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Hyperresolving phase-only filters with an optically addressable liquid crystal spatial light modulator
Micron 34 (2003) 327–332 www.elsevier.com/locate/micron
Hyperresolving phase-only filters with an optically addressable liquid crystal spatial light modulator J. McOrist, M.D. Sharma, C.J.R. Sheppard*,1, E. West, K. Matsuda Physical Optics Laboratory, School of Physics, University of Sydney, Sydney, NSW 2006, Australia
Abstract Hyperresolving (sometimes called ‘superresolving’ or ‘ultraresolving’) phase-only filters can be generated using an optically addressable liquid crystal spatial light modulator. This approach avoids the problems of low efficiency, and coupling between amplitude and phase modulation, that arise when using conventional liquid crystal modulators. When addressed by a programmed light intensity distribution, it allows filters to be changed rapidly to modify the response of a system or permit the investigation of different filter designs. In this paper we present experimental hyperresolved images obtained using an optically addressable parallel-aligned nematic LCD with two zone Toraldo type phase-only filters. The images are compared with theoretical predictions. q 2003 Elsevier Ltd. All rights reserved. Keywords: Hyperresolving; Filters; Liquid crystal spatial light modulator
1. Introduction Recently, Davis et al. demonstrated the use of a twisted nematic liquid crystal spatial light modulator (LCSLM) for generating hyperresolving and apodizing amplitude filters (Davis et al., 1999). Here we demonstrate the generation of phase-only filters using an optically addressable LCSLM. Experimental results clearly demonstrate hyperresolution using a two zone phase-only filter. Applications for hyperresolving filters include increasing the recording density in optical disks (Cox, 1984) and enhanced resolution in photolithographic (Hild et al., 1998) and confocal microscope systems (Hegedus, 1985; Sheppard, 1995; Martinez-Corral et al., 1998; Akduman, 1998; Deng et al., 2000). The term hyperresolution, in general, can be defined as the effect of reducing the width of the central lobe of a focal spot below the classical limit. In practice there is a trade-off, between the decrease in width of the central lobe and the increase in level of the outer lobes, that determines the effectiveness of the system and is driven by system requirements. The opposite effect of increasing * Corresponding author. Tel.: þ 31-15-27-89407; fax: þ31-15-27-88105. E-mail address: [email protected] (C.J.R. Sheppard). 1 On leave at: Optics Group, Department of Applied Physics, Technical University of Delft, Lorentzweg 1, Delft CD 2628, Netherlands 0968-4328/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0968-4328(03)00059-3
the width of the central lobe and reducing the level of the outer lobes is called apodization (Jacquinot and Roizen-Dossier, 1964). Hyperresolution is sometimes called ‘superresolution’ or ‘ultraresolution’, but the term superresolution is best reserved for the more restrictive case when the spatial frequency bandwidth of the system is increased above the classical limit (Cox and Sheppard, 1986). Both hyperresolution and apodization are typically achieved by using a filter (or mask) in the pupil of the system, the design and properties of the filter determining the point spread function of the focused spot. The first hyperresolving masks, based on concentric rings of oscillating sign, were proposed by Toraldo di Francia (Toraldo di Francia, 1952), who showed that the central lobe can be made arbitrarily narrow. The behaviour of the intensity in the focal plane (Cox et al., 1982) and along the optic axis (Sheppard et al., 1998) for Toraldo filters has been presented. A simple approach, based on pupil moments, to determine performance parameters for realvalued filters has also been given (Sheppard and Hegedus, 1988). Amplitude-only filters can give only a modest gain. Sales has described how use of phase-only filters, with phase changes of other than p radians, can result in improved overall performance (Sales and Morris, 1997). In this paper we describe the use of an optically addressable parallel-aligned nematic LCSLM to produce
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phase-only filters which are used to demonstrate hyperresolution. The experimentally obtained hyperresolved point spread functions are compared with theoretical predictions.
