Appendix Jones Calculus in Phase-Only Liquid Crystal Spatial Light Modulators
In Jones calculus, the polarization state of light is represented by a Jones vector,
Ex EEE ⇒ (A.1) E y where Ex and Ey are normalized complex amplitude components along x- and y-axes, respectively. An optical element is represented by a Jones matrix,
TT11 12 TTT ⇒ , (A.2) TT21 22 and light that impinges on it undergoes the transformation
(out) (in) Ex 11 12 ETT x E(out) = TEE=TE (in) ⇒ = . (A.3) (out) TT (in) E y 21 22 E y A phase-only liquid crystal spatial light modulator (SLM) is a two-dimensional array of independently addressable uniaxial birefringent pixels. The pixels are characterized by a common ordinary refractive index, no, along one axis and individually voltage- controllable extraordinary refractive index, ne(V), along an orthogonal axis. To under- stand the various operation modes of an SLM, consider the optical setup shown in Fig. A.1. The SLM orientation is specified by the angle θ between the extraordinary axis and the y-axis. The Jones matrix for an SLM oriented with its extraordinary axis parallel to the y-axis (θ=0) is
2π expj dno 0 (0) λ TSLM = . (A.4) 2π 0 exp dne () x , y λ 306 Appendix: Jones Calculus in Phase-Only Liquid Crystal Spatial Light Modulators
where d is the thickness of the SLM cell and the index modulation ne ( x, y) is controlled by the applied pixel voltages V= V( x, y) . To study the output polarization states, it is convenient to write the unrotated Jones matrix as 1 0 2π TTTSLM = expj dno , λ 0 expδ ()x , y 2π where δ ()x, y= d n() x, y− n describes a spatially varying phase retardation. λ e o Each SLM pixel acts as a programmable phase retarder, enabling various polarization states to be encoded on an incident light. Using this alternate expression with the rotation matrix, Rθ, we can apply a rotational transformation to get the new SLM Jones matrix for an arbitrary θ
()θ -1-1-1 (0) TRTRSLM= θ SLM θ
2 2 cosθ+ sin θ expj() δ sin θ cos θ− 1 + expj() δ . (A.5) = exp jφ ()o 2 2 sinθ cos θ− 1 + expj() δ sin θ+ cos θ expj() δ
Note that the uniform phase offset, φ0 =(2 π λ )dno , which does not affect the polariza- tion, is commonly neglected in Jones calculus.
polarization axis
y x SLM axis θθθ Input light
Polarizer SLM
FigFigFig A.1 Optical setup for realizing different SLM operation modes. A linearly polarized (along the y-axis) light is incident on a spatial light modulator (SLM). The director axis of the liquid crystal molecules in the SLM is oriented at an angle, θ, with the incident polarization.
A.1 Spatial Phase Modulation
Consider a vertically polarized input light
()in 0 E = , (A.6) 1 A.2 Spatial Polarization Modulation 307 incident on an SLM oriented at θ=0. Using the SLM Jones matrix in Eq. (A.4), the output after SLM is a new vector EEE (out) given by 0 ()out (0)(0)(0) ()in ETE=SLM = exp jφ ()x , y . (A.7) 1 This is a vertically polarized output that contains a spatial phase modulation, 2π φ ()x, y= dne () x, y . (A.8) λ Thus, the SLM operates as a spatial phase-only modulator when its extraordinary axis is aligned with the incident polarization.
A.2 Spatial Polarization Modulation
o For an SLM oriented at θ = 45 , the rotationally transformed TSLM in Eq. (A.4) becomes
1+ expjδ 1 − exp j δ −1 1 ( ) ( ) RTR()()θSLM θ= exp()j φo . (A.9) 2 1− exp()()jδ 1 + exp j δ
Using the same vertically polarized input EEE(in), the output field EEE(out) is
− δ (out ) (0) 0 1 1 exp( j ) EET= TETSLM = exp()jφo 1 2 1+ exp()jδ (A.10) sinδ x , y /2 2φo + δ− π ()() = exp j . 2 jcos()δ ()x , y /2
The uniform phase offset, which is not relevant to the polarization state, can be neglected and we can describe the output polarization simply as
sin(δ ( x , y) /2) EEE ()out = (A.11) jcos()δ ()x , y /2
In spatial amplitude modulation, an output polarizer isolates either the vertical or horizontal component of Eq. (A.11). Spatial polarization modulation utilizes the full output with both components. Spatially programmed polarization is achieved by controlling the spatial phase retardation of the SLM. Output polarization vectors for some values of δ are illustrated in Table A.1. Other δ -values yield elliptic polarization states with major axis aligned either along the horizontal or the vertical. 308 Appendix: Jones Calculus in Phase-Only Liquid Crystal Spatial Light Modulators
Table A.A.A.1A.111 Polarization vector in Eq. (A.11) for various retardation, δ.
