COMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, August 23, 2004, 12:14 PM)

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1 COMPUTER GENERATED HOLOGRAMS Optical Sciences 67 W.J. Dallas (Monday, August 3, 4, 1:14 PM) PART IV: CHAPTER FOUR OPTICAL TESTING Part IV: Chapter Four Page 1 of 1 Introduction In optical testing an element under test, the test-piece, is compared to an element of known quality, the reference-piece. The comparison may be done in several different ways; some involve test- and referencepieces that are intentionally made to be quite different from one another. The basic idea in using CGH=s for optical testing is to replace, or augment, the conventional optical reference-piece with a CGH. CGH=s can be used in either transmission or in reflection. The primary advantage of a CGH is its flexibility in producing wavefronts. There is no limitation to circular symmetry and the production of aspheric wavefronts does not require greatly increased effort as it does with conventionally fabricated reference-pieces. The CGH can either replace the reference piece, incorporating its focusing power as well as any aspheric components of the design, or it can augment a conventional test element. In the latter case, the focusing power is provided by a conventional spherical element that can be relatively easily designed and fabricated. Using this approach relieves the CGH of the focusing function which can involve the investment of considerable space-bandwidth product. The CGH is then responsible only for providing deviations from sphericity. For these hybrid test elements both the conventional and the diffractive elements are used in a way that utilizes the strengths of each while avoiding their weaknesses. For optical testing, we choose to modify the language that we have introduced for display holograms. We introduced the modulation-image decomposition for discussing display holograms. That decomposition consisted of the true-image and false-images. For optical testing we are not interested in images, but rather in waves and wavefronts. In addition, because the illuminating waves may be more complicated than those we considered for display, it is advantageous to concentrate our attention on the actual transmittance (reflectance) of the CGH rather than the wave it produces. A final difference is that the spatial carrier is sometimes not a separate term, but rather an integral part of the desired transmittance. We will therefore remain with the general term harmonic decomposition. We will refer to ideal-transmittance which is the harmonic containing the transmittance we wish to implement. In optical testing, the ideal-transmittance of the CGH generally modifies only the phase of the incident wave. In other words, it is phase-only CGH=s that are generally used in optical testing. In addition, the incident wave generally has (or is assumed to have) uniform amplitude. One of the consequences is that the transmitted wave also has uniform amplitude. The ideal-transmittance of CGH=s used in optical testing generally not derived from wave propagation equations such as the ones we have discussed for display holograms, but rather from lens-design programs. One common type of hologram that is used is the binary-synthetic interferogram. In this CGH, the phase of the CGH transmittance is calculated and then the Fresnel-zones are plotted alternately as opaque and transmissive. The result is exactly equivalent to the CGH we have been terming the square-wave phase-only CGH. It is this CGH type that we will concentrate on in our discussions of optical testing.

2 φ, e i ( x, Part IV: Chapter Four Page of 1 In the remainder of this chapter we will first review the bare essentials of interferometry. We will then look at two interferometers: the Michelson and Mach-Zehnder interferometers. For null testing we will touch on two related topics in wave propagation: back-propagation ( sending a wave back the way it came ), and phase-conjugation. For lossless systems, those consisting solely of phase-modifying elements, back-propagation is equivalent to phase-conjugation. We will then look at the particular example of testing a transmissive optical element in a Twyman- Green (modified Michelson) interferometer. Following that discussion we will comment on the testing of optical blanks, on polarization interferometry and on white-light interferometry. A word of caution, the test setups we use in our discussions are simplified and meant to illustrate principles. Actual testing can involve considerably more complicated arrangements. One of the differences was mentioned at the beginning of this chapter: it is common to combine conventional optical elements with CGH=s. Interferometry Interference involves two elements: combining waves and making their interactions visible. The pair of waves that are being combined consists of one from the object under test and one from the reference-piece. Combining the waves is done in an interferometer through the use of mirrors and beam-splitters. A light source, commonly a laser or laser diode, produces illumination which is collimated, split, and supplied to the two optical elements over two separate paths. The waves emerging from those elements are then recombined and detected. The detection is on a square-law detector such as a CCD where the deposited energy pattern, the interference pattern, becomes available for visual or computer analysis. Two of the basic interferometers are the Michelson interferometer and the Mach-Zehnder interferometer. See Figure 1. Mathematically, we consider two waves of uniform amplitude, the reference wave which is the ideal-wave from the CGH, e iφ ( x, and the wave from the element under test which we will term i ( x,. The irradiance on the detector from the superimposed waves is e φ { [ φ φ ]} iφ ( x, iφ ( x, (, ) = + = 1+ cos (, ) (, ) I xy e e xy xy In order to more easily interpret the interference pattern, it is common to introduce a relative tilt between the two waves. A tilt about the y-axis results in the irradiance pattern I( xy, ) = { 1+ cos[ πξx+ φ( xy, ) φ( xy, )]} We can simplify this expression somewhat by using the phase difference φ = φ( x, φ ( x, [ ]