2. The liquid crystal display There are ferroelectric, smectic and nematic liquid crystal displays (LCD). We will not dwell on ferroelectric or smectic LCD. However, smectic LCD have been discussed for phase modulation (De Bougrenet and Dupont, 1997) and ferroelectric LCSLMs can give phase-only modulation (Freeman et al., 1992), but suffer from low efficiency. The nematic LCD are available as either twisted or parallel-aligned. In addition, both kinds of nematic LCD can be either optically or electrically addressable. The electrically and optically addressable twisted nematic LCD usually produce coupling between amplitude and phase (Lu and Saleh, 1990; Konforti et al., 1988; Barnes et al., 1989; Marquez et al., 2001). However, under correct operating conditions, these devices are capable of phase-only operation (Pezzaniti and Chipman, 1993; Davis et al., 1998; Campos et al., 2002). On the other hand, the electrically addressable parallel-aligned nematic LCD do not exhibit coupling between amplitude and phase. However, Stolz et al. (2001) have theoretically investigated coupled phase and amplitude (spiral) domains with electrically addressable parallel-aligned nematic LCD. The optically addressable parallel-aligned nematic LCD, also called zero twist or electrically controlled birefringent devices, can produce pure phase modulation. Mukohzaka et al. (1994) investigated the diffraction efficiency, intensity versus phase transfer characteristics and response time for the optically addressable parallel-aligned nematic LCD. For a twisted nematic LCSLM, amplitude modulation occurs as a result of pixelation (Carcole´ et al., 1994). Other problems are dynamic range and poor efficiency. The device used here, an optically addressable parallel-aligned nematic LCD, is not pixelated, and can achieve continuously variable phase modulation. The LCSLM used was a parallel-aligned spatial light modulator (X5641 series, Hamamatsu Photonics KK) that has a light sensitive layer of amorphous silicon to control the amplitude of an ac electric field across a parallelaligned liquid crystal (LC) layer. A multilayer dielectric mirror is placed in between the amorphous silicon and the LC layer. The ac field comes from transparent electrodes placed on the outside of the amorphous silicon– mirror – LC layer stack. Write-light to the device impinges upon the amorphous layer and reduces its local resistivity. This reduced resistivity causes a rise in the electric field across the LC layer, changing the orientation of the molecules and the refractive index.
Read-light entering through the LC layer is reflected off the dielectric mirror and leaves the device after passing through the LC layer again. The double pass through the modulated layer causes spatial phase modulation to appear on the polarization component of the outgoing light beam parallel to the LC directors. The device has a spatial resolution of , 30 line pairs/mm over an 18 £ 18 mm2 square aperture with a maximum phase modulation of , 1:5l at 633 nm. The LCSLM has two modes of operation, both providing phase-only modulation; an intensity modulation mode and a phase modulation mode. Phase-only modulation of high accuracy is obtained with the intensity modulation mode as used in this experiment. In this mode the laser light is passed through a polarizer to align the polarization of the laser light. Prior to detection an analyzer is used to select only one polarization because the direction of alignment of the liquid crystal molecules is at 458 to the bench top while the laser light is polarized perpendicular to the bench top. In comparison, when using phase modulation mode, an analyzer is not necessary because both the laser light and direction of alignment of the liquid crystal molecules are perpendicular to the bench top. As such, the LCSLM itself does not change the polarization of light.
3. The optical system The optical system consists of two light sources, providing a read-beam and a write-beam, respectively. The read-beam is modulated in phase by the LCSLM. The write-beam, in conjunction with an input mask in our experiment, determines the modulation pattern of the readbeam. The optical system for attaining phase modulation using the LCSLM is quite simple and shown in Fig. 1. The read-beam from a 633 mm 10 mW HeNe laser, polarized perpendicular to the optical bench, is focused through a pinhole spatial filter, using a microscope objective ( £ 16). The light from the pinhole then expands, and is collimated using a 336 mm focal length doublet lens. The expanded beam passes through a beam splitter on to the LCSLM. Since an intensity modulation mode is being used, an analyzer is inserted after reflection from the LCSLM to select the appropriate polarization. The light is focused using a £ 10 microscope objective. An aperture stop is used to limit aberrations from the focusing lens. The focused spot is then magnified on to a CCD camera, coupled to a computer. The write-beam used to modulate the phase of the readbeam, is collimated from a point like white light source. The input filter and the LCSLM are placed very close to one another, ensuring that a copy of the input filter is projected onto the LCSLM. The Fresnel number of the write-beam setup is large, hence Fresnel diffraction is negligible and the polychromatic nature of the white light source does not
J. McOrist et al. / Micron 34 (2003) 327–332
329
Fig. 1. The optical system used to produce phase-only hyperresolving masks.