δ ( x, y) E ()out 0 0 j ; 1 1 1 π ; /2 2 j 1 π ; 0 1 1 ; 3π/2 2 − j 0 − j ; 2π 1
Table A.2 A graphical representation of the output polarization ellipticity and orientation as a function of the phase shift implemented in corresponding regions of SLM-1 and SLM-2
A.3 Spatial Polarization Modulation with Arbitrary Axis 309 A.3 Spatial Polarization Modulation with Arbitrary Axis
A polarizer and SLM can generate elliptically polarized light with major axis that can be vertical or horizontal. It is possible to generate an arbitrary state of polarization by taking this elliptically polarized light and rotating its major axis to a desired arbitrary orientation. This rotation can be realized using SLMs and two quarter wave plates, and the full polarization encoding system is schematically illustrated in Fig. A.2. Both SLMs are oriented at θ=45°.
Elliptical generator (T1) Elliptical rotator (T2)
2D polarization Input light encoded light Polariser SLM-1 λ/4 SLM-2 λ/4
Fig. A.2Fig. A.2 An optical system for converting incident polarized light into an arbitrary state of elliptically polarized light with the major axis of the elliptically polarized light rotated an arbitrary angle. The lines denote the extraordinary axis of the SLMs, the quarter wave plates (λ/4) and the polarization direction of the linear polarizer.
To verify the axis rotation, let us write the transformation matrix for the two quarter wave plates and SLM:
1 0 cos(δ2 (x ,/2 y) ) − j sin(δ2 ( x ,/2 y) ) 1 0 T = rotator 0 − j 0 j − jsin()δ2 () x , y /2 cos()δ2 ()x , y /2 (A.12) cos()δ2 ()x , y /2 sin()δ ()x , y /2 = . −sin()δ ()x , y /2 cos()δ2 ()x , y /2
This result shows that TTTrotator is, indeed, a rotation matrix with a rotation angle of
φ2(x,y)/2. The net transformation matrix for the optical system in Fig. A.2 is
(π /4) RTδ2 /2 SLM
cos()δ2 ()x ,/2 y sin()δ2 ()x ,/2 y cos()δ1 ()x ,/2 y− j sin()δ1 () x ,/2 y = − δ δ − δ δ sin()2 ()x , y /2 cos()2 ()x , y /2 j sin()1 () x , y /2 cos()1 ()x , y /2
cos()δ1 ()x , y /2cos()δ2 ()x , y /2− j sin()δ1 () x , y /2sin()δ2 ()x , y /2 = . δ δ δ δ jsin()1 () x , y /2sin()() x , y /2 cos()1 ()x , y /2cos()2 ()x , y /2 (A.13) 310 Appendix: Jones Calculus in Phase-Only Liquid Crystal Spatial Light Modulators
Output polarization states from this arbitrary polarization encoding system are graphically illustrated in Table A.2. The phase retardations of SLM-1 and SLM-2 determine the type and direction of the output polarization. It is interesting to note that the direction of the elliptically polarized light changes from left handed to right handed when the phase modulation of SLM-1 is above π. This table shows that it is possible to generate an arbitrary state of elliptical polarization if both SLM-1 and SLM-2 can produce a phase modulation of at least 2π. Experimental demonstrations of arbitrary polarization encoding are available in Ref. [1].
