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Wavefront correction using a self-referencing phase conjugation system based on a Zernike cell
1 May 2001
Optics Communications 191 (2001) 31±38
www.elsevier.com/locate/optcom
Wavefront correction using a self-referencing phase conjugation system based on a Zernike cell S.R. Dale a, G.D. Love a,b,*, R.M. Myers a, A.F. Naumov a a b
Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK School of Engineering, University of Durham, South Road, Durham DH1 3LE, UK Received 4 January 2001; accepted 15 February 2001
Abstract A wavefront correction system was produced using a phase conjugating method based on a point diraction interferometer, or Zernike cell. This has the main advantage that no separate reference beam is required. The PDI was constructed using an optically addressed liquid crystal spatial light modulator, which means that the system is also relatively easy to align, and insensitive to tip/tilt errors. We demonstrate phase correction of induced high spatial frequency aberrations, and show wavefront correction leading to a 3 improvement in Strehl ratio. We also discuss the utility of the spatial light modulator as a common-path point diraction interferometer Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Phase conjugation; Liquid crystals; Adaptive optics; Spatial light modulators
1. Introduction Optical phase conjugation can be used to correct aberrations in optical systems, whereby a beam of light is distorted by an aberrating medium, passes through the phase conjugate system to generate a beam with the opposite phase distortion, and then re-passes through the distorting media to produce a ¯at unaberrated wavefront. The general technique of optical phase conjugation is described, for example, by Goodman [1]. The work in this paper is based on an optically
* Corresponding author. Address: Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK. Fax: +44-191-374-3749. E-mail address: [email protected] (G.D. Love).
addressed liquid crystal (LC) spatial light modulator (OA-SLM) and a number of authors have published work on their use in phase conjugate systems [2±6]. All these systems achieve phase conjugation based on the holographic technique described by Goodman [1]. In this paper we present results of a system ®rst described theoretically and numerically by Sherstobitov and coworkers [7±11] in which phase conjugation is achieved using a phase-shifting 1 point diraction interferometer (PDI) [12]. Some initial experimental results using 1 Some authors use the term, ``point diraction interferometer'' to refer only to the type of interferometer with an amplitude mask, whereas they use the term ``Zernike interferometer'' to refer to the type with a phase mask. In this paper we use the terms interchangeably, and we explicitly state that it is a phase-shifting point diraction interferometer.
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 1 1 6 - 6
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S.R. Dale et al. / Optics Communications 191 (2001) 31±38
the technique were presented in Ref. [13]. Here we show explicitly results of phase conjugation using the Zernike cell, and also demonstrate the use of an OA-SLM as a PDI. This means that the system has a number of advantages. The general technique of using the Zernike cell has the advantages that it operates with low-power beams, it can correct high spatial frequency aberrations, and it does not require a separate reference beam (selfreferencing wavefront correction using a dierent technique was also recently described in Ref. [14]). It has the disadvantages that it can only correct light from a single point source (i.e. zero ®eld of view) and is therefore suitable for laser correction, rather than imaging, and it introduces amplitude modulation into the beam (although we show this is not necessarily a problem). The use of an OASLM as the phase spot means that the system is relatively simple to align and is insensitive to wavefront slopes, and also means that the size of the phase diracting spot adjusts automatically to account for varying aberration amplitudes.
2. System analysis The basic optical con®guration for phase reversal is shown in Fig. 1. The system consists of a conventional phase-shifting PDI in a double-pass arrangement. The ®rst lens focuses the incident light onto the phase ®lter, which is produced by the OA-SLM. The second lens recollimates the beam, and the mirror re¯ects the light back through the system. LC PDIs have been described by others [15,16] which allow quantitative measurements to be made by phase-shifting interferometry. These designs, along with the conventional PDI, all re-
quire accurate alignment of spatial ®lter and input beams. The use of an OA-SLM means that the phase spot position and size automatically coincides with the focal spot from the lens. An analytical analysis of the system is presented by Sherstobitov [7], but here we present a simpler heuristical treatment which gives a clearer physical insight into the phase conjugation mechanism. The aberrated input wavefront can be represented, for small values of the aberration, h, as win Aeih A 1 ih:
1
The ®rst lens Fourier transforms this input onto the spatial ®lter. Since the Fourier transform, I, is functionally distributive, then, IfWin g If Ag If Aihg:
2
The plane wave component of the input transforms passes through the Airy disk which falls on the p=2 phase-shifting central spot. The second lens and mirror combination means that the beam passes through the phase spot twice, giving a total phase shift of p, and therefore we can write the output beam as, 2 IfWout g eip=2 If Ag If Aihg
If Ag If Aihg:
3
Therefore, wout A 1 ih;
4
which (from the point of view of imaging) is equivalent to wout A 1
ih win :
5
Hence the output beam is the conjugate of the input.