cause any deterioration of the filter pattern written onto the LCSLM. The value of the phase change induced by the transparent regions of the input filter can be altered continuously by adjusting the brightness of the writebeam. The filters used at first were simply photocopied transparencies, and were binary in nature (black/white). To demonstrate the method, initially a two zone Toraldo filter was used, with an opaque inner zone and a transparent outer zone. As the phase change induced by the LCSLM is dependent only on the write-light intensity, the transparent zone is a region of phase change that was set to p radians, while the opaque, inner zone is a region of no phase change.
4. Results and discussion The input filters are characterized in terms of normalized inner radii, as that is given by the diameter of the inner opaque region divided by the full diameter of the input filter. The normalized inner radius a of the input filter can be varied from a ¼ 0; when the input filter is totally transparent, to a ¼ 1; when the input filter is totally opaque. Fig. 2 shows the input filter with a ¼ 0:3 and 0.6. The experimental input filters had a full diameter of 10 mm. Hence the diameter of the opaque regions were 3 mm for the a ¼ 0:3 input filter and 6 mm for the a ¼ 0:6 input filter. The active area of the LCSLM is 20 £ 20 mm2. Images taken by the CCD camera with input filters of a ¼ 0; 0:3 and 0.6 are shown in Fig. 3. The CCD images are 6 £ 6 mm2 corresponding to 230 £ 230 mm2 in the focal plane of the microscope objective. We note that
Fig. 2. The input filters with: (a) a ¼ 0:3; (b) a ¼ 0:6: An input filter with a ¼ 0 corresponds to an unobstructed aperture.
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Fig. 3. Measured intensity point spread functions: (a) a ¼ 0; (b) a ¼ 0:3; (c) a ¼ 0:6:
the width of the central lobe reduces as the value of a increases, giving hyperresolved images. However, the strength of the side lobes increases. This is just visible in the image for a ¼ 0:3:
Fig. 4. Calculated intensity point spread functions: (a) a ¼ 0; (b) a ¼ 0:3; (c) a ¼ 0:6:
For comparison purposes, the theoretical intensity pattern has been calculated using 2 ð1 ða J0 ðvrÞr dr 2 2 J0 ðvrÞr dr 0 0 I¼ 1 ð1 2 2a2 Þ2 4
ð1Þ
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Fig. 5. A comparison of a line scan through the experimental results with theoretical results for a ¼ 0:6: We have used five point averages of a single line scan of the experimental results.
where r is a normalized radial coordinate and n is a radial optical coordinate given by v ¼ kr sin a
ð2Þ
with k ¼ 2p=l; a numerical aperture of sin a and a radial distance of r: The theoretical intensity patterns for a ¼ 0; 0:3 and 0.6 are shown in Fig. 4 and are in agreement with experimental results shown in Fig. 3. We note that the effect of the aperture stop in the detection arm has been taken into consideration. The dimensions of the images in Figs. 3 and 4 are directly comparable. To further confirm the agreement, Fig. 5 compares a line scan through the experimental results with theoretical results for a ¼ 0:6: Due to apodization effects, the side lobes are depressed in the experimental results in comparison to theoretical calculations. From theoretical considerations the size of the central lobe decreases as the normalized inner radius a of the input filter is increased in the range 0 – 221/2. This comes at the expense of an increased intensity in the side lobes. For values of a greater than 221/2 the filter becomes an apodizer. Fig. 6 shows the normalized position of the first zero as a function of a for a two zone Toraldo phase filter
Fig. 6. Normalized position of the first zero as a function of the normalized inner radius, a; for a two zone Toraldo phase filter with a phase change of p: The width of the central lobe according to classical theory is indicated by the horizontal dashed line while the transition from hyperresolution to apodization is indicated by the dashed vertical line.
