- Email: [email protected]
Reconfigurable ternary-phase array illuminator based on the generalised phase contrast method
1 January 2000
Optics Communications 173 Ž2000. 169–175 www.elsevier.comrlocateroptcom
Reconfigurable ternary-phase array illuminator based on the generalised phase contrast method ) Jesper Gluckstad , Paul C. Mogensen ¨ Optics and Fluid Dynamics Department, Risø National Laboratory, P.O. Box 49, DK-4000 Roskilde, Denmark Received 18 August 1999; accepted 19 October 1999
Abstract A new technique is presented for implementing a dynamic array illuminator scheme based on the generalised phase contrast imaging method. It is shown that by using a ternary-phase rather than a binary-phase approach for encoding input patterns a larger range of array compression factors can be achieved. We demonstrate how a graphical phase chart can be used for the optimisation of the array illumination. Due to the low requirements for both spatial and temporal bandwidth products the encoding method is ideal for implementation on reconfigurable phase-only spatial light modulating devices with diverse applications such as dynamic laser tweezer arrays and generation of structured light for machine vision. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Array illuminator; Structured light; Phase contrast; Spatial light modulator
1. Introduction In this paper we present a new approach for the realisation of an array illuminator based on the generalised phase contrast method w1–5x. An array illuminator provides a way of splitting a uniformly expanded laser beam into multiple bright spots or a periodic pattern with the smallest possible loss of energy from the incoming laser beam. The array illuminator scheme that we propose and analyse is based on ternary-phase encoding of the input wavefront to produce a binarised intensity pattern at the output. This implementation of an array illuminator could offer considerable advantages over alternative
) Corresponding author. Fax: q45-46-7745-65; e-mail: [email protected]
architectures such as diffractive optics based systems where the generation of a dynamic intensity profile is required, in for example reconfigurable illumination of arrays of photonic devices, the generation of dynamic laser tweezer arrays and also the coding of structured light in machine vision applications w6–10x. The theoretical basis for this ternary-phase array illuminator has been developed from the analysis of a phase-only optical projection system w1x, which is an extension of the original Zernike phase contrast scheme w11x. Using our approach the small phase angle limitation of Zernike’s method is no longer a limiting constraint and the complete range of phase values from zero to 2 p can be utilised. It has been shown theoretically w1x and experimentally w2x, that a system in which the spatial average value of the input phase modulated light is carefully matched to the phase shifting value of a phase contrast filter, can
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 3 2 - X
170
J. Gluckstad, P.C. Mogensenr Optics Communications 173 (2000) 169–175 ¨
produce an essentially lossless phase to intensity mapping of the input light. When generating array illumination it is highly desirable that the light conversion is performed with the smallest possible loss of energy. It is therefore only really practical for such a system to be based on diffractive or refractive phase-only spatial transforms. Previously proposed phase-only encoding techniques can be complicated to perform, requiring for example, the generation of phase-only diffractive elements w6,7x. Our method offers a significant reduction in the space–bandwidth product normally associated with phase encoding since we apply a simple pixel to pixel imaging operation from the input phase pattern to the output intensity distribution. For this reason it is feasible to use a dynamic and relatively coarse-grained spatial light modulator as the input phase modulator, without seriously compromising the image reconstruction quality. In the following sections of this paper we discuss the generic model for the ternary phase array illuminator that we are proposing. In Section 3 we show in detail the important mathematical analysis for the system which leads to a generic description of the optimum operating space for the general class of phase contrast filter ŽPCF. based array illuminators. In conjunction with the mathematical treatment we also introduce a graphical method, in which a phase chart can be used for the optimisation of such array illuminators. Finally, in Section 4 we treat some specific array illuminator cases and compare our results with those available in the literature. We show that our mathematical treatment provides a very successful approach for dealing with all types of PCF based illuminator systems.
2. The generic model The set-up we consider in this paper is based on the phase contrast filtering architecture illustrated in Fig. 1. An output intensity distribution produced from an input phase pattern is obtained by applying a truncated, on-axis, phase-filtering operation in the spatial frequency domain between two Fourier transforming lenses. The first lens performs a spatial Fourier transform, so that direct propagated light is
Fig. 1. An array illuminator set-up based on a 4-f imaging system with a phase-only spatial light modulator ŽPOSLM. generating the input wavefront. Where u is the zero-order phase shift of the filter, the factor h describes the fraction of filter radius measured with respect to the size of the mainlobe of the zero-order diffracted light from the input aperture and a is the complex spatial average of the input wavefront.
focused onto this on-axis filtering region whereas spatially varying phase information generates light scattered to locations outside this central region. Applying a different phase shift to the two filtering regions we can obtain an intensity pattern at the observation plane by use of the second Fourier lens. The generalised phase contrast method works very well in the case where there is a large separation between the on-axis, low spatial frequency light and the higher spatial frequencies in the Fourier plane. The periodicity inherent in the array illuminator gives the good separation of the high and lower spatial frequencies required for optimum performance in this system. 3. Mathematical analysis The first step in the mathematical analysis is the derivation of a relationship between the input phase values and the output intensity. This approach is based directly on detailed derivations previously obtained for the generalised phase contrast method w1,4,5x. We can write an expression for the intensity at the observation plane of the set-up shown in Fig. 1, such that: I Ž xX , yX ; f˜ . s
Ž 1.
with
° ~a s A HHA exp Ž i f Ž x , y . . d x d y s < a < exp Ž i f y1
a
.