Reference
1. R. L. Eriksen, P. C. Mogensen, and J. Glückstad, “Elliptical polarisation encoding in two dimensions using phase-only spatial light modulators,” Opt. Commun. 187187187, 187 325-336 (2001). Index
3D optical trapping and manipulation, 176 Brownian motion, 200 4f system, 80, 106, 109, 183, 220, 225, 229, 233, chrome-on-glass mask, 256, 259 242, 244, 257, 264, 277, 285, 287, 288, circ-function, 181, 249, 286 289, 291, 293 colloidal constellations, 176 filtering, 157, 287, 288 colour-based sorting, 156 combined filter parameter, 2, 27, 30–32, 36–38, aberration, 1, 35, 176, 186, 233, 235, 255, 278, 40, 43–44, 46, 250–254, 257, 286, 288, 300 280 common-path interferometer (CPI), 1–4, 8, 14– Airy disc, 51 24, 27–32, 35–59, 61, 235, 248–255, 263, Airy function, 29 296, 300–304 annular intensity profile, 156, 159, 161 pair configuration, 48 aperture matching, 257 types, 2, 23, 300 Argand diagram, 250, 251–252, 255, 260–261, common-path interferometry, 1, 3, 8, 258 287–288 complex conjugate, 237, 274 array illumination, 220, 243 complex filter space, 37, 41, 58, 300 array illuminator, 221 plot, 37, 41, 58, 300 artefacts, 51 compression factor, 237, 243 Ashkin, Arthur, 151, 156, 167, 180, 203 compromise filter, 39 automation, 191, 196, 197, 212 computational axial and transverse stiffness, 209 cost, 58 axial control, 168, 171–172, 175, 211 load, 163 axial dependency of the trap stiffness, 194 overhead, 176, 238 axial dynamic range of manipulation, 209 computer-controlled polarization landscape, 174 axial force curve, 203, 207 computer-generated holography (CGH), 85–95, axial manipulation, 209–210 119, 123, 240 concurrent top- and side-view imaging, 178–179 bacteriorhodopsin, 218, 224–225 confinement stress, 165 binary coupling grating, 277, 280 constant intensity criterion, 250, 252 binary diffractive optical element, 234 contrast, see generalized phase contrast (GPC) binary phase mask, 288 phase contrast, reverse phase contrast (RPC) binary phase modulation, 247, 253, 263, 288 convolution, 51, 238 binary phase patterns counter-propagating beam trap, 168, 169 decrypting by amplitude, 296 C-parameter, 45 biologically safe operating wavelength, 171 critical separation, 208, 210 broadband spatial spectrum, 240 crosstalk, 230 312 Index cryptography, 245, 270, 273–275, 278, 283, fringe accuracy, 37, 43, 46–49, 58, 300 284, 296, 303 fringe visibility, 35, 260 functionalization, 202 dark ground absorption, 46 design freedom, 244, 300–301 Gabor, 8, 255 in pattern projection, 244 Gaussian detection algorithm, 196 beam, 156, 186, 203, 218 device constraints, 32, 176 illumination, 222 differential power, 170, 204, 210–211 generalized Henning method, 44, 45, 46, 47 diffractive microlenses, 232, 235, 277 generalized phase contrast (GPC) diffractive optical elements, 233, 235, 277 alternative schemes and implementations, digital micromirror-array device, 256, 218–245, 302–303 263–268 customized phase landscapes using, 303 Dirac-delta assumption, 43 foundation of, 13–25 direct search optimization, 241 GPC-SPM technique, 187 doughnut trap, 158, 161 hybrid-GPC filter, 236 drag-and-drop user interface, 201 integrated planar device implementation, 233 dual imaging, 179 introduced, 1–5, 7–11 dual-beam system, 183 matched filtering, 236 dual-path system, 283 miniaturized implementation, 5 dynamic filter, 244 optical cryptography using, 273–297, 303– dynamic range of axial position control, 203 304 programmable optical micromanipulation, encoding fill factor, 218, 223–224 151–212, 301–302 encoding, advantage of a straightforward, reversal of the method, see reverse phase 156 contrast (RPC) energy efficiency, 226, 236 shaping light by, 103–144 energy ratio, 53 wavefront engineering, GPC-based, error-free decryption, 295 61–100 escape velocities, 194 wavefront sensing and analysis, use in, 35–59 fabrication artifacts, 280 Gerchberg-Saxton method, 241 far-field diffraction pattern, 268 graphical user-interface (GUI), 185, 242 field absorption, 2, 41, 44, 46, 49, 59, 300 growth arrest, 164 field distribution, 206, 218, 220 field of view, 3, 51, 52, 58, 161, 168, 176–177, halo 179, 181, 200–202, 280 intensity, 