331
Fig. 7. The normalized height of the first side lobe as a function of a for a two zone Toraldo phase filter with a phase change of p: The height of the first side lobe according to classical theory is indicated by the horizontal dashed line while the transition from hyperresolution to apodization is indicated by the dashed vertical line.
with a phase change of p: The graph has a singularity along a ¼ 221=2 when the filter transits from being a hyperresolving one to an apodizing one. For a , 221=2 ; the filter has a central lobe smaller than predicted by classical theory, i.e. that predicted when not considering pupil plane phase modifications. The side lobes increase in intensity as the inner radius increases. Fig. 7 shows the normalized height of the first side lobe as a function of a for a two zone Toraldo phase filter with a phase change of p: As the hyperresolving effect of the input filter increases, the height of the first side lobe increases. For such filters the height of the first side lobe never goes below the value predicted by classical theory. The optimum value of a for a particular hyperresolving application depends on the trade-off between required width of central lobe and the acceptable height of first side lobe.
5. Conclusion We have used an optically addressable parallel-aligned nematic LCD with two zone Toraldo type filters to produce hyperresolved images which have been compared with theoretical predictions. The optically addressable LCSLM has the advantage for generating hyperresolving filters that it can be operated in a continuously variable phase-only mode, at high efficiency, and can be controlled dynamically using amplitude-modulated white light. The parallelaligned device exhibits decoupled phase and amplitude modulation. In addition because of optical write there is no amplitude modulation of the read beam resulting from no pixelation effects.
Acknowledgements The authors would like to thank the Science Foundation for Physics within the University of Sydney.
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References Akduman, I., Brand, U., Hester, G., Grochmalicki, J., Pike, R., Bertero, M., 1998. Superresolving masks for incoherent high-numerical-aperture scanning microscopy in three dimensions. J. Opt. Soc. A 15, 2275–2287. Barnes, T.H., Eiju, T., Matsuda, K., Ooyama, N., 1989. Phase-only modulation using a twisted nematic liquid crystal television. Appl. Opt. 28, 4845–4852. Campos, J., Marquez, A., Nicolas, J., Moreno, I., Iemmi, C., Davis, J.A., Yzuel, M.J., 2002. Characterization of the liquid crystal display modulation: optimization for some applications. ACTA Physica Polonica 101, 189–200. Carcole´, E., Campos, J., Bosch, S., 1994. Diffraction theory of Fresnel lenses encoded in low-resolution devices. Appl. Opt. 33, 162– 174. Cox, I.J., 1984. Increasing the bit packing densities of optical disk systems. Appl. Opt. 23, 3260–3261. Cox, I.J., Sheppard, C.J.R., 1986. Information capacity and resolution in an optical system. J. Opt. Soc. Am. A 3, 1152–1158. Cox, I.J., Sheppard, C.J.R., Wilson, T., 1982. Reappraisal of arrays of concentric annuli as superresolving filters. J. Opt. Soc. Am. 72, 1287–1291. Davis, J.A., Moreno, I., Tsai, P., 1998. Polarization eigenstates for twistednematic liquid-crystal displays. Appl. Opt. 37, 937 –945. Davis, J.A., Escalera, J.C., Marquez, A., 1999. Programmable axial apodizing and hyperresolving amplitude filters with a liquid-crystal spatial light modulator. Opt. Lett. 24, 628 –630. De Bougrenet de la Tocnaye, J.L., Dupont, L., 1997. Complex amplitude modulation by use of liquid-crystal spatial light modulators. Appl. Opt. 36, 1730–1741. Deng, X., Liu, L., Wang, G., Xu, Z., 2000. Superresolving phase-only filters in confocal scanning imaging system. Optik 111, 503 –507. Freeman, M.O., Brown, T.A., Walba, D.M., 1992. Quantized complex ferroelectric liquid crystal spatial light modulators. Appl. Opt. 31, 3917–3929. Hegedus, Z.S., 1985. Annular pupil arrays: applications to confocal microscopy. Opt. Acta 32, 815–826.