3. System simulation
Fig. 1. Experimental arrangement of the basic phase conjugating method based on the phase-shifting PDI, or Zernike cell.
We simulated the performance of the system to investigate the quality of the phase conjugation. The input wavefront was described by a constant amplitude within the aperture multiplied by the complex phase term. This was Fourier transformed to give the amplitude distribution in the
S.R. Dale et al. / Optics Communications 191 (2001) 31±38
®lter plane, which was multiplied by the phase factor to allow for the central phase-shifting spot, /spot , given by /spot eip=2 /spot 1
for r < rspot ;
for r P rspot ;
6
where rspot is the phase-shifting spot radius, and r is the radial polar coordinate in the plane of the phase mask. This modi®ed amplitude was then Fourier transformed, and then back propagated through the system. The phase values of the input and output signals should be opposite in sign if phase conjugation has occurred, and therefore they were added together to give a wavefront error array. In the simulation we investigated the conjugation quality as a function of both the amplitude and the spatial frequency of the aberration. The input aberrations were azimuthally symmetric functions of the form, /in A cos 2pur;
7
Fig. 2. Plots of the input and output beam intensity and phases. Top left: input beam intensity (i.e. a top-hat function). Top right: input beam phase. Bottom left: output beam intensity pattern. The system has clearly amplitude modulated the beam. However this has a relatively weak eect on the beam quality. Bottom right: output beam phase. Note that it is the conjugate of the input phase. The spatial frequency of the aberration u 3 cycles/aperture radius, and amplitude 0:4 waves P±V.
33
where A is the applied amplitude, r is the normalised radial polar coordinate in the pupil plane, and u is the spatial frequency. We display u in units of cycles/aperture radius so that the results do not depend on the overall aperture size. Fig. 2 demonstrates the basic results by plotting input and output beam intensities and phases. It is clear that the output phase has the conjugate shape to the input phase. It is also clear that the beam has been intensity modulated. This is not surprising, since this type of optical system is usually used to visualise phase. However, because the interferometer is of the phase-shifting type, then energy is conserved, and because amplitude modulation has a relatively weak eect on the ®nal focussed spot quality, then the system can still produce useful wavefront correction. Quantitative results are shown in Figs. 3 and 4, where the uncorrected and corrected Strehl ratios (calculated by using the Marechal approximation) are shown as a function of the amplitude, A, and spatial frequency, u of the aberration. Fig. 3 shows that for very small (<0.2 waves P±V) then the system actually degrades the Strehl ratio, and this is caused by the induced amplitude modulation. For intermediate amplitudes, then the system produces an increase in Strehl ratio. For large values of the aberration, then the approximation
Fig. 3. Conjugation quality versus aberration amplitude. The dotted line shows the Strehl ratio of the uncorrected beam, and the solid line shows the corrected Strehl ratio. The results are for a ®xed value of the spatial frequency of the input aberration (u 3 cycles/aperture radius).
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S.R. Dale et al. / Optics Communications 191 (2001) 31±38
quency that the system can correct for is governed only by the overall aperture of the system. 4. The OA-SLM as a Zernike cell
Fig. 4. Conjugation quality versus spatial frequency of the aberration (in cycles/aperture radius). The dotted line shows the Strehl ratio of the uncorrected beam, and the solid line shows the corrected Strehl ratio. The results are for a ®xed value of the amplitude of the input aberration (0.4 waves, P±V).