3 Part IV: Chapter Four Page 3 of 1 and making the spatial dependence implicit. The resulting equation is I ( πξ x φ) = 1 + cos + If the two wavefronts match perfectly, φ =, and a straight-line fringe pattern results I( xy, ) = 1+ cos( πξx) In another configuration, the CGH may be used as a compensating element to a wave rather than i ( x, generating a test wave directly. For this case, we have the wave under test, e φ, being multiplied by the CGH transmittance e iφ ( x,. Notice that for this application, the CGH complex amplitude transmittance is the complex conjugate of the transmittance for the first form of testing. The wave exiting the CGH is the product of the test wave and the CGH transmittance [ (, ) (, )] i x y x y u ( x, = e φ φ exit This wave is then interfered with a simple wave, such as a tilted plane wave to give the result πξ i x πξ i x exit [ (, ) (, )] I( x, = e + u ( x, = e + e φ φ o o i x y x y or I = 1 { + cos [ πξx+ φ] } which is exactly the result we saw in the first implementation. Schematic diagrams of the Michelson and of the Mach-Zehnder interferometers are shown in Figure 1. The Michelson interferometer is easily adapted to testing optical elements by replacing one of the arms by a transmissive reference piece and a mirror to be tested. It then becomes one variety of Twyman-Green interferometer. See Figure. If the element under test is perfect, the wave exiting the test element and traveling to the left is a plane wave. This wave is deflected downward by the beam splitter and interferes with the reference plane wave that was reflected from the upper mirror. If the reference and test waves are co-linear, then there will ideally be a uniform irradiance pattern on the detector. If the waves are tilted relative to one-another, then there would be a straight-line, cosine, interference pattern. If the element under test were to deviate from perfection, then the simple interference pattern at the output would be distorted. Phase Conjugation The common use of phase conjugation is for realtime adaptive elements pre-compensating for beam propagation through distorting phase media. For our present consideration, we need only one fixed realization of the conjugated phase. Phase conjugation is a useful consideration when the CGH is used in a null-test, i.e., as a reflective compensating element that is meant to send the rays defining the ideal-wave back along the paths they arrived on. If the complex amplitude of the

4 ideal arriving wave is (, ) = (, ) i u x y A x y e φ ( x, Part IV: Chapter Four Page 4 of 1 then the exiting wave should be i ( x, u ( x, = A( x, e φ See Appendix B for a more extensive discussion of the operation of phase conjugation. The CGH reflectance, for the true-wave, that implements this operation is (, ) r x y ( x, (, ) reflected wave u = = = e incident wave u x y, iφ ( x See Figure 4. Note that even if the incident wave does not have a constant amplitude, the reflectance is still phase-only. Radial Carrier CGH=s We focus our attention on the phase-only square-wave CGH. The primary mathematical tool we use for this type of CGH is generalized harmonic analysis, in particular the Fourier seriers expansion of the squarewave which is Sq ( p; α) = αsinc( mα) e πimp The usual technique for producing a radial carrier CGH is to use a quadratic phase (chirp) as the carrier. In the squarewave expansion, this involves the substitution r (, r ) p = φ θ ; r x y λ f + π = + Notice that we have moved to polar coordinates. The generalized harmonic expansion then becomes iπ r λ( f m) imφ ( r, θ ) r φ(, r θ) Sq + ; α = αsinc ( mα ) e e λ f π We recognize the quadratic phase factors as lenses in the parabolic approximation. Each harmonic will be focused to a different z-plane. The harmonics can be separated, to some degree, by placing a small aperture on the z-axis at the position that the desired harmonic comes to a focus. The size of the aperture is determined by the phase modulation. A variation of this formulation would to use a spherical phase rather a parabolic phase. In this case we have 1 φ( r, θ ) p= r + f + λ π Less usual is to use a linear, radial, phase corresponding to the substitution

