Abstract

By superimposing a tunable binary phase grating with a conventional computer-generated hologram, the total power of multiple holographic 3D spots can be easily controlled by changing the phase depth of grating with high accuracy to a random power value for real-time optical manipulation without extra power loss. Simulation and experiment results indicate that a resolution of 0.002 can be achieved at a lower time cost for normalized total spot power.

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  20. S. Serati and J. Harriman, “Spatial light modulator considerations for beam control in optical manipulation applications,” Proc. SPIE 6326, 63262W, 63262W-11 (2006), http://link.aip.org/link/?PSI/6326/63262W/1 .
    [CrossRef]
  21. L. Xu, L. Y. Wu, J. Zhang, and X. Liu, “Effect of phase valley on diffraction efficiency of liquid crystal optical phased array,” Proc. SPIE 7133, 71333L, 71333L-8 (2008), http://dx.doi.org/10.1117/12.821247 .
    [CrossRef]
  22. L. Xu, J. Zhang, and L. Y. Wu, “Influence of phase delay profile on diffraction efficiency of liquid crystal optical phased array,” Opt. Laser Technol. 41(4), 509–516 (2009), http://dx.doi.org/10.1016/j.optlastec.2008.07.003 .
    [CrossRef]
  23. X. D. Xun and R. W. Cohn, “Phase calibration of spatially nonuniform spatial light modulators,” Appl. Opt. 43(35), 6400–6406 (2004), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-43-35-6400 .
    [CrossRef] [PubMed]

2010 (1)

2009 (4)

D. Engström, A. Frank, J. Backsten, M. Goksör, and J. Bengtsson, “Grid-free 3D multiple spot generation with an efficient single-plane FFT-based algorithm,” Opt. Express 17(12), 9989–10000 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-12-9989 .
[CrossRef] [PubMed]

J. Pine and G. Chow, “Moving live dissociated neurons with an optical tweezer,” IEEE Trans. Biomed. Eng. 56(4), 1184–1188 (2009), http://dx.doi.org/10.1109/TBME.2008.2005641 .
[CrossRef] [PubMed]

M. Funk, S. J. Parkin, A. B. Stilgoe, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Constant power optical tweezers with controllable torque,” Opt. Lett. 34(2), 139–141 (2009), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-34-2-139 .
[CrossRef] [PubMed]

L. Xu, J. Zhang, and L. Y. Wu, “Influence of phase delay profile on diffraction efficiency of liquid crystal optical phased array,” Opt. Laser Technol. 41(4), 509–516 (2009), http://dx.doi.org/10.1016/j.optlastec.2008.07.003 .
[CrossRef]

2008 (2)

L. Xu, L. Y. Wu, J. Zhang, and X. Liu, “Effect of phase valley on diffraction efficiency of liquid crystal optical phased array,” Proc. SPIE 7133, 71333L, 71333L-8 (2008), http://dx.doi.org/10.1117/12.821247 .
[CrossRef]

O. Akcakir, C. R. Knutson, C. Duke, E. Tanner, D. M. Mueth, J. S. Plewa, and K. F. Bradley, “High-sensitivity measurement of free-protein concentration using optical tweezers,” Proc. SPIE 6863, 686305, 686305-11 (2008), http://dx.doi.org/10.1117/12.763924 .
[CrossRef]

2007 (2)

R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15(4), 1913–1922 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-4-1913 .
[CrossRef] [PubMed]

A. R. Moradi, E. Ferrari, V. Garbin, E. Di Fabrizio, and D. Cojoc, “Strength control in multiple optical traps generated by means of diffractive optical elements,” Optoelectron. Adv. Mater. 1, 158–161 (2007), http://www.infim.ro/~oamrc/index.php?option=magazine&op=view&idu=315&catid=12 .

2006 (1)

S. Serati and J. Harriman, “Spatial light modulator considerations for beam control in optical manipulation applications,” Proc. SPIE 6326, 63262W, 63262W-11 (2006), http://link.aip.org/link/?PSI/6326/63262W/1 .
[CrossRef]

2005 (2)

2004 (2)

M. Škereň, I. Richter, and P. Fiala, “Design and optimization considerations of multi-focus phase-only diffractive elements,” Proc. SPIE 5182, 233–242 (2004), http://dx.doi.org/10.1117/12.505573 .
[CrossRef]

X. D. Xun and R. W. Cohn, “Phase calibration of spatially nonuniform spatial light modulators,” Appl. Opt. 43(35), 6400–6406 (2004), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-43-35-6400 .
[CrossRef] [PubMed]

2002 (2)