¢f˜ Ž x , y . s f Ž x , y . y f X
X
X
X
a
Ž 2.
J. Gluckstad, P.C. Mogensenr Optics Communications 173 (2000) 169–175 ¨
In this formulation, a is the spatial average of the input wavefront and as such is generally complex with absolute value < a < and phase fa . The input phase pattern, f Ž x, y ., is addressed on the phaseonly spatial light modulator ŽPOSLM. with an aperture area A. Eq. Ž1. also includes the parameters u for the zero-order phase shift of the filter and K which depends directly on J0 , the zero-order Bessel function, such that w5x: K s 1 y J0 Ž 1.22 p P h .
Ž 3.
The factor h describes the fraction of filter radius measured with respect to the size of the mainlobe of the zero-order diffracted light from the input aperture. The value of h should be chosen to completely encompass the zero-order light with the result that the value of K tends to unity as the Bessel function tends to zero. We have therefore taken K as unity for the remainder of this analysis. Using the fact that we wish to generate a dark background for the array illumination at the observation plane as our principal design criterion, from Eq. Ž1., we can write: I Ž xX0 , yX0 ; f˜ 0 . s 0
Ž 4.
where Ž xX0 , yX0 . indicates observation plane coordinates of the background of the array illumination, and f˜ 0 is the phase shift generating a zero-intensity level at the observation plane. Applying the dark background condition of Eq. Ž4. to Eq. Ž1. we obtain the following expression for the phase term: K < a < Ž 1 y exp Ž i u . . s exp Ž i f˜ 0 .
171
Before applying the previously derived expressions to a further analysis we can deduce the range of valid phase parameters fulfilling our design criteria from Eq. Ž4.. The largest possible absolute value that the complex spatial average value, a , can have is unity, this leads to the following solution intervals for Eqs. Ž6. and Ž7.:
½
u s w pr3; 5pr3 x f˜ 0 s w ypr3; pr3 x
Ž 8.
These intervals can be depicted graphically as indicated in the chart shown in Fig. 2. This phase chart visualisation was originally proposed in Ref. w4x. The chart provides a graphical method of relating the intensity at a point in the output plane, directly to the phase at a point in the input plane. This mapping is dependent on the filtering parameters and the spatial average value of the input phase pattern. On the example phase chart shown in Fig. 2, the largest possible range of valid filter phase values, u , indicated as the dark dotted arc, arises from the first interval given by Eq. Ž8.. The corresponding range for the potential f˜ 0 , shown as the wide dashed arc, is directly determined by our dark background criterion. Thus the arc lengths constrained by the two intersection points of the largest diameter K < a <-circle and the f˜ unity-circle indicate the possible solution intervals for u and f˜ 0 . The general applicability of
Ž 5.
A significant consequence arising from Eq. Ž5. is that we now have a simple way of expressing a new design criterion relating the spatial average value of any input phase pattern to the zero-order phase shift of a matched Fourier phase filter. Since K is by definition positive and by taking the modulus of Eq. Ž5. we obtain: K < a < s <2 sin Ž ur2 .
Ž 6.
Isolating terms related to the phasor components of Eq. Ž5., we get:
f˜ 0 s Ž u y p . r2
Ž 7.
Fig. 2. The phase chart for graphical visualisation of the phase to intensity mapping. The wide dashed line corresponds to a zero-intensity generating phase shift in the input plane. The dotted line indicates the maximum range of the phase filtering parameter u . The quadratic intensity scale is used to determine the output intensity corresponding to a given input phase.
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J. Gluckstad, P.C. Mogensenr Optics Communications 173 (2000) 169–175 ¨
the phase chart has been previously described with the focus on the optimisation of a generalised set of common path filtering parameters. Here, we have a simpler situation with only one filter parameter, u , constrained by Eq. Ž6. to the interval described by Eq. Ž8.. The zero-point of the quadratic intensity scale of Fig. 2 is then fixed at the intersection point of a desired K < a <-circle and an estimated f˜ 0-value corresponding to a point constrained by the circular arc indicated by the dashed line in Fig. 2. The corresponding u-value is implicitly governed by Eq. Ž6.. Keeping the quadratic intensity scale fixed about this point and rotating it, we can directly estimate the phase-to-intensity mapping through the system of Fig. 1 from the intersection points of the f˜-unity circle and the quadratic intensity scale. An example showing the application of the phase chart is illustrated in Fig. 3. With the dark background criterion the highest intensity value that can be achieved is four times that of the input and is obtained for f˜ s f˜ 0 q p. At the other extremity where f˜ s f˜ 0 we have zero intensity corresponding to our design criterion expressed by Eq. Ž4.. We also note that symmetrically about the point f˜ s f˜ 0 q p we can achieve identical intensity levels Žillustrated by phase values f˜ s f˜ 1 and f˜ s f˜ 2 . which will be utilised in the subsequent analysis. Having explained the basic principle of operation and shown how the phase chart can be used, we can derive the specific parameters for an array illumina-
Fig. 4. Schematic representation of the spatial light modulator aperture area A. For the ternary phase case this can be considered as consisting of three generalised subareas A n with phase shifts f˜ n .