53 fill-factor, 55 light, 54 filter parameters, 11, 27, 28–30, 35, 37, 41–47, Hankel transform, 241 49, 226, 247, 249, 251–257, 283, 286 Hanseniaspora uvarum, 165–166 filter phase shift, 226 harmonic potentials, 207 fix point, 29, 32 helical phase front, 55 fluid drag force, 193, 194 Henning phase contrast, 3, 23, 41–47, 59, 300 force curves, 209, 211 higher orders, 219, 227, 234 Fourier high-power applications, 231 decomposition, 10 holographic optical tweezers, 186 filter, 28–29, 38, 41, 44, 49, 181, 222, 236, holography, 301 264, 266–268, 283, 285 lens, 51, 217, 220, 224, 238, 302 independent 3D control of multiple particles, transforming lens, 287–288 172, 174 Fresnel integral, 206 induced refractive index change, 219 Index 313 information microscope, 1, 3, 4, 7, 8, 50, 56, 152, 157–158, capacity, 212, 255, 301 160, 162, 165, 167–169, 173, 176–181, inhomogeneous mixture, 154, 163 185–187, 190, 192–193, 198–199, 207, input phase distribution, 8, 9, 27, 28, 37, 38, 40, 212, 233, 277, 301 42 microstructure, 151, 187, 189–190, 200, integrated planar-optical device, 277, 282 202 intensity roll-off, 219 microtessellation, 198 intensity-to-phase mapping, 41 microtool, 187, 191, 302 unambiguous, 39, 40 minimum critical separation, 203 interactive operation, 154 mixed culture, 164–165 interference pattern, 7, 8, 35, 255 modified phasor chart, 30–32 interferogram, 52–54, 260 momentum components, 204 inversion, 182 multi-cell laser-manipulation in a microfluidic iterative design, 242 environment, 196 iterative Fourier transform, 241 multi-particle optical manipulation, 153 multiple beam illumination, 229–230 Kerr see also generalized phase contrast coefficient, 219, 225 medium, 218–219, 221, 223–245 negative phase contrast, 9 noise lab-on-a-chip system, 180 factors, 291, 295–296 laser-pattern-writing, 199 tolerance, 296 light normalized zero-order, 219, 228, 249 efficiency, 3, 9, 170, 277, 291, 301 numerical aperture (NA), 50, 104, 152, 165, intensity, 44, 159, 179, 196, 219 167, 169–170, 173, 176–177, 179–180, phase, 7 184–186, 192, 242, 280, 302 synthesis, 185, 192, 212 throughput, 49, 157, 258, 268, 278 on-axis filtering, 248 linear phase-to-intensity mapping regime, operating regime, 44, 45, 220 35, 46 optical linearity, 41–46, 59, 238, 242, 244, 300 actuation, 190 liquid crystal correlation system, 237, 238 display, 176, 256 decryption, 5, 231, 236, 275, 278–279, projector, 157, 278 291, 303 long working distance, 177, 192, 212, 302 elevator, 167–172 lookup table, 53 flat, 152, 183, 265, 278 lossless operating curve, 59, 300 force, 4, 152, 167, 191, 198, 203, 205, low-index particle, 156–164 212, 301 low-NA implementation, 177 Fourier transform, 265, 268, 286 low-pass filter, 249 raking, 160 tweezers, 156, 170, 177, 179–180, 186 Mach-Zender interferometer, 256 vortex, 156 matched filtering, 236 optimal filter phase shift, 226 maximum contrast visibility, 223 optimal linearity, 44 maximum phase error, 55 optimal visibility, 38, 39, 230 mGPC method, 239, 242 optimization algorithm, 58, 176, 241 micro-assembly, all-optical directed, 197 optimization procedure, 268 microfabricated optimum separation, 208 structures, 188, 191, 203 output interferogram, 27, 52, 229 microfluidic system, 191–196 output phase modulation, 248, 250, 252–253, microfluidics, 177, 180 255, 262, 287 314 Index parabolic flow profile, 193 potential well, 159, 167–168, 186, 203 parallel-aligned nematic liquid crystal, 157, 278 power ratio, 180, 184–185, 188– 189, 203, 207, particle size, role of, 209 210 pattern projection, 4, 238, 242, 244 power tolerance, 191, 263, 302 peak irradiance, 2, 32, 35–41, 58, 173, 300 phase quantitative phase imaging, 35, 50, 58–59, 301 ambiguities, 39 quantitative phase microscopy, 3–4, 49, 231 encoding, 3, 152, 160, 184–185, 223, 226, quantization errors, 51 237, 243, 273 quarter-wave-shifting, 9 error, 56, 57 modulation depth, 37, 227, 252–254 radial force constant, 207 offset, 27, 36, 306–307 radiation pressure, 151, 153, 180 perturbation, 7–8, 10, 27, 35–37, 39, 41–42, reactive ion etching, 232, 276 44, 59, 255, 260, 300 real-time, interactive optical manipulation, 165, quantization, 