Hild, R., Yzuel, M.J., Escalera, J.C., Campos, J., 1998. Influence of nonuniform pupils in imaging periodical structures by photolithographic systems. Opt. Engng 37, 1353– 1363. Jacquinot, P., Roizen-Dossier, M.B., 1964. Apodisation. Progr. Opt. 3, 29 –186. Konforti, N., Marom, E., Wu, S.-T., 1988. Phase-only modulation with twisted nematic liquid-crystal spatial light modulators. Opt. Lett. 13, 251 –253. Lu, K., Saleh, B.E.A., 1990. Theory and design of the liquid crystal TV as an optical spatial phase modulator. Opt. Engng 29, 240–246. Marquez, A., Lemmi, C., Escalera, J.C., Campos, J., Ledesma, S., Davis, J.A., Yzuel, M.J., 2001. Amplitude apodizers encoded onto Fresnel lenses implemented on a phase-only spatial light modulator. Appl. Opt. 40, 2316–2322. Martinez-Corral, M., Andre`s, P., Zapata-Rodriguez, C.J., Sheppard, C.J.R., 1998. Improvement of three-dimensional resolution in confocal scanning micrsocopy by combination of two pupil filters. Optik 107, 145 –148. Mukohzaka, N., Yoshida, N., Toyoda, H., Kobayashi, Y., Hara, Tsutomu, 1994. Diffraction efficiency analysis of a parallel-aligned nematicliquid-crystal spatial light modulator. Appl. Opt. 33, 2804–2811. Pezzaniti, J.L., Chipman, R.A., 1993. Phase-only modulation of a twisted nematic liquid-crystal TV by use of the eigenpolarization states. Opt. Lett. 18, 1567–1569. Sales, T.R.M., Morris, G.M., 1997. Diffractive superresolving elements. J. Opt. Soc. Am. A 14, 1637–1646. Sheppard, C.J.R., 1995. Leaky annular pupils for improved axial imaging. Optik 99, 32–34. Sheppard, C.J.R., Hegedus, Z.S., 1988. Axial behavior of pupil-plane filters. J. Opt. Soc. Am. A 5, 643 –647. Sheppard, C.J.R., Calvert, G., Wheatland, M., 1998. Focal distribution for superresolving toraldo filters. J. Opt. Soc. Am. A 15, 849–855. Stolz, C., Bigue, L., Ambs, P., 2001. Implementation of high-resolution diffractive optical elements on coupled phase and amplitude spatial light modulators. Appl. Opt. 40, 6415– 6423. Toraldo di Francia, G., 1952. Nuovo pupille superrisolvente. Atti. Fond. Giorgio Ronchi 7, 366–372.
Hyperresolving phase-only filters with an optically addressable liquid crystal spatial light modulator J. McOrist, M.D. Sharma, C.J.R. Sheppard*,1, E. West, K. Matsuda Physical Optics Laboratory, School of Physics, University of Sydney, Sydney, NSW 2006, Australia
Abstract Hyperresolving (sometimes called ‘superresolving’ or ‘ultraresolving’) phase-only filters can be generated using an optically addressable liquid crystal spatial light modulator. This approach avoids the problems of low efficiency, and coupling between amplitude and phase modulation, that arise when using conventional liquid crystal modulators. When addressed by a programmed light intensity distribution, it allows filters to be changed rapidly to modify the response of a system or permit the investigation of different filter designs. In this paper we present experimental hyperresolved images obtained using an optically addressable parallel-aligned nematic LCD with two zone Toraldo type phase-only filters. The images are compared with theoretical predictions. q 2003 Elsevier Ltd. All rights reserved. Keywords: Hyperresolving; Filters; Liquid crystal spatial light modulator
1. Introduction Recently, Davis et al. demonstrated the use of a twisted nematic liquid crystal spatial light modulator (LCSLM) for generating hyperresolving and apodizing amplitude filters (Davis et al., 1999). Here we demonstrate the generation of phase-only filters using an optically addressable LCSLM. Experimental results clearly demonstrate hyperresolution using a two zone phase-only filter. Applications for hyperresolving filters include increasing the recording density in optical disks (Cox, 1984) and enhanced resolution in photolithographic (Hild et al., 1998) and confocal microscope systems (Hegedus, 1985; Sheppard, 1995; Martinez-Corral et al., 1998; Akduman, 1998; Deng et al., 2000). The term hyperresolution, in general, can be defined as the effect of reducing the width of the central lobe of a focal spot below the classical limit. In practice there is a trade-off, between the decrease in width of the central lobe and the increase in level of the outer lobes, that determines the effectiveness of the system and is driven by system requirements. The opposite effect of increasing * Corresponding author. Tel.: þ 31-15-27-89407; fax: þ31-15-27-88105. E-mail address: [email protected] (C.J.R. Sheppard). 1 On leave at: Optics Group, Department of Applied Physics, Technical University of Delft, Lorentzweg 1, Delft CD 2628, Netherlands 0968-4328/03/$ - see front matter q 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0968-4328(03)00059-3