tor. In the remainder of this section we derive the expressions for the implementation of a ternary-phase array illuminator and show that for the special case of a binary phase illuminator these expressions agree with previous work w12,13x. We can consider the available spatial light modulator aperture area, A, as divided into subareas A 0 , A1 and A 2 with respective phase values f˜ 0 , f˜ 1 and f˜ 2 Žsee Fig. 4.. In this figure we indicate the accumulated subareas for each of the ternary phase values keeping in mind that the periodic spatial structuring of the subareas is optional depending on the specific needs for array illumination, whether periodic or irregular w6x. What is of interest is to derive general expressions relating the addressing parameters for the input spatial light modulator to the range of possible phase shift values of the Fourier filter obeying the design criterion we have already set out. From Fig. 4 we can express the total area defined by the POSLM aperture and its average phase value as the sum of the phase-weighted sub-areas: A 0 exp Ž i f˜ 0 . q A1 exp Ž i f˜ 1 . q A 2 exp Ž i f˜ 2 . s A < a < Ž 9. This can be further simplified by expressing the subareas as fractions of the total area, A, such that R 1 s A1rA and R 2 s A 2rA:
Fig. 3. The phase chart for a ternary array case showing that two equal intensities I1 and I2 can be produced from phase shifts f˜ 1 and f˜ 2 , respectively. The dark background criterion determines the location of the zero-intensity point for the quadratic intensity scale at f˜ 0 .
Ž 1 y R1 y R 2 . exp Ž i f˜ 0 . q R1 exp Ž i f˜ 1 . q R 2 exp Ž i f˜ 2 . s < a <
Ž 10 .
For the array illuminator considered here, we are interested in binary output intensity patterns, the
J. Gluckstad, P.C. Mogensenr Optics Communications 173 (2000) 169–175 ¨
levels of which correspond to the input phase values. In this case the dark background region is defined by Ž A 0 , f˜ 0 . and the bright output level of intensity, I, is determined by Ž A1 , f˜ 1 . and Ž A 2 , f˜ 2 . in the input plane. For the binary output intensity condition it follows that I Ž f˜ 1 . s I Ž f˜ 2 .
Ž 11 .
This equality corresponds to the symmetry condition explained previously and shown in Fig. 3. Due to this symmetry we can simplify the analysis by applying the following substitution:
f s f˜ 1 y f˜ 0 s f˜ 0 y f˜ 2
Ž 12 .
so that Eq. Ž10. can be reformulated as: R 1 Ž exp Ž i f . y 1 . q R 2 Ž exp Ž yi f . y 1 . s Ž exp Ž yi u . y 1 .
y1
Ž 13 .
It is now a straightforward task to solve Eq. Ž13. for the real part and the imaginary part respectively, to obtain the following sets of equations:
~°R q R 1
2s
1
2
Ž 2 sin Ž fr2. .
173
and by combining this with Eq. Ž6., we obtain an expression for R 1 where: 2
y2 R 1 s Ž 2 sin Ž ur2 . . s Ž K < a < . Ž 18 . This result turns out to be the special case that corresponds to the set of solutions found by Zhou et al. w12x where a binary phase pattern served as the input. In the next section we will illustrate graphically how this set of solutions is related to a specific subset in our solution space. We finish this section by summarising the sequence of steps to synthesise a given array illuminator configuration from the previously derived parameters.
PROCEDURE 1. Choose the desired compression factor, C, and phase filter parameter u . 2. Estimate R 1 q R 2 s Cy1 . 3. Calculate the phase f s 2 siny1 Ž Cr4.. 4. Calculate the fractional areas R 1 and R 2 from Eq. Ž15.. 5. Address the spatial layout of array illuminator with the calculated ratio of the phases.
y2
¢R y R s Ž 2 sinŽ f . tanŽ ur2. .
y1
Ž 14 .
This can also be expressed in terms of the fractional areas, such that:
½
R1 s 1r8 w Ž sin Ž fr2.. y2 q 2 Ž sin Ž f . tan Ž ur2.. y1 x R 2 s 1r8 w Ž sin Ž fr2.. y2 y 2 Ž sin Ž f . tan Ž ur2.. y1 x
Ž 15 .
Since we have focused on solutions where identical intensity levels are obtained in both the R 1-region and the R 2-region we can define an array illuminator compression factor, C, in the following way: C s Ž R1 q R 2 .
y1
s 4 sin2 Ž fr2 .
Ž 16 .
The minimum compression factor corresponds to uniform illumination of the whole observation plane such that R 1 q R 2 s 1, whereas the maximum compression factor is found from Eq. Ž16. to be: C s 4. An interesting special case can be deduced from Eq. Ž14. by setting R 2 s 0 where we find that:
usf
Ž 17 .
Fig. 5. Three-dimensional plot showing the functional dependence of the inverse compression factor, Cy1 on the parameters u and f˜ 1. This plot maps the complete solution space for the ternaryphase array illuminator.
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J. Gluckstad, P.C. Mogensenr Optics Communications 173 (2000) 169–175 ¨
4. Examples In this section we will look at some examples illustrating the versatility of our ternary-phase encoding scheme. First, however, it is helpful to outline the potential solution space available to us. For that purpose we have generated the three-dimensional mesh plot shown in Fig. 5, illustrating the functional dependence of the inverse compression factor, Cy1 s R 1 q R 2 , on parameters u and f˜ 1 Žfrom Eq. Ž12. it can be seen that f or f˜ 2 could also have been chosen instead of f˜ 1 .. We notice that our solution space is a triangular plane in the Ž u , f˜ 1 .-domain. The inverse compression factor ranges from a maximum value of one to a minimum value of 0.25 obtained at the single point Ž u , f˜ 1 . s Ž p, p .. Keeping the surface shape of the 3D-plot in mind it is useful to take a closer look at the solution domain defined by Ž u , f˜ 1 . as illustrated in Fig. 6 Žcorresponding to a view along the vertical axis of Fig. 5.. From Fig. 6 it is clear that the available operating space for the array illuminator lies within the triangular region in the Ž u , f˜ 1 .-plane. Choosing a large
Fig. 7. Phase chart representations of the three special cases Ža., Žb. and Žc. in Fig. 6, showing graphically how different design parameters are visualised for a ternary-phase array illuminator with a compression factor C s 2.