C. J. Kennedy, “Model for variation of laser power with M2.,” Appl. Opt. 41(21), 4341–4346 (2002), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-41-21-4341 .
[CrossRef] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002), http://dx.doi.org/10.1016/S0030-4018(02)01524-9 .
[CrossRef]

1997 (1)

M. L. Scott, L. A. Bieber, and T. S. Kalkur, “Gray scale deformable grating spatial light modulator for high speed optical processing,” Proc. SPIE 3046, 129–136 (1997), http://dx.doi.org/10.1117/12.276600 .
[CrossRef]

1994 (1)

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23(1), 247–285 (1994), http://dx.doi.org/10.1146/annurev.bb.23.060194.001335 .
[CrossRef] [PubMed]

1986 (1)

1972 (1)

1970 (1)

H. Dammann, “Blazed synthetic phase-only holograms,” Optik (Stuttg.) 31, 95–104 (1970).

Akahori, H.

Akcakir, O.

O. Akcakir, C. R. Knutson, C. Duke, E. Tanner, D. M. Mueth, J. S. Plewa, and K. F. Bradley, “High-sensitivity measurement of free-protein concentration using optical tweezers,” Proc. SPIE 6863, 686305, 686305-11 (2008), http://dx.doi.org/10.1117/12.763924 .
[CrossRef]

Backsten, J.

Bengtsson, J.

Bieber, L. A.

M. L. Scott, L. A. Bieber, and T. S. Kalkur, “Gray scale deformable grating spatial light modulator for high speed optical processing,” Proc. SPIE 3046, 129–136 (1997), http://dx.doi.org/10.1117/12.276600 .
[CrossRef]

Block, S. M.

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23(1), 247–285 (1994), http://dx.doi.org/10.1146/annurev.bb.23.060194.001335 .
[CrossRef] [PubMed]

Bradley, K. F.

O. Akcakir, C. R. Knutson, C. Duke, E. Tanner, D. M. Mueth, J. S. Plewa, and K. F. Bradley, “High-sensitivity measurement of free-protein concentration using optical tweezers,” Proc. SPIE 6863, 686305, 686305-11 (2008), http://dx.doi.org/10.1117/12.763924 .
[CrossRef]

Chow, G.

J. Pine and G. Chow, “Moving live dissociated neurons with an optical tweezer,” IEEE Trans. Biomed. Eng. 56(4), 1184–1188 (2009), http://dx.doi.org/10.1109/TBME.2008.2005641 .
[CrossRef] [PubMed]

Cohn, R. W.

Cojoc, D.

A. R. Moradi, E. Ferrari, V. Garbin, E. Di Fabrizio, and D. Cojoc, “Strength control in multiple optical traps generated by means of diffractive optical elements,” Optoelectron. Adv. Mater. 1, 158–161 (2007), http://www.infim.ro/~oamrc/index.php?option=magazine&op=view&idu=315&catid=12 .

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002), http://dx.doi.org/10.1016/S0030-4018(02)01524-9 .
[CrossRef]

Dammann, H.

H. Dammann, “Blazed synthetic phase-only holograms,” Optik (Stuttg.) 31, 95–104 (1970).

Di Fabrizio, E.

A. R. Moradi, E. Ferrari, V. Garbin, E. Di Fabrizio, and D. Cojoc, “Strength control in multiple optical traps generated by means of diffractive optical elements,” Optoelectron. Adv. Mater. 1, 158–161 (2007), http://www.infim.ro/~oamrc/index.php?option=magazine&op=view&idu=315&catid=12 .

Di Leonardo, R.

Duke, C.

O. Akcakir, C. R. Knutson, C. Duke, E. Tanner, D. M. Mueth, J. S. Plewa, and K. F. Bradley, “High-sensitivity measurement of free-protein concentration using optical tweezers,” Proc. SPIE 6863, 686305, 686305-11 (2008), http://dx.doi.org/10.1117/12.763924 .
[CrossRef]

Eckerle, K. L.

Engström, D.

Ferrari, E.

A. R. Moradi, E. Ferrari, V. Garbin, E. Di Fabrizio, and D. Cojoc, “Strength control in multiple optical traps generated by means of diffractive optical elements,” Optoelectron. Adv. Mater. 1, 158–161 (2007), http://www.infim.ro/~oamrc/index.php?option=magazine&op=view&idu=315&catid=12 .

Fiala, P.

M. Škereň, I. Richter, and P. Fiala, “Design and optimization considerations of multi-focus phase-only diffractive elements,” Proc. SPIE 5182, 233–242 (2004), http://dx.doi.org/10.1117/12.505573 .
[CrossRef]

Frank, A.

Funk, M.

Garbin, V.