the width of the central lobe and reducing the level of the outer lobes is called apodization (Jacquinot and Roizen-Dossier, 1964). Hyperresolution is sometimes called ‘superresolution’ or ‘ultraresolution’, but the term superresolution is best reserved for the more restrictive case when the spatial frequency bandwidth of the system is increased above the classical limit (Cox and Sheppard, 1986). Both hyperresolution and apodization are typically achieved by using a filter (or mask) in the pupil of the system, the design and properties of the filter determining the point spread function of the focused spot. The first hyperresolving masks, based on concentric rings of oscillating sign, were proposed by Toraldo di Francia (Toraldo di Francia, 1952), who showed that the central lobe can be made arbitrarily narrow. The behaviour of the intensity in the focal plane (Cox et al., 1982) and along the optic axis (Sheppard et al., 1998) for Toraldo filters has been presented. A simple approach, based on pupil moments, to determine performance parameters for realvalued filters has also been given (Sheppard and Hegedus, 1988). Amplitude-only filters can give only a modest gain. Sales has described how use of phase-only filters, with phase changes of other than p radians, can result in improved overall performance (Sales and Morris, 1997). In this paper we describe the use of an optically addressable parallel-aligned nematic LCSLM to produce
328
J. McOrist et al. / Micron 34 (2003) 327–332
phase-only filters which are used to demonstrate hyperresolution. The experimentally obtained hyperresolved point spread functions are compared with theoretical predictions.
2. The liquid crystal display There are ferroelectric, smectic and nematic liquid crystal displays (LCD). We will not dwell on ferroelectric or smectic LCD. However, smectic LCD have been discussed for phase modulation (De Bougrenet and Dupont, 1997) and ferroelectric LCSLMs can give phase-only modulation (Freeman et al., 1992), but suffer from low efficiency. The nematic LCD are available as either twisted or parallel-aligned. In addition, both kinds of nematic LCD can be either optically or electrically addressable. The electrically and optically addressable twisted nematic LCD usually produce coupling between amplitude and phase (Lu and Saleh, 1990; Konforti et al., 1988; Barnes et al., 1989; Marquez et al., 2001). However, under correct operating conditions, these devices are capable of phase-only operation (Pezzaniti and Chipman, 1993; Davis et al., 1998; Campos et al., 2002). On the other hand, the electrically addressable parallel-aligned nematic LCD do not exhibit coupling between amplitude and phase. However, Stolz et al. (2001) have theoretically investigated coupled phase and amplitude (spiral) domains with electrically addressable parallel-aligned nematic LCD. The optically addressable parallel-aligned nematic LCD, also called zero twist or electrically controlled birefringent devices, can produce pure phase modulation. Mukohzaka et al. (1994) investigated the diffraction efficiency, intensity versus phase transfer characteristics and response time for the optically addressable parallel-aligned nematic LCD. For a twisted nematic LCSLM, amplitude modulation occurs as a result of pixelation (Carcole´ et al., 1994). Other problems are dynamic range and poor efficiency. The device used here, an optically addressable parallel-aligned nematic LCD, is not pixelated, and can achieve continuously variable phase modulation. The LCSLM used was a parallel-aligned spatial light modulator (X5641 series, Hamamatsu Photonics KK) that has a light sensitive layer of amorphous silicon to control the amplitude of an ac electric field across a parallelaligned liquid crystal (LC) layer. A multilayer dielectric mirror is placed in between the amorphous silicon and the LC layer. The ac field comes from transparent electrodes placed on the outside of the amorphous silicon– mirror – LC layer stack. Write-light to the device impinges upon the amorphous layer and reduces its local resistivity. This reduced resistivity causes a rise in the electric field across the LC layer, changing the orientation of the molecules and the refractive index.