5 Part IV: Chapter Four Page 5 of 1 φ(, r θ ) p= ρ r+ π The resulting binary CGH transmittance is φ(, r θ) (, ) ; sinc ( m πimρ r imφ r θ Sq ρ r + α = α α) e e π The exponential terms coming from the carrier are now axicons rather than lenses. The phase of an ideal axicon is 1 φaxicon = πρ ; r = x + y ; ρ = =a constant r The equi-phase rings of the axicon are equally spaced and have a separation = 1 ρ. Some CGH=s used for testing null-correctors for large telescope mirrors are accurately modeled by this axicon carrier. For annular CGH=s the diffraction orders for this carrier can be separated by constructing a stop having an annular aperture. See Figure 4. Given a phase function at the CGH, a minimum mean square error MMSE fit can be used to determine the linear and quadratic phase components. Refer to Appendix C for explicit formulas. Phase Conjugation using a Binary Phase-Only Square-Wave CGH in a Twyman-Green Interferometer: An Example The harmonic decomposition for a phase-only square-wave CGH of duty-cycle is φ ( xy, ) im ( x, rcgh ( x, = Sq ; α = αsinc( mα) e φ π The carrier is not explicitly shown in this expression. In the case we are examining, the carrier is part of the wave we are encoding. This expansion in modulation images is valid, the drawback is that, because we see no explicit carrier, we can=t predict the location of the images. There is no reason to believe that the modulation images will be spatially separated. If however, the phase can be split into a carrier and a change, we can proceed. We denote the ideal incident wave by. For our present example, let us assume that the phase is close to that of an axicon. The modulated axicon phase can be obtained from polynomial fitting and then written as φ = πρ r + φ Normally we would encode a phase φ in the CGH and find that the ideal-reflectance is produced in the m =+ 1harmonic of the CGH. For the present example, we will use the phase φ directly in the CGH and find that the, desired, ideal-reflectance is associated with the m = diffraction order. We know that for reconstructions in higher harmonic orders, the duty cycle of 1 the square wave decreases. Anticipating the results, we chooseα =. Substituting the phase and 4 the duty-cycle into the modulation-image decomposition gives

6 Part IV: Chapter Four Page 6 of 1 φ ( xy, ) 1 1 m imφ ( x, rcgh ( x, = Sq ; = sinc e = π m sinc e πimρr e im φ ( x, 4 4 Remembering that the conjugating phase isφ = φ we see that this phase distribution occurs in the minus second order. For m = we have the ideal-reflectance of the CGH which is 1 iφ( x, 1 iφ( x, 1 4πρ i r i φ( x, sinc e = e = e e 4 4 π π The middle expression shows that the m = diffraction order contains exactly the conjugating phase we desire. The expression on the right indicates how we can separate the undesired orders from the desired order by leading us to identifying the constant ρ with a radial spatial frequency and therefore with a diffraction angle. In order for the CGH to be useful, it is important that ideal-wave can be separated from the other harmonic waves. This separation is fulfilled for an annular axicon. Each wave is confined to an annulus. The positions of inner and outer boundaries of the annulus corresponding to each diffraction order can be calculated using the diffraction angle supplied by the radial spatial frequency. At distances greater than a certain minimum from the CGH, the annulus corresponding to the ideal-wave no longer overlaps the annular support of the other harmonics. See Figure 4. Figure 5 shows the test configuration. A plane wave enters from the left and is split. The wave traveling to the right passes through the element under test and is reflected by the CGH compensating element. If the element under test is perfect, the wave exiting the test element and traveling to the left is a plane wave. This wave is deflected downward by the beam splitter and interferes with the reference plane wave that was reflected from the upper mirror. If the reference and test waves are co-linear, then there would ideally be a uniform irradiance on the detector. If the waves are tilted relative to one-another, then there would be a straight-line interference pattern. If the element under test were to deviate from perfection, then the simple interference pattern at the output would be distorted. Blanks (rough surfaces) We generally consider the wavefront from the object under test to have come from an optically smooth object. Computer holography can, however, be applied in an earlier stage of production by examining the optical blanks for the proper figure and to determine the initial stages of the polishing program. Changes in the blank can be seen, even though the blank itself may be too rough to allow direct testing of the figure. Polarization Not only the surfaces under test, but also the wave properties can vary from the scalar monochromatic simplicity of basic interferometry. One of the properties that can be used to advantage is the property of polarization. This property can be used not only to test polarizing