Fig. 6. A plot of the available Ž u , f˜ 1 . operating domain for the ternary-phase array illuminator. This is effectively a planar crosssection of the plot in Fig. 5. The boundaries for this triangular operating domain are defined by the R 2 s 0 line Žthe binary-phase array case., the R1 q R 2 s1 line Žthe limiting case for no background, A 0 s 0 in Fig. 4. and the u s p boundary.
phase filter value, u , provides a large range of potential f˜ 1 values indicated by D f˜ 1. In fact, having u s p Žthe vertical edge of the triangle in Fig. 6. provides the largest range of options for the parameter f˜ 1 and is the obvious choice if the phase of the Fourier filter has to be fixed w2x. Along the second edge of the solution domain we have R 1 q R 2 s 1 and along the final edge we have the subset of solutions derived in Ref. w12x corresponding to having R 2 s 0 in our formulation Žsee Eq. Ž18... Finally, at the point Ž u , f˜ 1 . s Ž p, p . in Fig. 6 we
J. Gluckstad, P.C. Mogensenr Optics Communications 173 (2000) 169–175 ¨
show the single solution corresponding to the derivations of Lohmann et al. w13x. Now that we have discussed the potential solution space we can choose some examples for comparison and apply the previously described phase chart to illustrate the functionality of different ternary-phase encoding schemes. Assume that we are looking for solutions having an inverse compression factor equal to a 1r2. In Fig. 6 we have drawn a dotted line to indicate the possible sets of phase parameters Ž u , f˜ 1 . that can be used for obtaining this compression factor. On this dotted line we have indicated three specific examples that we can illustrate graphically using the phase chart scheme. In Fig. 7 we show the phase chart representations of these three cases Ža., Žb. and Žc. highlighted in Fig. 6. For the example in Fig. 7Ža. we have u s p and f˜ 1 s pr2. A comparison with our mathematical analysis from the preceding section shows that this corresponds to R 1 s R 2 s 1r4. Fig. 7Žb. and Žc. correspond to different values of u and f˜ 1 indicated on Fig. 6 that also give the desired compression ratio of 2. It can be seen that Ža. and Žb. correspond to a ternary mapping whilst Žc. corresponds to the limiting condition of a binary mapping as derived by Zhou and co-workers w12x. These examples show how our phase chart can be used as a tool to visualise the operation parameters for the array illumination. 5. Conclusions We have demonstrated that a ternary phase array illuminator based on the generalised phase contrast method can be used for the generation of binary intensity patterns. We have developed a mathematical analysis, which can be applied to the optimisation of this type of array illuminator and we also demonstrate that the construction of a phase chart can provide a very useful tool for the optimisation of the illuminator design. We have also shown that there exists an optimum operating region and we have
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parameterised this in such a fashion as to simplify the analysis of these systems. Comparisons with the literature reveal an excellent agreement with some specific array illuminator cases which can also be completely described by our mathematical approach. Acknowledgements This work has been funded as part of an award from the Danish Technical Scientific Research Council. References w1x J. Gluckstad, Phase contrast image synthesis, Opt. Commun. ¨ 130 Ž1996. 225–230. w2x J. Gluckstad, L. Lading, H. Toyoda, T. Hara, Lossless light ¨ projection, Opt. Lett. 22 Ž1997. 1373–1375. w3x J. Gluckstad, Pattern generation by inverse phase contrast – ¨ comment, Opt. Commun. 147 Ž1998. 16–19. w4x J. Gluckstad, Graphic method for analyzing common path ¨ interferometers, Appl. Opt. 37 Ž34. Ž1998. 8151–8152. w5x J. Gluckstad, Image decrypting common path interferometer, ¨ Proc. of SPIE 3715 Ž1999. 152–159. w6x J.-N. Gillet, Y. Sheng, Irregular spot array generator with trapezoidal apertures of varying heights, Opt. Commun. 166 Ž1999. 1–7. w7x V. Arrizon, E. Carreon, M. Testorf, Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations, Opt. Commun. 160 Ž1999. 207–213. w8x M. Reicherter, T. Haist, E.U. Wagemann, H.J. Tiziani, Opticle particle trapping with computer-generated holograms written on a liquid-crystal display, Opt. Lett. 24 Ž9. Ž1999. 608–610. w9x J. Batlle, E. Mouaddib, J. Salvi, Recent progress in coded structured light as a technique to solve the correspondence problem: a survey, Pattern Recognition 31 Ž7. Ž1998. 963– 982. w10x J. Gluckstad, Adaptive array illumination and structured light ¨ generated by spatial zero-order self-phase modulation in a Kerr medium, Opt. Commun. 120 Ž1995. 194–203. w11x F. Zernike, How I discovered phase contrast, Science 121 Ž1955. 345–349. w12x C. Zhou, L. Liu, Zernike array illuminator, Optik 102 Ž2. Ž1996. 75–78. w13x A.W. Lohmann, J. Schwider, N. Streibl, J. Thomas, Array illuminator based on phase contrast, Appl. Opt. 27 Ž14. Ž1988. 2915–2921.
