLD STUFF During hologram construction light from a reference and object point sources expose the holographic material as illustrated in Figure XXa The resulting field at a point H on the hologram plane is Consider the phase of the eld from a point source P that propagates to a point Q on the xy plane containing the holographic recording material The phase of a spherical wavefront originating at point P on a hologram plane referenced to the origin of the hologram plane oxy21 d PQP0 1xx02y ygt2z ltxyzigt zilzo1x xo2 y yo2Z12 1x ag211 12 Expanding the square root terms using the binomial expansion gives the following expression for the phase 27239 l 170x7y 7Zx2 J2 2xxo 200 1 8 3 x4 y4 2x2y2 4x3x0 4y3y0 4x2yy0 4xy2x0 Z0 6x2x 6y2y 2x2y 2y2x 8xyx0y0 4xx 4yy 4xx0y 4xx0y 4xy y0 HOTS Similar relations can be written for the reference and reconstruction waves During hologram construction light propagates from a reference point source r and an object point source 0 to a hologram plane As mentioned previously the phase relations for the paraxial images are formed by the combination of the phase values for the object reference and reconstruction waves 1 U3 p0r gtlt03 600 0 U or gt D 4 p 04 op 00 or The superscript 1 implies that the terms represent the first order terms in 12 obtained from the binomial expansion of the phase term When p r an accurate reconstruction of the virtual image is obtained while when p r an accurate reconstruction of the conjugate image of the object results The first order terms in 12 for the phase of the reconstructed wavefront at the hologram plane is 0 zzlxz y2 2xxp 2ny 27rlx2 y2 2xx0 2yyo 27rlx2 y2 2xxr 2yyr 3 22 2 2P 21 2 20 21 2 Zr This expression can be rewritten as 3 2 Z Z Z Z Z Z Z Z Z 12 p 0 r p o r p o r x lt12 lx2y2 1 2x pxoxr 2y yp yoyr W A first order term represents the phase of the Gaussian reference sphere converging to the paraXial image points X3 y3 23 and X4 y4 Z4 and can now be defined as 0 22 7239l x2 y2 2xx4 2yy4 4 2 2 z x xpzozr uxozpzr uxrzpzo 4 4 202 2132 21320 zpzozr zozr uzpzr uzpzo The phase terms go and gol nrepresent Gaussian wavefronts converging to the paraXial 3 image locations X3 y3 23 and X4 y4 24 The thirdorder term in terms of 1233 for the phase of the Gaussian reference sphere is 27239 1 033 QW y4 2ny2 4x3x3 4y3y3 4x2yy3 4xy2x3 3 6x2x32 6y2y32 2x2y32 2y2x32 8xyx3y3 4xx33 4yy 4xxgy 4xx3y 4xy y3 Replacing X and y with p x2 y2 and t9 tan391yx results in an expression for the phase of the wavefront corresponding to the thirdorder aberrations becomes 27r 1 4 1 3 1 2 2 2 W Tz pSEp Cxcos6Cys1n6 Ep Axcos 6Ays1n 62Axycos6sm6 p2FpDx cos Dy sin6 S is the spherical aberration coef cient Cx and Cy are the coma aberration coef cients Ax Ay and Axy are the astigmatism aberration coef cients F is the eld curvature coef cient and DK and Dy are the distortion aberration coef cients Since holographic lenses are typically more useful when used to form real images the aberration coef cients are formulated in terms of p4 the conjugate reconstruction rather than the primary reconstruction phase p3 The aberration coef cients for p3 can be obtained by changing the signs for 20 and Zr Spherical Aberration Coef cient 1 y y 1 22 32 33 327 P 0 r 4 1 1 31 1 1 1 2 1 1 61 JOE 1 7 7 77 3 7 l 20 Zr 2p 20 Zr 2p ZOZr 20 Zr 202er When zr amp zp oo ie plane wavesS z Therefore when 21 x12 gt S 0 z 0 When 2 27 S 0 regardless of the values of 2C and u This implies that the reference and object points at the same distance from the hologram plane Coma Coef cient C xaxr x 3 3 3 3 Zp Z0 Zr Z4 2 2 2 2 xp uxa xr x4 Ax 3 3 33 zp z 2 z4 0 739 Field Curvature Coef cient xyltxygtltxsysgtxy 3 3 3 3 2P Zo Z Z4 Distortion Coef cient 3 2 3 2 3 2 3 2 xpxpyp xo x0y0 xr 99 x4x4y4 x Z 3 3 3 3 2p Zo Zr 24 Aberrated Image Formation during Hologram Reconstruction Hologram Plane PX1y121 Q X y d o Aberrated ParaXial Wavefront Image Point 0 Reference y Wavefront PXoy05Z0 Qxy y The reconstructed image can be referenced to the paraXial image converging to the paraXial image point When the reconstruction conditions deviate from the ideal case the wavefront will no longer converge to the ideal image point The phase of the non spherical wavefront can be described using aberration coefficients
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