7 Part IV: Chapter Four Page 7 of 1 elements, but also to incorporate robustness into the measurement of non-polarizing surfaces. Phase-shifting interferometry is such method. White Light One of the disadvantages of coherent illumination is coherent noise. This noise has its origins in practically all parts of the interferometer. A way to minimize this noise, while still maintaining the capability of producing interferograms is through the use of white-light interferometry. In order to produce interference patterns, the path lengths through the test and reference arms of the interferometer must be equalized. The conditions for equal path interferometry are satisfied, simultaneously, only for small areas of the interferogram. A path-length compensation element, such as a mirror mounted on a piezo-electric axial-translation unit, is introduced to sweep through the possible path-lengths and make facilitate formation a composite image. The result is a low- noise, but low-contrast fringe pattern in the interferogram. The fringe pattern is then contrast-enhanced for easy viewing. Appendix A: Reflection from a Plane Mirror This appendix was written in order to supply an easy entry into the propagator-based portion of the next appendix which concerns back-propagation and phase conjugation. Here we consider a plane mirror located in the plane z =. We look at an incident wave which begins in the original plane z = zand travels to the mirror. At the mirror, the incident wave is uxy (,,). We can relate it to the wave in the original plane through the Fourier transforms of the waves by using the transfer function of free space. The result is where 1 πiz ( ξ + η ) λ U ξη,, = U ξη,, z e = U ξη,, z P( ξη,, z ) 1 iz P(,, z ) e π λ ξη ξ + = η Perhaps the most confusing elements of both simple reflection and of phase conjugation is the change of coordinates. If we have a plane mirror in a plane parallel to the x-y plane and the incident wave traveling to the right in the positive z-direction, then the exiting wave, traveling in the negative z-direction enters a new coordinate system. We denote the new coordinates with primes. One way of correctly describing the results is by the transformations x ' =+ x, y' = y, z' = z The salient elements of this transformation is reflection in the z-coordinate and maintenance of a right-handed coordinate system. If we denote those coordinate with primes then the reflected wave is v x', y', = u ( x, y, ) We have ignored the additive constant phase change on reflection. If we transform the

8 expressions on both sides of the equality, we get Part IV: Chapter Four Page 8 of 1 V ( ξη,,) = U ( ξη,,) We expect that if we propagate AV@ to z' = z, which is to the right of the mirror, we will obtain the mirror image of the original wave. The propagation is V ξη,, z = V ξη,, P( ξη,, z ) = U ξη,, P( ξη,, z ) = U( ξη,,) P( ξη,, z ) = U( ξη,, z ) = U ( ξη,, z ) We have made use of the fact that the propagator (transfer function of free space) is circularly symmetric. Inverse transforming yields v( x', y', z) = u ( x, y, z) = u ( x', y', z) Which is the expected mirror image. The complex conjugate comes from the fact that the mirror image is propagating in the negative z -direction. Appendix B: Phase Conjugation Phase conjugation is a powerful principle. It states that if we produce a wave at a surface that has the complex conjugate amplitude of an incident wave, then the resulting wave will retrace the path of the incident wave. The resulting wave will everywhere be the complex conjugate of the incident wave. Again we restrict our attention to a plane parallel geometry, and a flat CGH. We will look at a wave Au@ propagating from an original plane z = z to the plane of the CGH at z =. The reflected wave Av@ will propagate from the CGH back to the original plane. See Figure 3. We will approach the problem in two different ways. First through time reversal, and then using the transfer-function of free space. The time reversal argument goes as follows. The time-varying complex amplitude of the incident monochromatic wave-field in 3-space is i t u( x, y, z, t) = u( x, y, z) e π ν Reversing time gives πν (,,, ) (,, ) (,, ) + i t πν i t = = u xyz t u xyze u xyze What this equation tells us is that we can move a wave backwards in time by conjugating the wave, moving forward in time, then taking the complex conjugate of the result. In our case we are considering the plane of the CGH. The time reversed spatial part of the wave at the plane of the CGH will be (,,) u x y The reflectance of the CGH will be