Optics Communications 173 Ž2000. 169–175 www.elsevier.comrlocateroptcom
Reconfigurable ternary-phase array illuminator based on the generalised phase contrast method ) Jesper Gluckstad , Paul C. Mogensen ¨ Optics and Fluid Dynamics Department, Risø National Laboratory, P.O. Box 49, DK-4000 Roskilde, Denmark Received 18 August 1999; accepted 19 October 1999
Abstract A new technique is presented for implementing a dynamic array illuminator scheme based on the generalised phase contrast imaging method. It is shown that by using a ternary-phase rather than a binary-phase approach for encoding input patterns a larger range of array compression factors can be achieved. We demonstrate how a graphical phase chart can be used for the optimisation of the array illumination. Due to the low requirements for both spatial and temporal bandwidth products the encoding method is ideal for implementation on reconfigurable phase-only spatial light modulating devices with diverse applications such as dynamic laser tweezer arrays and generation of structured light for machine vision. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Array illuminator; Structured light; Phase contrast; Spatial light modulator
1. Introduction In this paper we present a new approach for the realisation of an array illuminator based on the generalised phase contrast method w1–5x. An array illuminator provides a way of splitting a uniformly expanded laser beam into multiple bright spots or a periodic pattern with the smallest possible loss of energy from the incoming laser beam. The array illuminator scheme that we propose and analyse is based on ternary-phase encoding of the input wavefront to produce a binarised intensity pattern at the output. This implementation of an array illuminator could offer considerable advantages over alternative
) Corresponding author. Fax: q45-46-7745-65; e-mail: [email protected]
architectures such as diffractive optics based systems where the generation of a dynamic intensity profile is required, in for example reconfigurable illumination of arrays of photonic devices, the generation of dynamic laser tweezer arrays and also the coding of structured light in machine vision applications w6–10x. The theoretical basis for this ternary-phase array illuminator has been developed from the analysis of a phase-only optical projection system w1x, which is an extension of the original Zernike phase contrast scheme w11x. Using our approach the small phase angle limitation of Zernike’s method is no longer a limiting constraint and the complete range of phase values from zero to 2 p can be utilised. It has been shown theoretically w1x and experimentally w2x, that a system in which the spatial average value of the input phase modulated light is carefully matched to the phase shifting value of a phase contrast filter, can
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 3 2 - X
170
J. Gluckstad, P.C. Mogensenr Optics Communications 173 (2000) 169–175 ¨
produce an essentially lossless phase to intensity mapping of the input light. When generating array illumination it is highly desirable that the light conversion is performed with the smallest possible loss of energy. It is therefore only really practical for such a system to be based on diffractive or refractive phase-only spatial transforms. Previously proposed phase-only encoding techniques can be complicated to perform, requiring for example, the generation of phase-only diffractive elements w6,7x. Our method offers a significant reduction in the space–bandwidth product normally associated with phase encoding since we apply a simple pixel to pixel imaging operation from the input phase pattern to the output intensity distribution. For this reason it is feasible to use a dynamic and relatively coarse-grained spatial light modulator as the input phase modulator, without seriously compromising the image reconstruction quality. In the following sections of this paper we discuss the generic model for the ternary phase array illuminator that we are proposing. In Section 3 we show in detail the important mathematical analysis for the system which leads to a generic description of the optimum operating space for the general class of phase contrast filter ŽPCF. based array illuminators. In conjunction with the mathematical treatment we also introduce a graphical method, in which a phase chart can be used for the optimisation of such array illuminators. Finally, in Section 4 we treat some specific array illuminator cases and compare our results with those available in the literature. We show that our mathematical treatment provides a very successful approach for dealing with all types of PCF based illuminator systems.
2. The generic model The set-up we consider in this paper is based on the phase contrast filtering architecture illustrated in Fig. 1. An output intensity distribution produced from an input phase pattern is obtained by applying a truncated, on-axis, phase-filtering operation in the spatial frequency domain between two Fourier transforming lenses. The first lens performs a spatial Fourier transform, so that direct propagated light is
Fig. 1. An array illuminator set-up based on a 4-f imaging system with a phase-only spatial light modulator ŽPOSLM. generating the input wavefront. Where u is the zero-order phase shift of the filter, the factor h describes the fraction of filter radius measured with respect to the size of the mainlobe of the zero-order diffracted light from the input aperture and a is the complex spatial average of the input wavefront.
focused onto this on-axis filtering region whereas spatially varying phase information generates light scattered to locations outside this central region. Applying a different phase shift to the two filtering regions we can obtain an intensity pattern at the observation plane by use of the second Fourier lens. The generalised phase contrast method works very well in the case where there is a large separation between the on-axis, low spatial frequency light and the higher spatial frequencies in the Fourier plane. The periodicity inherent in the array illuminator gives the good separation of the high and lower spatial frequencies required for optimum performance in this system. 3. Mathematical analysis The first step in the mathematical analysis is the derivation of a relationship between the input phase values and the output intensity. This approach is based directly on detailed derivations previously obtained for the generalised phase contrast method w1,4,5x. We can write an expression for the intensity at the observation plane of the set-up shown in Fig. 1, such that: I Ž xX , yX ; f˜ . s
Ž 1.
with
° ~a s A HHA exp Ž i f Ž x , y . . d x d y s < a < exp Ž i f y1
a
.