Read-light entering through the LC layer is reflected off the dielectric mirror and leaves the device after passing through the LC layer again. The double pass through the modulated layer causes spatial phase modulation to appear on the polarization component of the outgoing light beam parallel to the LC directors. The device has a spatial resolution of , 30 line pairs/mm over an 18 £ 18 mm2 square aperture with a maximum phase modulation of , 1:5l at 633 nm. The LCSLM has two modes of operation, both providing phase-only modulation; an intensity modulation mode and a phase modulation mode. Phase-only modulation of high accuracy is obtained with the intensity modulation mode as used in this experiment. In this mode the laser light is passed through a polarizer to align the polarization of the laser light. Prior to detection an analyzer is used to select only one polarization because the direction of alignment of the liquid crystal molecules is at 458 to the bench top while the laser light is polarized perpendicular to the bench top. In comparison, when using phase modulation mode, an analyzer is not necessary because both the laser light and direction of alignment of the liquid crystal molecules are perpendicular to the bench top. As such, the LCSLM itself does not change the polarization of light.
3. The optical system The optical system consists of two light sources, providing a read-beam and a write-beam, respectively. The read-beam is modulated in phase by the LCSLM. The write-beam, in conjunction with an input mask in our experiment, determines the modulation pattern of the readbeam. The optical system for attaining phase modulation using the LCSLM is quite simple and shown in Fig. 1. The read-beam from a 633 mm 10 mW HeNe laser, polarized perpendicular to the optical bench, is focused through a pinhole spatial filter, using a microscope objective ( £ 16). The light from the pinhole then expands, and is collimated using a 336 mm focal length doublet lens. The expanded beam passes through a beam splitter on to the LCSLM. Since an intensity modulation mode is being used, an analyzer is inserted after reflection from the LCSLM to select the appropriate polarization. The light is focused using a £ 10 microscope objective. An aperture stop is used to limit aberrations from the focusing lens. The focused spot is then magnified on to a CCD camera, coupled to a computer. The write-beam used to modulate the phase of the readbeam, is collimated from a point like white light source. The input filter and the LCSLM are placed very close to one another, ensuring that a copy of the input filter is projected onto the LCSLM. The Fresnel number of the write-beam setup is large, hence Fresnel diffraction is negligible and the polychromatic nature of the white light source does not
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329
Fig. 1. The optical system used to produce phase-only hyperresolving masks.
cause any deterioration of the filter pattern written onto the LCSLM. The value of the phase change induced by the transparent regions of the input filter can be altered continuously by adjusting the brightness of the writebeam. The filters used at first were simply photocopied transparencies, and were binary in nature (black/white). To demonstrate the method, initially a two zone Toraldo filter was used, with an opaque inner zone and a transparent outer zone. As the phase change induced by the LCSLM is dependent only on the write-light intensity, the transparent zone is a region of phase change that was set to p radians, while the opaque, inner zone is a region of no phase change.
4. Results and discussion The input filters are characterized in terms of normalized inner radii, as that is given by the diameter of the inner opaque region divided by the full diameter of the input filter. The normalized inner radius a of the input filter can be varied from a ¼ 0; when the input filter is totally transparent, to a ¼ 1; when the input filter is totally opaque. Fig. 2 shows the input filter with a ¼ 0:3 and 0.6. The experimental input filters had a full diameter of 10 mm. Hence the diameter of the opaque regions were 3 mm for the a ¼ 0:3 input filter and 6 mm for the a ¼ 0:6 input filter. The active area of the LCSLM is 20 £ 20 mm2. Images taken by the CCD camera with input filters of a ¼ 0; 0:3 and 0.6 are shown in Fig. 3. The CCD images are 6 £ 6 mm2 corresponding to 230 £ 230 mm2 in the focal plane of the microscope objective. We note that
Fig. 2. The input filters with: (a) a ¼ 0:3; (b) a ¼ 0:6: An input filter with a ¼ 0 corresponds to an unobstructed aperture.