win Aeih A 1 ih used in Section 2 is no longer valid, and again the system does not work. Note that the Strehl ratio of the uncorrected beam increases when the amplitude > 0:8 waves. This is a real eect. Consider, for example, the on-axis beam intensity of a diraction limited beam, as a function of axial distance (or alternatively induced defocus), in which case the intensity is an oscillating function. Fig. 4 shows the conjugation quality versus spatial frequency, and indicates that the spatial frequency must be greater than a certain value in order for useful conjugation to occur. This is because for very small values of the spatial frequency, then the diracted beam is barely displaced from the Airy disk, and both terms pass through the phase-shifting spot, and phase conjugation does not occur. In this simulation (and indeed the previous simulations as well), the spot size was two times the Airy disk size, and thus wavefront correction does not occur until the spatial frequency of the aberration > 2 cycles/aperture radius (a spatial frequency of n cycles per radius will diract light into the nth Airy ring). This size was chosen empirically because it gave the best conjugation quality. Other authors have also noted that an apparently non-optimum spot size gives improved fringe visibility in a Zernike interferometer [17]. The maximum spatial fre-
The OA-SLM was ®rst set up as shown in Fig. 5 in order to test its use as a PDI, or Zernike cell. The OA-SLM consists of a number of thin ®lm layers sandwiched between two glass substrates: a photoconductor, a LC layer, alignment layers, transparent electrodes and a dielectric mirror. When a DC voltage is applied to the electrodes, it is divided between the photoconductor and the LC according to the amount of (write) light incident on the device. Therefore the induced phase modulation is proportional to the intensity of light incident on the device. The speci®cations of the SLM are shown in Table 1. An electrically addressed LC SLM with 69 pixels (Meadowlark Optics Hex69 device [18]) was used to generate test phase patterns. The light was focused onto the
Fig. 5. Experimental apparatus to test the use of the OA-SLM as a PDI, or Zernike cell. Table 1 Speci®cations of the optically addressed SLM. Type M19-T, produced by F. Vladimirov of the State Optical Institute, St. Petersburg, Russia Sensitivity Resolution at 5% MTF Resolution at 50% MTF Contrast ratio Grayscales, 0.15ND per step Phase shift Rise time Decay time Write light spectral region Read light spectral region Aperture size Voltage
0.1±0.01 lW/cm2 100±250 lp/mm 60±70 lp/mm 100:1 8 0±3p (HeNe) 1±10 ms 50±100 ms 400±500 nm 600±1100 nm 6 cm2 5±20 V (DC)
S.R. Dale et al. / Optics Communications 191 (2001) 31±38
Fig. 6. Results showing the OA-SLM used as a PDI in order to visualise the phase distortion (second order astigmatism) produced by an electrically addressed SLM.
SLM, and the zero order light produces a phase diracting spot. When the device is used with low voltages then the threshold voltage of the LC means that the phase spot has an approximately sharp boundary. An example resulting interferogram is shown in Fig. 6, in which the phase aberration was second order astigmatism. The magnitude and form of the aberration aects the fringe contrast in all common-path interferometers of this type. In general, unity contrast is only obtained when the amount of light passing through the phase discontinuity is equal to the integrated intensity from the higher order terms (see for example, Ref. [19]), which cannot be achieved here. A full discussion of phase visualisation by this type of intereferometer is given in Ref. [20]. Furthermore, in the case of the OA-SLM the induced phase shift in the diracting spot is a function of the overall intensity. Fig. 7 shows how the fringe contrast varies with beam power, and gives an indication of the useful working range of the device. The intensities were measured using an optical power meter. These results show that an OA-SLM can be used as a common-path interferometer with two advantages over a usual PDI. Firstly, the interferometer is insensitive to the overall alignment, and therefore is more stable, and secondly the fringe contrast can be adjusted by controlling the voltage applied to the OA-SLM.
35
Fig. 7. Interferogram contrast versus beam intensity on the SLM.
5. The OA-SLM as a phase conjugation system A phase conjugation system based on the system shown in Fig. 1 was assembled into one arm of a Michelson interferometer as shown in Fig. 8 so that phase conjugation could be observed by interfering the corrected beam with a ¯at reference wavefront. Again, an electrically addressed LC-SLM was used to generate aberrations, and the OA-SLM phase conjugation system then corrected these aberrations. The applied voltage to the OA-SLM was adjusted in order to optimise the correction, as indicated by the straightness of the resulting fringes. An example result showing correction of a phase shift produced by the actuation of a single LC pixel is shown in Fig. 9. We have shown just a single pixel because it clearly shows very high resolution phase conjugation by the correction of discontinuous fringes. Furthermore, in this con®guration the resulting intensity modulation negligible, and one can see the eects of phase conjugation clearly. The experimental system was then modi®ed to that shown in Fig. 10 in order to measure Strehl gains. The output of the phase conjugation system after its double pass through the aberrator, was focussed onto a pinhole, the size of which was chosen to pass only the central portion of the Airy pattern. An optical power meter measured the power of the light passing through the pinhole. Measurements were recorded with the
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S.R. Dale et al. / Optics Communications 191 (2001) 31±38
Fig. 8. Experimental con®guration showing the phase conjugation system used in one arm of a Michelson interferometer, used to generate the results shown in Fig. 9.