¢f˜ Ž x , y . s f Ž x , y . y f X
X
X
X
a
Ž 2.
J. Gluckstad, P.C. Mogensenr Optics Communications 173 (2000) 169–175 ¨
In this formulation, a is the spatial average of the input wavefront and as such is generally complex with absolute value < a < and phase fa . The input phase pattern, f Ž x, y ., is addressed on the phaseonly spatial light modulator ŽPOSLM. with an aperture area A. Eq. Ž1. also includes the parameters u for the zero-order phase shift of the filter and K which depends directly on J0 , the zero-order Bessel function, such that w5x: K s 1 y J0 Ž 1.22 p P h .
Ž 3.
The factor h describes the fraction of filter radius measured with respect to the size of the mainlobe of the zero-order diffracted light from the input aperture. The value of h should be chosen to completely encompass the zero-order light with the result that the value of K tends to unity as the Bessel function tends to zero. We have therefore taken K as unity for the remainder of this analysis. Using the fact that we wish to generate a dark background for the array illumination at the observation plane as our principal design criterion, from Eq. Ž1., we can write: I Ž xX0 , yX0 ; f˜ 0 . s 0
Ž 4.
where Ž xX0 , yX0 . indicates observation plane coordinates of the background of the array illumination, and f˜ 0 is the phase shift generating a zero-intensity level at the observation plane. Applying the dark background condition of Eq. Ž4. to Eq. Ž1. we obtain the following expression for the phase term: K < a < Ž 1 y exp Ž i u . . s exp Ž i f˜ 0 .
171
Before applying the previously derived expressions to a further analysis we can deduce the range of valid phase parameters fulfilling our design criteria from Eq. Ž4.. The largest possible absolute value that the complex spatial average value, a , can have is unity, this leads to the following solution intervals for Eqs. Ž6. and Ž7.:
½
u s w pr3; 5pr3 x f˜ 0 s w ypr3; pr3 x
Ž 8.
These intervals can be depicted graphically as indicated in the chart shown in Fig. 2. This phase chart visualisation was originally proposed in Ref. w4x. The chart provides a graphical method of relating the intensity at a point in the output plane, directly to the phase at a point in the input plane. This mapping is dependent on the filtering parameters and the spatial average value of the input phase pattern. On the example phase chart shown in Fig. 2, the largest possible range of valid filter phase values, u , indicated as the dark dotted arc, arises from the first interval given by Eq. Ž8.. The corresponding range for the potential f˜ 0 , shown as the wide dashed arc, is directly determined by our dark background criterion. Thus the arc lengths constrained by the two intersection points of the largest diameter K < a <-circle and the f˜ unity-circle indicate the possible solution intervals for u and f˜ 0 . The general applicability of
Ž 5.
A significant consequence arising from Eq. Ž5. is that we now have a simple way of expressing a new design criterion relating the spatial average value of any input phase pattern to the zero-order phase shift of a matched Fourier phase filter. Since K is by definition positive and by taking the modulus of Eq. Ž5. we obtain: K < a < s <2 sin Ž ur2 .
Ž 6.
Isolating terms related to the phasor components of Eq. Ž5., we get:
f˜ 0 s Ž u y p . r2
Ž 7.
Fig. 2. The phase chart for graphical visualisation of the phase to intensity mapping. The wide dashed line corresponds to a zero-intensity generating phase shift in the input plane. The dotted line indicates the maximum range of the phase filtering parameter u . The quadratic intensity scale is used to determine the output intensity corresponding to a given input phase.
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the phase chart has been previously described with the focus on the optimisation of a generalised set of common path filtering parameters. Here, we have a simpler situation with only one filter parameter, u , constrained by Eq. Ž6. to the interval described by Eq. Ž8.. The zero-point of the quadratic intensity scale of Fig. 2 is then fixed at the intersection point of a desired K < a <-circle and an estimated f˜ 0-value corresponding to a point constrained by the circular arc indicated by the dashed line in Fig. 2. The corresponding u-value is implicitly governed by Eq. Ž6.. Keeping the quadratic intensity scale fixed about this point and rotating it, we can directly estimate the phase-to-intensity mapping through the system of Fig. 1 from the intersection points of the f˜-unity circle and the quadratic intensity scale. An example showing the application of the phase chart is illustrated in Fig. 3. With the dark background criterion the highest intensity value that can be achieved is four times that of the input and is obtained for f˜ s f˜ 0 q p. At the other extremity where f˜ s f˜ 0 we have zero intensity corresponding to our design criterion expressed by Eq. Ž4.. We also note that symmetrically about the point f˜ s f˜ 0 q p we can achieve identical intensity levels Žillustrated by phase values f˜ s f˜ 1 and f˜ s f˜ 2 . which will be utilised in the subsequent analysis. Having explained the basic principle of operation and shown how the phase chart can be used, we can derive the specific parameters for an array illumina-
Fig. 4. Schematic representation of the spatial light modulator aperture area A. For the ternary phase case this can be considered as consisting of three generalised subareas A n with phase shifts f˜ n .