A. R. Moradi, E. Ferrari, V. Garbin, E. Di Fabrizio, and D. Cojoc, “Strength control in multiple optical traps generated by means of diffractive optical elements,” Optoelectron. Adv. Mater. 1, 158–161 (2007), http://www.infim.ro/~oamrc/index.php?option=magazine&op=view&idu=315&catid=12 .

Goksör, M.

Grier, D. G.

Harriman, J.

S. Serati and J. Harriman, “Spatial light modulator considerations for beam control in optical manipulation applications,” Proc. SPIE 6326, 63262W, 63262W-11 (2006), http://link.aip.org/link/?PSI/6326/63262W/1 .
[CrossRef]

Heckenberg, N. R.

Ianni, F.

Kalkur, T. S.

M. L. Scott, L. A. Bieber, and T. S. Kalkur, “Gray scale deformable grating spatial light modulator for high speed optical processing,” Proc. SPIE 3046, 129–136 (1997), http://dx.doi.org/10.1117/12.276600 .
[CrossRef]

Kennedy, C. J.

Knutson, C. R.

O. Akcakir, C. R. Knutson, C. Duke, E. Tanner, D. M. Mueth, J. S. Plewa, and K. F. Bradley, “High-sensitivity measurement of free-protein concentration using optical tweezers,” Proc. SPIE 6863, 686305, 686305-11 (2008), http://dx.doi.org/10.1117/12.763924 .
[CrossRef]

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002), http://dx.doi.org/10.1016/S0030-4018(02)01524-9 .
[CrossRef]

Ladavac, K.

Lee, S. H.

Liu, X.

L. Xu, L. Y. Wu, J. Zhang, and X. Liu, “Effect of phase valley on diffraction efficiency of liquid crystal optical phased array,” Proc. SPIE 7133, 71333L, 71333L-8 (2008), http://dx.doi.org/10.1117/12.821247 .
[CrossRef]

Mielenz, K. D.

Moradi, A. R.

A. R. Moradi, E. Ferrari, V. Garbin, E. Di Fabrizio, and D. Cojoc, “Strength control in multiple optical traps generated by means of diffractive optical elements,” Optoelectron. Adv. Mater. 1, 158–161 (2007), http://www.infim.ro/~oamrc/index.php?option=magazine&op=view&idu=315&catid=12 .

Mueth, D. M.

O. Akcakir, C. R. Knutson, C. Duke, E. Tanner, D. M. Mueth, J. S. Plewa, and K. F. Bradley, “High-sensitivity measurement of free-protein concentration using optical tweezers,” Proc. SPIE 6863, 686305, 686305-11 (2008), http://dx.doi.org/10.1117/12.763924 .
[CrossRef]

Nieminen, T. A.

Parkin, S. J.

Persson, M.

Pine, J.

J. Pine and G. Chow, “Moving live dissociated neurons with an optical tweezer,” IEEE Trans. Biomed. Eng. 56(4), 1184–1188 (2009), http://dx.doi.org/10.1109/TBME.2008.2005641 .
[CrossRef] [PubMed]

Plewa, J. S.

O. Akcakir, C. R. Knutson, C. Duke, E. Tanner, D. M. Mueth, J. S. Plewa, and K. F. Bradley, “High-sensitivity measurement of free-protein concentration using optical tweezers,” Proc. SPIE 6863, 686305, 686305-11 (2008), http://dx.doi.org/10.1117/12.763924 .
[CrossRef]

Polin, M.

Richter, I.

M. Škereň, I. Richter, and P. Fiala, “Design and optimization considerations of multi-focus phase-only diffractive elements,” Proc. SPIE 5182, 233–242 (2004), http://dx.doi.org/10.1117/12.505573 .
[CrossRef]

Roichman, Y.

Rubinsztein-Dunlop, H.

Ruocco, G.

Scott, M. L.

M. L. Scott, L. A. Bieber, and T. S. Kalkur, “Gray scale deformable grating spatial light modulator for high speed optical processing,” Proc. SPIE 3046, 129–136 (1997), http://dx.doi.org/10.1117/12.276600 .
[CrossRef]

Serati, S.

S. Serati and J. Harriman, “Spatial light modulator considerations for beam control in optical manipulation applications,” Proc. SPIE 6326, 63262W, 63262W-11 (2006), http://link.aip.org/link/?PSI/6326/63262W/1 .
[CrossRef]

Škeren, M.

M. Škereň, I. Richter, and P. Fiala, “Design and optimization considerations of multi-focus phase-only diffractive elements,” Proc. SPIE 5182, 233–242 (2004), http://dx.doi.org/10.1117/12.505573 .
[CrossRef]

Stilgoe, A. B.

Svoboda, K.

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23(1), 247–285 (1994), http://dx.doi.org/10.1146/annurev.bb.23.060194.001335 .
[CrossRef] [PubMed]

Tanner, E.