Fig. 9. Interferograms produced by the Michelson interferometer shown in Fig. 8, showing the eect of phase conjugation of the phase shift produced by a single LC pixel. PCM ± phase conjugate mirror. The beam power was 4.4 lW, the control voltage applied to the OA-SLM 6.6 V and the aberration amplitude k=4 (single pass).
phase conjugation system turned on and o, and the increase in Strehl ratio, Sgain , was de®ned as, Sgain
Scorrected Suncorrected Icorrected Iuncorrected ; Suncorrected Iuncorrected 8
where I is the recorded power by the meter, and the subscripts refer to the system being on and o. A phase aberration was produced by actuating alternate SLM pixels to produce a checker-board like pattern. This ensured that the aberration did not contain very low spatial fre-
S.R. Dale et al. / Optics Communications 191 (2001) 31±38
37
Fig. 10. Experimental apparatus used to measure gains in Strehl ratio of the phase conjugation system.
quencies that the system was unable to correct. Results were recorded using with induced (single pass) phase shift amplitudes of k=4 and 3k=4 with dierent beam powers. For each combination of aberration amplitude and beam power, the control voltage was adjusted to give the highest possible conjugation quality. Results are shown in Fig. 11. Strehl gains up to 300% were recorded. Gains were higher for the lower amplitude aberration. As expected from the results of Section 4, the gains were highest for beam powers in the range 10±100 mW.
6. Conclusions There are two separate main conclusions to this paper. First, we have shown that an optically addressed SLM can produce a phase discontinuity which can be used to produce a phase-shifting PDI, or a Zernike cell with the advantages of being insensitive to the beam alignment, and of having an electrically tuneable fringe contrast. Second, we have shown that such a device can be used as part of a phase conjugation system, used to correct wavefronts with spatial frequencies greater than minimum value which depends on the aperture size, and the size of the diracting spot. This system has the advantage of all phase conjugation systems over adaptive optics, i.e. the wavefront conjugation is automatic. Furthermore it is insensitive to beam alignment. For small changes in the aberration the system will adapt, but for large changes, then the voltage applied to the OASLM will need to be altered. For this reason this type of system would be useful for controlling slowly varying aberrations in, for example, laser systems.
Acknowledgements Fig. 11. Increases in Strehl ratio versus optical power for a number of realisations of the same experiment. The diamonds are for a peak±valley aberration amplitude of k=4, and the crosses are for a peak±valley aberration amplitude of 3k=4. The solid lines are connected through the mean values.
This work was performed as an M.Sci. undergraduate project in the Department of Physics at the University of Durham. Thanks to Fedor Vladimirov of the State Optical Institute, St. Petersburg, Russia for constructing the SLM.
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S.R. Dale et al. / Optics Communications 191 (2001) 31±38
References [1] J.W. Goodman, Introduction to Fourier Optics, McGraw Hill, 1968. [2] D.M. Pepper, C.J. Gaeta, P.V. Mitchell, Real-time holography, innovative adaptive optics, and compensated optical processors using spatial light modulators, in: U. Etran (Ed.), Spatial Light Modulator Technology, Marcel Dekker, 1995, pp. 585±665. [3] C.C. Mao, K.M. Johnson, G. Moddel, Optical phase conjugation using optically addressed liquid crystal light modulators, Opt. Lett. 15 (20) (1991) 1114±1116. [4] K.M. Johnson, C.C. Mao, G. Moddel, M.A. Handschy, K. Arnett, High-speed, low-power optical phase conjugation using a hybrid amorphous silicon/ferroelectric-liquid-crystal device, Opt. Lett. 15 (1990) 1114±1116. [5] E. Marom, U. Efron, Phase conjugation of low-power optical beams using liquid crystal light valves, Opt. Lett. 12 (7) (1987) 504±506. [6] M.A. Kramer, C.J. Wetterer, T. Martinez, One-way imaging through an aberrator with spatially incoherent light using an optically addressed spatial light modulator, Appl. Opt. 30 (23) (1991) 3319±3323. [7] V.E. Sherstobitov, Feasibility of compensation of smallscale inhomogeneities using a Zernike cell, Sov. J. Quant. Electron. 15 (1985) 50±53. [8] S.I. Kliment'ev, V.I. Kuprenyuk, V.E. Sherstobitov, Numerical modelling of a linear wavefront reversal system utilizing a spatial phase ®lter, Sov. J. Quant. Electron. 17 (5) (1987) 640±643. [9] S.I. Kliment'ev, V.I. Kuprenyuk, V.E. Sherstobitov, Numerical simulation of a resonator with a phase-conjugate mirror based on a Fourier phase corrector in the case of a low Fresnel number, Sov. J. Quant. Electron. 18 (1) (1988) 102±104. [10] S.I. Kliment'ev, V.I. Kuprenyuk, V.V. Lyubimov, V.E. Sherstobitov, Numerical simulation of an unstable ring