tor. In the remainder of this section we derive the expressions for the implementation of a ternary-phase array illuminator and show that for the special case of a binary phase illuminator these expressions agree with previous work w12,13x. We can consider the available spatial light modulator aperture area, A, as divided into subareas A 0 , A1 and A 2 with respective phase values f˜ 0 , f˜ 1 and f˜ 2 Žsee Fig. 4.. In this figure we indicate the accumulated subareas for each of the ternary phase values keeping in mind that the periodic spatial structuring of the subareas is optional depending on the specific needs for array illumination, whether periodic or irregular w6x. What is of interest is to derive general expressions relating the addressing parameters for the input spatial light modulator to the range of possible phase shift values of the Fourier filter obeying the design criterion we have already set out. From Fig. 4 we can express the total area defined by the POSLM aperture and its average phase value as the sum of the phase-weighted sub-areas: A 0 exp Ž i f˜ 0 . q A1 exp Ž i f˜ 1 . q A 2 exp Ž i f˜ 2 . s A < a < Ž 9. This can be further simplified by expressing the subareas as fractions of the total area, A, such that R 1 s A1rA and R 2 s A 2rA:
Fig. 3. The phase chart for a ternary array case showing that two equal intensities I1 and I2 can be produced from phase shifts f˜ 1 and f˜ 2 , respectively. The dark background criterion determines the location of the zero-intensity point for the quadratic intensity scale at f˜ 0 .
Ž 1 y R1 y R 2 . exp Ž i f˜ 0 . q R1 exp Ž i f˜ 1 . q R 2 exp Ž i f˜ 2 . s < a <
Ž 10 .
For the array illuminator considered here, we are interested in binary output intensity patterns, the
J. Gluckstad, P.C. Mogensenr Optics Communications 173 (2000) 169–175 ¨
levels of which correspond to the input phase values. In this case the dark background region is defined by Ž A 0 , f˜ 0 . and the bright output level of intensity, I, is determined by Ž A1 , f˜ 1 . and Ž A 2 , f˜ 2 . in the input plane. For the binary output intensity condition it follows that I Ž f˜ 1 . s I Ž f˜ 2 .
Ž 11 .
This equality corresponds to the symmetry condition explained previously and shown in Fig. 3. Due to this symmetry we can simplify the analysis by applying the following substitution:
f s f˜ 1 y f˜ 0 s f˜ 0 y f˜ 2
Ž 12 .
so that Eq. Ž10. can be reformulated as: R 1 Ž exp Ž i f . y 1 . q R 2 Ž exp Ž yi f . y 1 . s Ž exp Ž yi u . y 1 .
y1
Ž 13 .
It is now a straightforward task to solve Eq. Ž13. for the real part and the imaginary part respectively, to obtain the following sets of equations:
~°R q R 1
2s
1
2
Ž 2 sin Ž fr2. .
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and by combining this with Eq. Ž6., we obtain an expression for R 1 where: 2
y2 R 1 s Ž 2 sin Ž ur2 . . s Ž K < a < . Ž 18 . This result turns out to be the special case that corresponds to the set of solutions found by Zhou et al. w12x where a binary phase pattern served as the input. In the next section we will illustrate graphically how this set of solutions is related to a specific subset in our solution space. We finish this section by summarising the sequence of steps to synthesise a given array illuminator configuration from the previously derived parameters.
PROCEDURE 1. Choose the desired compression factor, C, and phase filter parameter u . 2. Estimate R 1 q R 2 s Cy1 . 3. Calculate the phase f s 2 siny1 Ž Cr4.. 4. Calculate the fractional areas R 1 and R 2 from Eq. Ž15.. 5. Address the spatial layout of array illuminator with the calculated ratio of the phases.
y2
¢R y R s Ž 2 sinŽ f . tanŽ ur2. .
y1
Ž 14 .
This can also be expressed in terms of the fractional areas, such that:
½
R1 s 1r8 w Ž sin Ž fr2.. y2 q 2 Ž sin Ž f . tan Ž ur2.. y1 x R 2 s 1r8 w Ž sin Ž fr2.. y2 y 2 Ž sin Ž f . tan Ž ur2.. y1 x
Ž 15 .
Since we have focused on solutions where identical intensity levels are obtained in both the R 1-region and the R 2-region we can define an array illuminator compression factor, C, in the following way: C s Ž R1 q R 2 .
y1
s 4 sin2 Ž fr2 .
Ž 16 .
The minimum compression factor corresponds to uniform illumination of the whole observation plane such that R 1 q R 2 s 1, whereas the maximum compression factor is found from Eq. Ž16. to be: C s 4. An interesting special case can be deduced from Eq. Ž14. by setting R 2 s 0 where we find that:
usf
Ž 17 .
Fig. 5. Three-dimensional plot showing the functional dependence of the inverse compression factor, Cy1 on the parameters u and f˜ 1. This plot maps the complete solution space for the ternaryphase array illuminator.
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4. Examples In this section we will look at some examples illustrating the versatility of our ternary-phase encoding scheme. First, however, it is helpful to outline the potential solution space available to us. For that purpose we have generated the three-dimensional mesh plot shown in Fig. 5, illustrating the functional dependence of the inverse compression factor, Cy1 s R 1 q R 2 , on parameters u and f˜ 1 Žfrom Eq. Ž12. it can be seen that f or f˜ 2 could also have been chosen instead of f˜ 1 .. We notice that our solution space is a triangular plane in the Ž u , f˜ 1 .-domain. The inverse compression factor ranges from a maximum value of one to a minimum value of 0.25 obtained at the single point Ž u , f˜ 1 . s Ž p, p .. Keeping the surface shape of the 3D-plot in mind it is useful to take a closer look at the solution domain defined by Ž u , f˜ 1 . as illustrated in Fig. 6 Žcorresponding to a view along the vertical axis of Fig. 5.. From Fig. 6 it is clear that the available operating space for the array illuminator lies within the triangular region in the Ž u , f˜ 1 .-plane. Choosing a large
Fig. 7. Phase chart representations of the three special cases Ža., Žb. and Žc. in Fig. 6, showing graphically how different design parameters are visualised for a ternary-phase array illuminator with a compression factor C s 2.