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Fig. 3. Measured intensity point spread functions: (a) a ¼ 0; (b) a ¼ 0:3; (c) a ¼ 0:6:
the width of the central lobe reduces as the value of a increases, giving hyperresolved images. However, the strength of the side lobes increases. This is just visible in the image for a ¼ 0:3:
Fig. 4. Calculated intensity point spread functions: (a) a ¼ 0; (b) a ¼ 0:3; (c) a ¼ 0:6:
For comparison purposes, the theoretical intensity pattern has been calculated using 2 ð1 ða J0 ðvrÞr dr 2 2 J0 ðvrÞr dr 0 0 I¼ 1 ð1 2 2a2 Þ2 4
ð1Þ
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Fig. 5. A comparison of a line scan through the experimental results with theoretical results for a ¼ 0:6: We have used five point averages of a single line scan of the experimental results.
where r is a normalized radial coordinate and n is a radial optical coordinate given by v ¼ kr sin a
ð2Þ
with k ¼ 2p=l; a numerical aperture of sin a and a radial distance of r: The theoretical intensity patterns for a ¼ 0; 0:3 and 0.6 are shown in Fig. 4 and are in agreement with experimental results shown in Fig. 3. We note that the effect of the aperture stop in the detection arm has been taken into consideration. The dimensions of the images in Figs. 3 and 4 are directly comparable. To further confirm the agreement, Fig. 5 compares a line scan through the experimental results with theoretical results for a ¼ 0:6: Due to apodization effects, the side lobes are depressed in the experimental results in comparison to theoretical calculations. From theoretical considerations the size of the central lobe decreases as the normalized inner radius a of the input filter is increased in the range 0 – 221/2. This comes at the expense of an increased intensity in the side lobes. For values of a greater than 221/2 the filter becomes an apodizer. Fig. 6 shows the normalized position of the first zero as a function of a for a two zone Toraldo phase filter
Fig. 6. Normalized position of the first zero as a function of the normalized inner radius, a; for a two zone Toraldo phase filter with a phase change of p: The width of the central lobe according to classical theory is indicated by the horizontal dashed line while the transition from hyperresolution to apodization is indicated by the dashed vertical line.
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Fig. 7. The normalized height of the first side lobe as a function of a for a two zone Toraldo phase filter with a phase change of p: The height of the first side lobe according to classical theory is indicated by the horizontal dashed line while the transition from hyperresolution to apodization is indicated by the dashed vertical line.
with a phase change of p: The graph has a singularity along a ¼ 221=2 when the filter transits from being a hyperresolving one to an apodizing one. For a , 221=2 ; the filter has a central lobe smaller than predicted by classical theory, i.e. that predicted when not considering pupil plane phase modifications. The side lobes increase in intensity as the inner radius increases. Fig. 7 shows the normalized height of the first side lobe as a function of a for a two zone Toraldo phase filter with a phase change of p: As the hyperresolving effect of the input filter increases, the height of the first side lobe increases. For such filters the height of the first side lobe never goes below the value predicted by classical theory. The optimum value of a for a particular hyperresolving application depends on the trade-off between required width of central lobe and the acceptable height of first side lobe.
5. Conclusion We have used an optically addressable parallel-aligned nematic LCD with two zone Toraldo type filters to produce hyperresolved images which have been compared with theoretical predictions. The optically addressable LCSLM has the advantage for generating hyperresolving filters that it can be operated in a continuously variable phase-only mode, at high efficiency, and can be controlled dynamically using amplitude-modulated white light. The parallelaligned device exhibits decoupled phase and amplitude modulation. In addition because of optical write there is no amplitude modulation of the read beam resulting from no pixelation effects.
Acknowledgements The authors would like to thank the Science Foundation for Physics within the University of Sydney.
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