[11]
[12] [13]
[14] [15] [16] [17]
[18] [19]
[20]
resonator with a Fourier phase corrector, Sov. J. Quant. Electron. 19 (4) (1989) 538±542. S.I. Kliment'ev, V.I. Kuprenyuk, V.V. Lyubimov, V.E. Sherstobitov, Oscillation modes in a resonator with a selector based on a Fourier phase corrector, Sov. J. Quant. Electron. 19 (1) (1989) 73±77. R.N. Smartt, Special applications of the point diraction interferometer, Proc. SPIE 192 (1979) 35±40. S.I. Kliment'ev, V.V. Kononov, V.I. Kuprenyuk, L.D. Smirnova, V.V. Sergeev, V.E. Sherstobitov, Compensation for small-scale phase distortions using a Fourier phase corrector, Sov. J. Quant. Electron. 15 (12) (1985) 1657± 1659. T. Shirai, T.H. Barnes, T.G. Haskell, Adaptive wave-front correction by means of all-optical feedback interferometry, Opt. Lett. 25 (11) (2000) 773±775. C.R. Mercer, K. Creath, Liquid-crystal point-diraction interferometer for wave-front measurements, Appl. Opt. 35 (1996) 1633±1642. H. Kadono, M. Ogusu, S. Toyooka, Phase shifting common path interferometer using a liquid-crystal phase modulator, Opt. Commun. 110 (1994) 391±400. G.W. Carhart, M.A. Vorontsov, E.W. Justh, Opto-electronic Zernike ®lter for high resolution wavefront analysis using a phase only liquid crystal spatial light modulator, Proc. Soc. Photo-Opt. Instrum. Eng. 4124 (2000) 138± 147. G.D. Love, Wavefront correction and production of Zernike modes with a liquid crystal SLM, Appl. Opt . 36 (7) (1997) 1517±1524. P.M. Birch, J. Gourlay, G.D. Love, A. Purvis, Real time aberration correction with a ferroelectric liquid crystal spatial light modulator, Appl. Opt. 37 (11) (1998) 2164± 2169. J. Gl uckstad, P.C. Mogensen, Optimal phase contrast in common-path interferometry, Appl. Opt. 40 (2) (2001) 268±282.
Optics Communications 191 (2001) 31±38
www.elsevier.com/locate/optcom
Wavefront correction using a self-referencing phase conjugation system based on a Zernike cell S.R. Dale a, G.D. Love a,b,*, R.M. Myers a, A.F. Naumov a a b
Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK School of Engineering, University of Durham, South Road, Durham DH1 3LE, UK Received 4 January 2001; accepted 15 February 2001
Abstract A wavefront correction system was produced using a phase conjugating method based on a point diraction interferometer, or Zernike cell. This has the main advantage that no separate reference beam is required. The PDI was constructed using an optically addressed liquid crystal spatial light modulator, which means that the system is also relatively easy to align, and insensitive to tip/tilt errors. We demonstrate phase correction of induced high spatial frequency aberrations, and show wavefront correction leading to a 3 improvement in Strehl ratio. We also discuss the utility of the spatial light modulator as a common-path point diraction interferometer Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Phase conjugation; Liquid crystals; Adaptive optics; Spatial light modulators
1. Introduction Optical phase conjugation can be used to correct aberrations in optical systems, whereby a beam of light is distorted by an aberrating medium, passes through the phase conjugate system to generate a beam with the opposite phase distortion, and then re-passes through the distorting media to produce a ¯at unaberrated wavefront. The general technique of optical phase conjugation is described, for example, by Goodman [1]. The work in this paper is based on an optically
* Corresponding author. Address: Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK. Fax: +44-191-374-3749. E-mail address: [email protected] (G.D. Love).
addressed liquid crystal (LC) spatial light modulator (OA-SLM) and a number of authors have published work on their use in phase conjugate systems [2±6]. All these systems achieve phase conjugation based on the holographic technique described by Goodman [1]. In this paper we present results of a system ®rst described theoretically and numerically by Sherstobitov and coworkers [7±11] in which phase conjugation is achieved using a phase-shifting 1 point diraction interferometer (PDI) [12]. Some initial experimental results using 1 Some authors use the term, ``point diraction interferometer'' to refer only to the type of interferometer with an amplitude mask, whereas they use the term ``Zernike interferometer'' to refer to the type with a phase mask. In this paper we use the terms interchangeably, and we explicitly state that it is a phase-shifting point diraction interferometer.
