Abstract

Arbitrary two-dimensional complex modulation of an optical field is a powerful tool for coherent optical systems. No single spatial light modulator (SLM) offers true arbitrary complex modulation, but they can be combined in order to achieve this. In this work, two sides of a twisted nematic (TN) liquid crystal SLM are used sequentially to implement different arbitrary modulation schemes. In order to fully explore and exploit the rich modulation behavior offered by a TN device, a generalized Jones matrix approach is used. A method for in situ characterization of the SLM inside the two-pass system is demonstrated, where each side of the SLM is independently characterized. This characterization data is then used to design appropriate polarizer configurations to implement arbitrary complex modulation schemes (albeit without 100% efficiency). Finally, an in situ optimization technique that corrects states by applying a translation in the complex plane is demonstrated. This technique can correct both for variations across the SLM and bulk changes in the SLM behavior due to the changing temperature.

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References

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2012 (1)

2009 (1)

C. Kohler, “Model-free method for measuring the full Jones matrix of reflective liquid-crystal displays,” Opt. Eng. 48, 044002 (2009).
[Crossref]

2003 (1)

R. Tudela, E. Martín-Badosa, I. Labastida, S. Vallmitjana, I. Juvells, and A. Carnicer, “Full complex Fresnel holograms displayed on liquid crystal devices,” J. Opt. A 5, S189–S194 (2003).
[Crossref]

2001 (1)

2000 (1)

P. Birch, R. Young, C. Chatwin, M. Farsari, D. Budgett, and J. Richardson, “Fully complex optical modulation with an analogue ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 175, 347–352 (2000).
[Crossref]

1999 (2)

R. W. Cohn, “Analyzing the encoding range of amplitude-phase coupled spatial light modulators,” Opt. Eng. 38, 361–367 (1999).
[Crossref]

S. Serati and K. Bauchert, “Sampling technique for achieving full unit-circle coverage using a real-axis spatial light modulator,” Proc. SPIE 3715, 112–119 (1999).
[Crossref]

1998 (1)

1996 (1)

1995 (1)

M. Yamauchi and T. Eiju, “Optimization of twisted nematic liquid crystal panels for spatial light phase modulation,” Opt. Commun. 115, 19–25 (1995).
[Crossref]

1994 (2)

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33, 3018–3022 (1994).
[Crossref]

C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
[Crossref]

1993 (2)

C. Zeile and E. Lueder, “Complex transmission of liquid crystal spatial light modulators in optical signal processing applications,” Proc. SPIE 1911, 195–206 (1993).
[Crossref]

J. L. Pezzaniti and R. A. Chipman, “Phase-only modulation of a twisted nematic liquid-crystal TV by use of the eigenpolarization states,” Opt. Lett. 18, 1567–1569 (1993).
[Crossref]

1992 (1)

1991 (1)

R. Juday and J. Florence, “Full-complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
[Crossref]

1990 (1)

B. Saleh and K. Lu, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[Crossref]

1987 (1)

1984 (1)

1967 (1)

1966 (1)

1963 (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1963).
[Crossref]

1941 (1)

Allebach, J. P.

Bauchert, K.

S. Serati and K. Bauchert, “Sampling technique for achieving full unit-circle coverage using a real-axis spatial light modulator,” Proc. SPIE 3715, 112–119 (1999).
[Crossref]

Birch, P.

P. Birch, R. Young, C. Chatwin, M. Farsari, D. Budgett, and J. Richardson, “Fully complex optical modulation with an analogue ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 175, 347–352 (2000).
[Crossref]

Brown, B. R.

Bryngdahl, O.

Budgett, D.

P. Birch, R. Young, C. Chatwin, M. Farsari, D. Budgett, and J. Richardson, “Fully complex optical modulation with an analogue ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 175, 347–352 (2000).
[Crossref]

Carnicer, A.

R. Tudela, E. Martín-Badosa, I. Labastida, S. Vallmitjana, I. Juvells, and A. Carnicer, “Full complex Fresnel holograms displayed on liquid crystal devices,” J. Opt. A 5, S189–S194 (2003).
[Crossref]

Chatwin, C.

P. Birch, R. Young, C. Chatwin, M. Farsari, D. Budgett, and J. Richardson, “Fully complex optical modulation with an analogue ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 175, 347–352 (2000).
[Crossref]

Chipman, R. A.

Cohn, R. W.

R. W. Cohn, “Analyzing the encoding range of amplitude-phase coupled spatial light modulators,” Opt. Eng. 38, 361–367 (1999).
[Crossref]

Davis, J.

Eiju, T.