Fig. 6. A plot of the available Ž u , f˜ 1 . operating domain for the ternary-phase array illuminator. This is effectively a planar crosssection of the plot in Fig. 5. The boundaries for this triangular operating domain are defined by the R 2 s 0 line Žthe binary-phase array case., the R1 q R 2 s1 line Žthe limiting case for no background, A 0 s 0 in Fig. 4. and the u s p boundary.
phase filter value, u , provides a large range of potential f˜ 1 values indicated by D f˜ 1. In fact, having u s p Žthe vertical edge of the triangle in Fig. 6. provides the largest range of options for the parameter f˜ 1 and is the obvious choice if the phase of the Fourier filter has to be fixed w2x. Along the second edge of the solution domain we have R 1 q R 2 s 1 and along the final edge we have the subset of solutions derived in Ref. w12x corresponding to having R 2 s 0 in our formulation Žsee Eq. Ž18... Finally, at the point Ž u , f˜ 1 . s Ž p, p . in Fig. 6 we
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show the single solution corresponding to the derivations of Lohmann et al. w13x. Now that we have discussed the potential solution space we can choose some examples for comparison and apply the previously described phase chart to illustrate the functionality of different ternary-phase encoding schemes. Assume that we are looking for solutions having an inverse compression factor equal to a 1r2. In Fig. 6 we have drawn a dotted line to indicate the possible sets of phase parameters Ž u , f˜ 1 . that can be used for obtaining this compression factor. On this dotted line we have indicated three specific examples that we can illustrate graphically using the phase chart scheme. In Fig. 7 we show the phase chart representations of these three cases Ža., Žb. and Žc. highlighted in Fig. 6. For the example in Fig. 7Ža. we have u s p and f˜ 1 s pr2. A comparison with our mathematical analysis from the preceding section shows that this corresponds to R 1 s R 2 s 1r4. Fig. 7Žb. and Žc. correspond to different values of u and f˜ 1 indicated on Fig. 6 that also give the desired compression ratio of 2. It can be seen that Ža. and Žb. correspond to a ternary mapping whilst Žc. corresponds to the limiting condition of a binary mapping as derived by Zhou and co-workers w12x. These examples show how our phase chart can be used as a tool to visualise the operation parameters for the array illumination. 5. Conclusions We have demonstrated that a ternary phase array illuminator based on the generalised phase contrast method can be used for the generation of binary intensity patterns. We have developed a mathematical analysis, which can be applied to the optimisation of this type of array illuminator and we also demonstrate that the construction of a phase chart can provide a very useful tool for the optimisation of the illuminator design. We have also shown that there exists an optimum operating region and we have
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parameterised this in such a fashion as to simplify the analysis of these systems. Comparisons with the literature reveal an excellent agreement with some specific array illuminator cases which can also be completely described by our mathematical approach. Acknowledgements This work has been funded as part of an award from the Danish Technical Scientific Research Council. References w1x J. Gluckstad, Phase contrast image synthesis, Opt. Commun. ¨ 130 Ž1996. 225–230. w2x J. Gluckstad, L. Lading, H. Toyoda, T. Hara, Lossless light ¨ projection, Opt. Lett. 22 Ž1997. 1373–1375. w3x J. Gluckstad, Pattern generation by inverse phase contrast – ¨ comment, Opt. Commun. 147 Ž1998. 16–19. w4x J. Gluckstad, Graphic method for analyzing common path ¨ interferometers, Appl. Opt. 37 Ž34. Ž1998. 8151–8152. w5x J. Gluckstad, Image decrypting common path interferometer, ¨ Proc. of SPIE 3715 Ž1999. 152–159. w6x J.-N. Gillet, Y. Sheng, Irregular spot array generator with trapezoidal apertures of varying heights, Opt. Commun. 166 Ž1999. 1–7. w7x V. Arrizon, E. Carreon, M. Testorf, Implementation of Fourier array illuminators using pixelated SLM: efficiency limitations, Opt. Commun. 160 Ž1999. 207–213. w8x M. Reicherter, T. Haist, E.U. Wagemann, H.J. Tiziani, Opticle particle trapping with computer-generated holograms written on a liquid-crystal display, Opt. Lett. 24 Ž9. Ž1999. 608–610. w9x J. Batlle, E. Mouaddib, J. Salvi, Recent progress in coded structured light as a technique to solve the correspondence problem: a survey, Pattern Recognition 31 Ž7. Ž1998. 963– 982. w10x J. Gluckstad, Adaptive array illumination and structured light ¨ generated by spatial zero-order self-phase modulation in a Kerr medium, Opt. Commun. 120 Ž1995. 194–203. w11x F. Zernike, How I discovered phase contrast, Science 121 Ž1955. 345–349. w12x C. Zhou, L. Liu, Zernike array illuminator, Optik 102 Ž2. Ž1996. 75–78. w13x A.W. Lohmann, J. Schwider, N. Streibl, J. Thomas, Array illuminator based on phase contrast, Appl. Opt. 27 Ž14. Ž1988. 2915–2921.