0030-4018/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 1 ) 0 1 1 1 6 - 6
32
S.R. Dale et al. / Optics Communications 191 (2001) 31±38
the technique were presented in Ref. [13]. Here we show explicitly results of phase conjugation using the Zernike cell, and also demonstrate the use of an OA-SLM as a PDI. This means that the system has a number of advantages. The general technique of using the Zernike cell has the advantages that it operates with low-power beams, it can correct high spatial frequency aberrations, and it does not require a separate reference beam (selfreferencing wavefront correction using a dierent technique was also recently described in Ref. [14]). It has the disadvantages that it can only correct light from a single point source (i.e. zero ®eld of view) and is therefore suitable for laser correction, rather than imaging, and it introduces amplitude modulation into the beam (although we show this is not necessarily a problem). The use of an OASLM as the phase spot means that the system is relatively simple to align and is insensitive to wavefront slopes, and also means that the size of the phase diracting spot adjusts automatically to account for varying aberration amplitudes.
2. System analysis The basic optical con®guration for phase reversal is shown in Fig. 1. The system consists of a conventional phase-shifting PDI in a double-pass arrangement. The ®rst lens focuses the incident light onto the phase ®lter, which is produced by the OA-SLM. The second lens recollimates the beam, and the mirror re¯ects the light back through the system. LC PDIs have been described by others [15,16] which allow quantitative measurements to be made by phase-shifting interferometry. These designs, along with the conventional PDI, all re-
quire accurate alignment of spatial ®lter and input beams. The use of an OA-SLM means that the phase spot position and size automatically coincides with the focal spot from the lens. An analytical analysis of the system is presented by Sherstobitov [7], but here we present a simpler heuristical treatment which gives a clearer physical insight into the phase conjugation mechanism. The aberrated input wavefront can be represented, for small values of the aberration, h, as win Aeih A 1 ih:
1
The ®rst lens Fourier transforms this input onto the spatial ®lter. Since the Fourier transform, I, is functionally distributive, then, IfWin g If Ag If Aihg:
2
The plane wave component of the input transforms passes through the Airy disk which falls on the p=2 phase-shifting central spot. The second lens and mirror combination means that the beam passes through the phase spot twice, giving a total phase shift of p, and therefore we can write the output beam as, 2 IfWout g eip=2 If Ag If Aihg
If Ag If Aihg:
3
Therefore, wout A 1 ih;
4
which (from the point of view of imaging) is equivalent to wout A 1
ih win :
5
Hence the output beam is the conjugate of the input.
3. System simulation
Fig. 1. Experimental arrangement of the basic phase conjugating method based on the phase-shifting PDI, or Zernike cell.
We simulated the performance of the system to investigate the quality of the phase conjugation. The input wavefront was described by a constant amplitude within the aperture multiplied by the complex phase term. This was Fourier transformed to give the amplitude distribution in the
S.R. Dale et al. / Optics Communications 191 (2001) 31±38
®lter plane, which was multiplied by the phase factor to allow for the central phase-shifting spot, /spot , given by /spot eip=2 /spot 1
for r < rspot ;
for r P rspot ;
6
where rspot is the phase-shifting spot radius, and r is the radial polar coordinate in the plane of the phase mask. This modi®ed amplitude was then Fourier transformed, and then back propagated through the system. The phase values of the input and output signals should be opposite in sign if phase conjugation has occurred, and therefore they were added together to give a wavefront error array. In the simulation we investigated the conjugation quality as a function of both the amplitude and the spatial frequency of the aberration. The input aberrations were azimuthally symmetric functions of the form, /in A cos 2pur;
7
Fig. 2. Plots of the input and output beam intensity and phases. Top left: input beam intensity (i.e. a top-hat function). Top right: input beam phase. Bottom left: output beam intensity pattern. The system has clearly amplitude modulated the beam. However this has a relatively weak eect on the beam quality. Bottom right: output beam phase. Note that it is the conjugate of the input phase. The spatial frequency of the aberration u 3 cycles/aperture radius, and amplitude 0:4 waves P±V.