233, 235, 277 195 shift, 8–9, 11, 41, 49, 55, 59, 181, 183, 218– rectangular PCF, 181 223, 226–230, 233, 245, 249, 253–265, refresh rates, 163, 268, 301, 303 277– 278, 281, 283, 285–291, 302, 308 relative phase, 8, 27, 36, 227, 248, 260, 275 phase contrast, 1–3, 8–10, 32, 35, 41–42, 58–59, removal of particle stacking, 154 151–152, 226, 229, 234, 247, 279, 296, reverse phase contrast (RPC), 3–5, 247–270, 299–300, 304 265, 268, 283–285, 303–304 phase contrast filter (PCF), 42, 50–51, 55–57, design procedure, 253 62–64, 70, 72, 79–80, 83–85, 92, 95–99, optimization, 253 107–143, 152, 157–159, 181–184, 217– reversibility, 254, 273 224, 227–235, 245, 277, 280–283, 290, robustness, 4, 35, 54, 220, 236, 263, 291 302 of the optical decryption approach, 291 self-induced, 219 phase-only Saccharomyces cerevisiae, 164–166, 193 correlation, 239–240, 242 saturation phase shift, 223 correlation filter, 239–240, 242 scaling factor, 249 cryptography, 276, 278 self-phase modulation, 220 filters, 238, 240 Shack-Hartmann wavefront sensor, 51 modulation, 3, 247–248, 256, 260, 275, 283 shift invariance, 242 optical cryptography, 4, 274 side-view imaging, 177, 186, 302 optical decryption, 278–279 side-view observations, 179 phase-shifting interferometry, 3, 301 signal-to-noise ratio (SNR), 36, 52, 87–88 phase-to-intensity mapping, 3, 9, 27, 29, 32, 35, simultaneous monitoring, 177 36, 39–41, 45–49, 59, 300 sinc envelope, 219 phasor, 4, 10, 27–32, 42, 45–49, 55, 59, 252, 300 sinc-function, 37, 219 addition, 28 small-scale phase approximation, 10 chart, 27–32, 45–49, 59, 300 space invariance, 238 diagram, 252 space-bandwidth product, 176 photolithographic procedure, 188 spatial average, 37, 234, 249–250, 264, 287 photon efficiency, 173, 176 spatial filter, 2, 10, 235, 243–244, 247, 249, 257, photoresist, 152, 187, 232, 276, 278 283, 285–286, 290, 292, 296, 299 planar integrated micro-optics, 4, 231, 235, 276, spatial frequency, 10–11, 55, 58, 157, 181, 218– 280 219, 242, 248, 286, 302 planar SRW, assumption of a, 51 coordinates, 219, 286 polarization encoding, 168–169, 173, 175, 309 spatial light modulator (SLM), 50, 62, 76, 80– polystyrene microspheres, 153, 160, 162, 164, 81, 86–96, 99–100, 103–124, 138–139, 194, 207 143, 152–153, 157–159, 161, 163, 169, Index 315
173, 176–178, 180–181, 184, 186, 193, stiffness, 163, 193, 207, 211–212 196, 217, 228, 242–243, 256, 262–263, stiffness calibration, 193 267, 278–279, 281–289, 305–310 trapping spatial polarization modulator (SPM), 169–170, with lower NA objectives, stable, 176 173–174, 176, 184, 187–188 trapping and sorting spatial variations, 50 of an inhomogeneous mixture of dyed beads, spiral phase contrast microscopy, 56 155 structured light illumination, 218 of inhomogeneous size-mixture, 155 SU8 negative photoresist, 198 superposition, 9, 28, 225, 229–231, 238, 243, uniform intensity criterion, 252–253, 259 249–250, 255, 273, 291 unstable axial equilibrium point, 207 synthetic reference wave (SRW), 2–3, 14–21, unstable equilibrium, 159, 161 29, 35, 43, 49–50, 63–64, 72–83, 92, 96– user-interactive sorting, 163 100, 107, 115, 131–138, 141, 182, 220, 222, 226, 230, 233, 249–250, 253–255, Van der Lugt optical correlator, 236 258, 260, 263–267, 300 van der Waals forces, 197 profile, 2, 50–52, 58, 64–65, 79, 92, 97–98, visibility, 2, 3, 9, 32, 36–41, 58, 173, 223, 228– 132, 141, 230 229, 281, 300–301 spatial profile, 19, 64, 76, 78, 98–99, voltage-controlled phase shift, 227 222 volume of manipulation, 179 volumetric particle position, 179 Taylor series expansion, 10 vortex phase, 55, 57 TE and TM components, 205 three-dimensional weak phase perturbations, 43, 300 trapping, 172 widefield phase imaging, 49 top-hat trapping, 159 topographic image, 233, 277 yeast, 164, 168, 171–172, 193–195, 302 topological charge, 55, 57 translational control, 189 Zernike phase contrast, 1–2, 7–9, 11, 41, 56, 274 transverse force curves, 207–211 range of linearity, 42 transverse intensity gradient, 207 zero point, 29, 30, 32, 45–46, 49 trap zero-order, 43, 55, 58, 176, 219, 220, 222, 227, arrays, 187 234, 240, 249, 255, 262, 268, 277, 302 efficiency, 203 phase parameter, 220