9 (, ) r x y ( x, y,) (,,) u = u x y Part IV: Chapter Four Page 9 of 1 Let us now move on to the propagator explanation. From the discussion in Appendix A we know that having the waves which are propagating in opposite direction be identical means that vx ( ', y', z) = u( x, y, z) Again we emphasize that the lateral coordinates in these two functions are specified in different coordinate systems. We begin with the original wave. Its Fourier transform at the CGH is 1 πiz ( ξ + η ) λ U ξη,, = U ξη,, z e = U ξη,, z P ξη,, z where we have used the propagator ( transfer-function of free-space ) relation. The reflected wave at the CGH,is propagated back to the original plane using the relation 1 πiz ( ξ + η ) λ V ( ξη,, z ) = V ( ξη,,) e = V( ξη,,) P( ξη,, z ) The Aequality@ relation is V ξη,, z = U ξη,, z which, when propagated to the CGH plane becomes or ( ξη,,) ( ξη,, ) = ( ξη,,) ( ξη,, ) = ( ξη,,) ( ξη,, ) V P z U P z U P z V ( ξη,,) = U ( ξη,,) Inverse transforming back to direct space leaves us with In consistent coordinates ( ', ', ) = (,, ) v x y u x y (,,) = (,,) v x y u x y The complex amplitude reflectance of the CGH should then be (, ) r x y v x y u x y = = u x, y, u x, y,,,,, We have arrived at the same answer given by the time-reversal argument. Going one step i further, writing the complex amplitude in polar form, we haveu = Ae φ. The result is that for

10 Part IV: Chapter Four Page 1 of 1 phase conjugation, the reflectance should be i ( x, r( x, = e φ Interestingly enough, we have used no approximations. The limitation comes from the same source in both derivations: absorption. For the time-reversal argument to be valid, we must be dealing with a conservative system, one that does not dissipate energy. For the propagator argument, we are assuming propagation through free-space, or at least piecewise constant regions of imaginary refractive index. Appendix C: Polynomial Fitting of the Phase for Circular-Carrier CGH=s The way we many times think about a CGH is that we impress our desired signal on a carrier. In some cases the carriers arise naturally as a component of an existing phase. Suppose we have a phase distribution φ ( r, θ ) that we wish to implement in a CGH. We can express this phase as φ = φcarrier + φmodulation One way of implementing this separation is by polynomial fitting. The result is φ = a + ar 1 + a r + φ r, θ For minimum mean-squared error fitting, the formulas are well known. Expressed in continuous form π rmax π rmax φ ( r, θ) rdrdθ φ( r, θ) a r drdθ π r= π r= a = ; a 1 = π r max 3 π rmax 3 π rmax 3 φ( r, θ) a r ar 1 r drdθ π = a = 1 4 π rmax and, of course, ( r, ) ( r, ) ( a ar 1 ar ) φ θ = φ θ + +

11 Part IV: Chapter Four Page 11 of 1 FIGURES Twyman-Green Interferometer Testing Spherical Mirrors Figure 1 Figure PHASE CONJUGATION OF THE TEST WAVE Phase-Conjugated Wave Incident Wave Source Element Under Test Waves from Other CGH Diffraction Orders CGH IV_4_PhaseConjugation Figure 3 Figure 4

12 igu re 5 Part IV: Chapter Four Page 1 of 1 F Twyman-Green Interferometer Testing an Aspheric Lens D:\Dallas\67_Cgh\\Cgh_Book\CghChapterIV4.wpd August 3, 4 (1:14PM) W.J. Dallas Plane Mirror Element under Test CGH