33
where A is the applied amplitude, r is the normalised radial polar coordinate in the pupil plane, and u is the spatial frequency. We display u in units of cycles/aperture radius so that the results do not depend on the overall aperture size. Fig. 2 demonstrates the basic results by plotting input and output beam intensities and phases. It is clear that the output phase has the conjugate shape to the input phase. It is also clear that the beam has been intensity modulated. This is not surprising, since this type of optical system is usually used to visualise phase. However, because the interferometer is of the phase-shifting type, then energy is conserved, and because amplitude modulation has a relatively weak eect on the ®nal focussed spot quality, then the system can still produce useful wavefront correction. Quantitative results are shown in Figs. 3 and 4, where the uncorrected and corrected Strehl ratios (calculated by using the Marechal approximation) are shown as a function of the amplitude, A, and spatial frequency, u of the aberration. Fig. 3 shows that for very small (<0.2 waves P±V) then the system actually degrades the Strehl ratio, and this is caused by the induced amplitude modulation. For intermediate amplitudes, then the system produces an increase in Strehl ratio. For large values of the aberration, then the approximation
Fig. 3. Conjugation quality versus aberration amplitude. The dotted line shows the Strehl ratio of the uncorrected beam, and the solid line shows the corrected Strehl ratio. The results are for a ®xed value of the spatial frequency of the input aberration (u 3 cycles/aperture radius).
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S.R. Dale et al. / Optics Communications 191 (2001) 31±38
quency that the system can correct for is governed only by the overall aperture of the system. 4. The OA-SLM as a Zernike cell
Fig. 4. Conjugation quality versus spatial frequency of the aberration (in cycles/aperture radius). The dotted line shows the Strehl ratio of the uncorrected beam, and the solid line shows the corrected Strehl ratio. The results are for a ®xed value of the amplitude of the input aberration (0.4 waves, P±V).
win Aeih A 1 ih used in Section 2 is no longer valid, and again the system does not work. Note that the Strehl ratio of the uncorrected beam increases when the amplitude > 0:8 waves. This is a real eect. Consider, for example, the on-axis beam intensity of a diraction limited beam, as a function of axial distance (or alternatively induced defocus), in which case the intensity is an oscillating function. Fig. 4 shows the conjugation quality versus spatial frequency, and indicates that the spatial frequency must be greater than a certain value in order for useful conjugation to occur. This is because for very small values of the spatial frequency, then the diracted beam is barely displaced from the Airy disk, and both terms pass through the phase-shifting spot, and phase conjugation does not occur. In this simulation (and indeed the previous simulations as well), the spot size was two times the Airy disk size, and thus wavefront correction does not occur until the spatial frequency of the aberration > 2 cycles/aperture radius (a spatial frequency of n cycles per radius will diract light into the nth Airy ring). This size was chosen empirically because it gave the best conjugation quality. Other authors have also noted that an apparently non-optimum spot size gives improved fringe visibility in a Zernike interferometer [17]. The maximum spatial fre-
The OA-SLM was ®rst set up as shown in Fig. 5 in order to test its use as a PDI, or Zernike cell. The OA-SLM consists of a number of thin ®lm layers sandwiched between two glass substrates: a photoconductor, a LC layer, alignment layers, transparent electrodes and a dielectric mirror. When a DC voltage is applied to the electrodes, it is divided between the photoconductor and the LC according to the amount of (write) light incident on the device. Therefore the induced phase modulation is proportional to the intensity of light incident on the device. The speci®cations of the SLM are shown in Table 1. An electrically addressed LC SLM with 69 pixels (Meadowlark Optics Hex69 device [18]) was used to generate test phase patterns. The light was focused onto the
Fig. 5. Experimental apparatus to test the use of the OA-SLM as a PDI, or Zernike cell. Table 1 Speci®cations of the optically addressed SLM. Type M19-T, produced by F. Vladimirov of the State Optical Institute, St. Petersburg, Russia Sensitivity Resolution at 5% MTF Resolution at 50% MTF Contrast ratio Grayscales, 0.15ND per step Phase shift Rise time Decay time Write light spectral region Read light spectral region Aperture size Voltage
0.1±0.01 lW/cm2 100±250 lp/mm 60±70 lp/mm 100:1 8 0±3p (HeNe) 1±10 ms 50±100 ms 400±500 nm 600±1100 nm 6 cm2 5±20 V (DC)
S.R. Dale et al. / Optics Communications 191 (2001) 31±38
Fig. 6. Results showing the OA-SLM used as a PDI in order to visualise the phase distortion (second order astigmatism) produced by an electrically addressed SLM.
SLM, and the zero order light produces a phase diracting spot. When the device is used with low voltages then the threshold voltage of the LC means that the phase spot has an approximately sharp boundary. An example resulting interferogram is shown in Fig. 6, in which the phase aberration was second order astigmatism. The magnitude and form of the aberration aects the fringe contrast in all common-path interferometers of this type. In general, unity contrast is only obtained when the amount of light passing through the phase discontinuity is equal to the integrated intensity from the higher order terms (see for example, Ref. [19]), which cannot be achieved here. A full discussion of phase visualisation by this type of intereferometer is given in Ref. [20]. Furthermore, in the case of the OA-SLM the induced phase shift in the diracting spot is a function of the overall intensity. Fig. 7 shows how the fringe contrast varies with beam power, and gives an indication of the useful working range of the device. The intensities were measured using an optical power meter. These results show that an OA-SLM can be used as a common-path interferometer with two advantages over a usual PDI. Firstly, the interferometer is insensitive to the overall alignment, and therefore is more stable, and secondly the fringe contrast can be adjusted by controlling the voltage applied to the OA-SLM.