M. Yamauchi and T. Eiju, “Optimization of twisted nematic liquid crystal panels for spatial light phase modulation,” Opt. Commun. 115, 19–25 (1995).
[Crossref]

Farsari, M.

P. Birch, R. Young, C. Chatwin, M. Farsari, D. Budgett, and J. Richardson, “Fully complex optical modulation with an analogue ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 175, 347–352 (2000).
[Crossref]

Florence, J.

R. Juday and J. Florence, “Full-complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
[Crossref]

Fütterer, G.

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 2008).

Gregory, D. A.

Gu, C.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays, Pure and Applied Optics (Wiley, 2010).

Hauck, R.

Häussler, R.

Jones, R.

Juday, R.

R. Juday and J. Florence, “Full-complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
[Crossref]

Juday, R. D.

Juvells, I.

R. Tudela, E. Martín-Badosa, I. Labastida, S. Vallmitjana, I. Juvells, and A. Carnicer, “Full complex Fresnel holograms displayed on liquid crystal devices,” J. Opt. A 5, S189–S194 (2003).
[Crossref]

Kanbayashi, Y.

Kato, H.

Kirsch, J. C.

Kohler, C.

C. Kohler, “Model-free method for measuring the full Jones matrix of reflective liquid-crystal displays,” Opt. Eng. 48, 044002 (2009).
[Crossref]

Kumar, B. V. K. V.

B. V. K. V. Kumar, A. Mahalanobis, and R. D. Juday, Correlation Pattern Recognition (Cambridge University, 2005).

Labastida, I.

R. Tudela, E. Martín-Badosa, I. Labastida, S. Vallmitjana, I. Juvells, and A. Carnicer, “Full complex Fresnel holograms displayed on liquid crystal devices,” J. Opt. A 5, S189–S194 (2003).
[Crossref]

Leister, N.

Lohmann, A. W.

Lu, G.

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33, 3018–3022 (1994).
[Crossref]

Lu, K.

C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
[Crossref]

B. Saleh and K. Lu, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[Crossref]

Lueder, E.

C. Zeile and E. Lueder, “Complex transmission of liquid crystal spatial light modulators in optical signal processing applications,” Proc. SPIE 1911, 195–206 (1993).
[Crossref]

Mahalanobis, A.

B. V. K. V. Kumar, A. Mahalanobis, and R. D. Juday, Correlation Pattern Recognition (Cambridge University, 2005).

Martín-Badosa, E.

R. Tudela, E. Martín-Badosa, I. Labastida, S. Vallmitjana, I. Juvells, and A. Carnicer, “Full complex Fresnel holograms displayed on liquid crystal devices,” J. Opt. A 5, S189–S194 (2003).
[Crossref]

Moreno, I.

Neto, L. G.

Paris, D. P.

Pezzaniti, J. L.

Reichelt, S.

Richardson, J.

P. Birch, R. Young, C. Chatwin, M. Farsari, D. Budgett, and J. Richardson, “Fully complex optical modulation with an analogue ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 175, 347–352 (2000).
[Crossref]

Roberge, D.

Saleh, B.

B. Saleh and K. Lu, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[Crossref]

Seldowitz, M. A.

Serati, S.

S. Serati and K. Bauchert, “Sampling technique for achieving full unit-circle coverage using a real-axis spatial light modulator,” Proc. SPIE 3715, 112–119 (1999).
[Crossref]

Sheng, Y.

Soutar, C.

C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
[Crossref]

Sweeney, D. W.

Tam, E. C.

Tsai, P.

Tudela, R.

R. Tudela, E. Martín-Badosa, I. Labastida, S. Vallmitjana, I. Juvells, and A. Carnicer, “Full complex Fresnel holograms displayed on liquid crystal devices,” J. Opt. A 5, S189–S194 (2003).
[Crossref]

Usukura, N.

Vallmitjana, S.

R. Tudela, E. Martín-Badosa, I. Labastida, S. Vallmitjana, I. Juvells, and A. Carnicer, “Full complex Fresnel holograms displayed on liquid crystal devices,” J. Opt. A 5, S189–S194 (2003).
[Crossref]

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1963).
[Crossref]

Yamauchi, M.

M. Yamauchi and T. Eiju, “Optimization of twisted nematic liquid crystal panels for spatial light phase modulation,” Opt. Commun. 115, 19–25 (1995).
[Crossref]

Yeh, P.

P. Yeh and C. Gu, Optics of Liquid Crystal Displays, Pure and Applied Optics (Wiley, 2010).

Young, R.