35
Fig. 7. Interferogram contrast versus beam intensity on the SLM.
5. The OA-SLM as a phase conjugation system A phase conjugation system based on the system shown in Fig. 1 was assembled into one arm of a Michelson interferometer as shown in Fig. 8 so that phase conjugation could be observed by interfering the corrected beam with a ¯at reference wavefront. Again, an electrically addressed LC-SLM was used to generate aberrations, and the OA-SLM phase conjugation system then corrected these aberrations. The applied voltage to the OA-SLM was adjusted in order to optimise the correction, as indicated by the straightness of the resulting fringes. An example result showing correction of a phase shift produced by the actuation of a single LC pixel is shown in Fig. 9. We have shown just a single pixel because it clearly shows very high resolution phase conjugation by the correction of discontinuous fringes. Furthermore, in this con®guration the resulting intensity modulation negligible, and one can see the eects of phase conjugation clearly. The experimental system was then modi®ed to that shown in Fig. 10 in order to measure Strehl gains. The output of the phase conjugation system after its double pass through the aberrator, was focussed onto a pinhole, the size of which was chosen to pass only the central portion of the Airy pattern. An optical power meter measured the power of the light passing through the pinhole. Measurements were recorded with the
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S.R. Dale et al. / Optics Communications 191 (2001) 31±38
Fig. 8. Experimental con®guration showing the phase conjugation system used in one arm of a Michelson interferometer, used to generate the results shown in Fig. 9.
Fig. 9. Interferograms produced by the Michelson interferometer shown in Fig. 8, showing the eect of phase conjugation of the phase shift produced by a single LC pixel. PCM ± phase conjugate mirror. The beam power was 4.4 lW, the control voltage applied to the OA-SLM 6.6 V and the aberration amplitude k=4 (single pass).
phase conjugation system turned on and o, and the increase in Strehl ratio, Sgain , was de®ned as, Sgain
Scorrected Suncorrected Icorrected Iuncorrected ; Suncorrected Iuncorrected 8
where I is the recorded power by the meter, and the subscripts refer to the system being on and o. A phase aberration was produced by actuating alternate SLM pixels to produce a checker-board like pattern. This ensured that the aberration did not contain very low spatial fre-
S.R. Dale et al. / Optics Communications 191 (2001) 31±38
37
Fig. 10. Experimental apparatus used to measure gains in Strehl ratio of the phase conjugation system.
quencies that the system was unable to correct. Results were recorded using with induced (single pass) phase shift amplitudes of k=4 and 3k=4 with dierent beam powers. For each combination of aberration amplitude and beam power, the control voltage was adjusted to give the highest possible conjugation quality. Results are shown in Fig. 11. Strehl gains up to 300% were recorded. Gains were higher for the lower amplitude aberration. As expected from the results of Section 4, the gains were highest for beam powers in the range 10±100 mW.
6. Conclusions There are two separate main conclusions to this paper. First, we have shown that an optically addressed SLM can produce a phase discontinuity which can be used to produce a phase-shifting PDI, or a Zernike cell with the advantages of being insensitive to the beam alignment, and of having an electrically tuneable fringe contrast. Second, we have shown that such a device can be used as part of a phase conjugation system, used to correct wavefronts with spatial frequencies greater than minimum value which depends on the aperture size, and the size of the diracting spot. This system has the advantage of all phase conjugation systems over adaptive optics, i.e. the wavefront conjugation is automatic. Furthermore it is insensitive to beam alignment. For small changes in the aberration the system will adapt, but for large changes, then the voltage applied to the OASLM will need to be altered. For this reason this type of system would be useful for controlling slowly varying aberrations in, for example, laser systems.
Acknowledgements Fig. 11. Increases in Strehl ratio versus optical power for a number of realisations of the same experiment. The diamonds are for a peak±valley aberration amplitude of k=4, and the crosses are for a peak±valley aberration amplitude of 3k=4. The solid lines are connected through the mean values.
This work was performed as an M.Sci. undergraduate project in the Department of Physics at the University of Durham. Thanks to Fedor Vladimirov of the State Optical Institute, St. Petersburg, Russia for constructing the SLM.
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