P. Birch, R. Young, C. Chatwin, M. Farsari, D. Budgett, and J. Richardson, “Fully complex optical modulation with an analogue ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 175, 347–352 (2000).
[Crossref]

Yu, F. T. S.

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33, 3018–3022 (1994).
[Crossref]

Zeile, C.

C. Zeile and E. Lueder, “Complex transmission of liquid crystal spatial light modulators in optical signal processing applications,” Proc. SPIE 1911, 195–206 (1993).
[Crossref]

Zhang, Z.

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33, 3018–3022 (1994).
[Crossref]

Appl. Opt. (6)

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1963).
[Crossref]

J. Opt. A (1)

R. Tudela, E. Martín-Badosa, I. Labastida, S. Vallmitjana, I. Juvells, and A. Carnicer, “Full complex Fresnel holograms displayed on liquid crystal devices,” J. Opt. A 5, S189–S194 (2003).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

P. Birch, R. Young, C. Chatwin, M. Farsari, D. Budgett, and J. Richardson, “Fully complex optical modulation with an analogue ferroelectric liquid crystal spatial light modulator,” Opt. Commun. 175, 347–352 (2000).
[Crossref]

M. Yamauchi and T. Eiju, “Optimization of twisted nematic liquid crystal panels for spatial light phase modulation,” Opt. Commun. 115, 19–25 (1995).
[Crossref]

Opt. Eng. (5)

C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid crystal cell,” Opt. Eng. 33, 2704–2712 (1994).
[Crossref]

B. Saleh and K. Lu, “Theory and design of the liquid crystal TV as an optical spatial phase modulator,” Opt. Eng. 29, 240–246 (1990).
[Crossref]

R. W. Cohn, “Analyzing the encoding range of amplitude-phase coupled spatial light modulators,” Opt. Eng. 38, 361–367 (1999).
[Crossref]

Z. Zhang, G. Lu, and F. T. S. Yu, “Simple method for measuring phase modulation in liquid crystal televisions,” Opt. Eng. 33, 3018–3022 (1994).
[Crossref]

C. Kohler, “Model-free method for measuring the full Jones matrix of reflective liquid-crystal displays,” Opt. Eng. 48, 044002 (2009).
[Crossref]

Opt. Lett. (2)

Proc. SPIE (3)

S. Serati and K. Bauchert, “Sampling technique for achieving full unit-circle coverage using a real-axis spatial light modulator,” Proc. SPIE 3715, 112–119 (1999).
[Crossref]

R. Juday and J. Florence, “Full-complex modulation with two one-parameter SLMs,” Proc. SPIE 1558, 499–504 (1991).
[Crossref]

C. Zeile and E. Lueder, “Complex transmission of liquid crystal spatial light modulators in optical signal processing applications,” Proc. SPIE 1911, 195–206 (1993).
[Crossref]

Other (4)

B. V. K. V. Kumar, A. Mahalanobis, and R. D. Juday, Correlation Pattern Recognition (Cambridge University, 2005).

http://dx.doi.org/10.17863/CAM.5955 .

P. Yeh and C. Gu, Optics of Liquid Crystal Displays, Pure and Applied Optics (Wiley, 2010).

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 2008).

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Figures (7)

Fig. 1.
Fig. 1. Two-pass SLM configuration used in this work. Collimated 633 nm linearly polarized light from a SMF passes through one half of the SLM before being imaged onto the other half. The relay system is a 4f system. A linear analyzer is applied at the output. The red arrow indicates light proceeding downstream into the next stage of the system, such as through an imaging system, or an optical Fourier transform lens. Inset: A schematic of the LC director profile in such a device. In one state (in this case, “off”), the birefringent LC molecules form a quarter-turn helix between the two electrodes. Application of an electric field re-orientates the molecules in the bulk, but the molecules near the surfaces are held near their original positions by surface anchoring.
Fig. 2.
Fig. 2. (a) The normalized transmitted intensity of different systems under varying polarizer and analyzer angles used to characterize the SLM and mirror. The first panel shows the intensity measured by a detector located at the position of the relay mirror in Fig. 1 used to characterize the 45° mirror. The second panel shows the transmitted intensity with the detector in the same place through the SLM at a uniform gray level 0. The third panel shows the configuration of Fig. 1, with the detector at the focal point of a lens situated optically downstream. Both halves of the SLM are at a uniform gray level 0. From these measurements, estimates of the Jones matrix representation of the SLM can be found. (b) Further measurements made to find the global phase delay of the Jones matrices by displaying a Ronchi grating on the SLM. The detector is again situated at the focal point of a lens downstream, and a pinhole is used to only select the DC component of the optical Fourier transform. In the data shown, only the first half of the SLM is swept; the second half remains at gray level 0. Using the cosine rule, as shown, the global phase delay θ relative to the 0 state can be found from knowledge of the amplitudes of the states separately |s| and the Ronchi amplitude R.
Fig. 3.
Fig. 3. (a) The final measured Jones matrices that represent the two halves of the SLM across gray levels (level 0 is represented by ×). (b) The two sets of eigenvalues (black lines, × represents level 0) and polarization ellipses representing the polarization eigenstates (from blue to red) for the first and second halves of the SLM.
Fig. 4.
Fig. 4. (a) The optimized complex amplitude modulation scheme for full complex modulation. All of the accessible states are shown, surrounded by the concave hull. The maximum modulation amplitude rmax defines a circle on the Argand plane within this point set. Inset (left): The states targeted (lines) and used (red points) within this circle. Inset (right): The SLM gray levels on the first and second sides of the SLM used by these states. Throughout, the colors indicate corresponding states. (b) A measurement of the transmitted amplitude for the different states. The different color lines correspond to the different phase angles in the unit circle, with the same amplitude function. (c) Calculated and experimental transmitted amplitudes for Ronchi gratings measured relative to the state marked by ∘. There is good agreement, and the desired phase modulation has been implemented.
Fig. 5.
Fig. 5. (a) The result of the optimization targeting continuous amplitude, binary phase. There are states in all directions around the span chosen to accommodate further optimization. Inset (top): The relative error plotted is the amplitude error of the target point from the ideal point divided by the inter-point spacing. Inset (bottom): The SLM gray levels used to achieve each of the 32 different states on the first and second sides. (b) The characterization of these states. The measured transmitted amplitude of each state for flat-field measurements is shown, as well as the zero-order amplitude when a Ronchi grating is displayed. Two different Ronchi gratings are considered for the cases where the first and last states in the sequence are held constant. The dotted lines are best-fit lines for the Ronchi gratings. The dashed line is at twice the gradient and agrees with the flat-field measurement, consistent with binary-phase modulation.
Fig. 6.
Fig. 6. (a) A schematic of combined image of both sides of the SLM. The heat map shows the Gaussian beam profile illuminating the SLM, overlaid with the aperture used for the initial characterization of the SLM. The vectors sample the displacement in the Argand plane described by Eq. (13), with the coefficients having been found by direct optimization. (b) The same Eq. (13), this time with the displacements shown in the Argand plane. All of the model-predicted states are shown as blue dots, with the green region showing the states used across the SLM for the optimized zero amplitude state. (c) The actual image displayed on the SLM. The red boxes correspond to the area in (a). The regions of level 255 are due to the displacement moving outside of the area of complex points. Due to their being negligible intensity there, this is essentially a free parameter. (d) The corrected zero level in the context of all of the other uncorrected states. Clearly, we now obtain a more ideal bilinear function when using this corrected level.
Fig. 7.
Fig. 7. (a) The normalized amplitudes for the different states for the characterization temperature of 35°C—as in previous results—and at a lower temperature of 26°C, with and without the same correction applied to each state. The minimum should be at level 0. (b) The zero-order intensity for a Ronchi grating measured relative to level 15 (∘), and a straight fit line which would correspond to constant phase, for both the corrected case and uncorrected case at 26°. The residuals of these fits are also plotted. (c) The displacement vectors applied across the illuminated region of the SLM. (d) All of the states used (green) when this displacement is applied to each point in the results of Fig. 5.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

Eout=P(θana)·Jmirror·J2(n2)·(1001)·J1(n1)·Jmirror·(E0cosθpolE0sinθpol).
J(n)=c(fgihjihjif+gi),
Eout=P(θana)·J(n)·(E0cosθpolE0sinθpol),
Eout=P(θana)·Jmirror·E0(θpol),
argminrp,rsθpol,θana|Imodel(θpol,θana)Iexpt.(θpol,θana)|2,
Eout=P(θana)·Jmirror·J1(0)·E0(θpol),
J(n)=ceiϕ(fgihjihjif+gi),
E=σNND·NND¯N,
argminθpol,θana,rmaxE,
R(L,φ)=12|A(15,0)+A(L,φ)|,
E=kdkS,
argminθpol,θana,θArgE
R(x,y)=A00+A10(xx0)+A01(yy0)+A20(xx0)2+A11(xx0)(yy0)+A02(yy0)2I(x,y)=B00+B10(xx0)+B01(yy0)+B20(xx0)2+B11(xx0)(yy0)+B02(yy0)2,

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