O. Akcakir, C. R. Knutson, C. Duke, E. Tanner, D. M. Mueth, J. S. Plewa, and K. F. Bradley, “High-sensitivity measurement of free-protein concentration using optical tweezers,” Proc. SPIE 6863, 686305, 686305-11 (2008), http://dx.doi.org/10.1117/12.763924 .
[CrossRef]

Wu, L. Y.

L. Xu, J. Zhang, and L. Y. Wu, “Influence of phase delay profile on diffraction efficiency of liquid crystal optical phased array,” Opt. Laser Technol. 41(4), 509–516 (2009), http://dx.doi.org/10.1016/j.optlastec.2008.07.003 .
[CrossRef]

L. Xu, L. Y. Wu, J. Zhang, and X. Liu, “Effect of phase valley on diffraction efficiency of liquid crystal optical phased array,” Proc. SPIE 7133, 71333L, 71333L-8 (2008), http://dx.doi.org/10.1117/12.821247 .
[CrossRef]

Xu, L.

L. Xu, J. Zhang, and L. Y. Wu, “Influence of phase delay profile on diffraction efficiency of liquid crystal optical phased array,” Opt. Laser Technol. 41(4), 509–516 (2009), http://dx.doi.org/10.1016/j.optlastec.2008.07.003 .
[CrossRef]

L. Xu, L. Y. Wu, J. Zhang, and X. Liu, “Effect of phase valley on diffraction efficiency of liquid crystal optical phased array,” Proc. SPIE 7133, 71333L, 71333L-8 (2008), http://dx.doi.org/10.1117/12.821247 .
[CrossRef]

Xun, X. D.

Zhang, J.

L. Xu, J. Zhang, and L. Y. Wu, “Influence of phase delay profile on diffraction efficiency of liquid crystal optical phased array,” Opt. Laser Technol. 41(4), 509–516 (2009), http://dx.doi.org/10.1016/j.optlastec.2008.07.003 .
[CrossRef]

L. Xu, L. Y. Wu, J. Zhang, and X. Liu, “Effect of phase valley on diffraction efficiency of liquid crystal optical phased array,” Proc. SPIE 7133, 71333L, 71333L-8 (2008), http://dx.doi.org/10.1117/12.821247 .
[CrossRef]

Annu. Rev. Biophys. Biomol. Struct. (1)

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23(1), 247–285 (1994), http://dx.doi.org/10.1146/annurev.bb.23.060194.001335 .
[CrossRef] [PubMed]

Appl. Opt. (4)

IEEE Trans. Biomed. Eng. (1)

J. Pine and G. Chow, “Moving live dissociated neurons with an optical tweezer,” IEEE Trans. Biomed. Eng. 56(4), 1184–1188 (2009), http://dx.doi.org/10.1109/TBME.2008.2005641 .
[CrossRef] [PubMed]

Opt. Commun. (1)

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207(1-6), 169–175 (2002), http://dx.doi.org/10.1016/S0030-4018(02)01524-9 .
[CrossRef]

Opt. Express (5)

Opt. Laser Technol. (1)

L. Xu, J. Zhang, and L. Y. Wu, “Influence of phase delay profile on diffraction efficiency of liquid crystal optical phased array,” Opt. Laser Technol. 41(4), 509–516 (2009), http://dx.doi.org/10.1016/j.optlastec.2008.07.003 .
[CrossRef]

Opt. Lett. (1)

Optik (Stuttg.) (1)

H. Dammann, “Blazed synthetic phase-only holograms,” Optik (Stuttg.) 31, 95–104 (1970).

Optoelectron. Adv. Mater. (1)

A. R. Moradi, E. Ferrari, V. Garbin, E. Di Fabrizio, and D. Cojoc, “Strength control in multiple optical traps generated by means of diffractive optical elements,” Optoelectron. Adv. Mater. 1, 158–161 (2007), http://www.infim.ro/~oamrc/index.php?option=magazine&op=view&idu=315&catid=12 .

Proc. SPIE (5)

O. Akcakir, C. R. Knutson, C. Duke, E. Tanner, D. M. Mueth, J. S. Plewa, and K. F. Bradley, “High-sensitivity measurement of free-protein concentration using optical tweezers,” Proc. SPIE 6863, 686305, 686305-11 (2008), http://dx.doi.org/10.1117/12.763924 .
[CrossRef]

M. Škereň, I. Richter, and P. Fiala, “Design and optimization considerations of multi-focus phase-only diffractive elements,” Proc. SPIE 5182, 233–242 (2004), http://dx.doi.org/10.1117/12.505573 .
[CrossRef]

S. Serati and J. Harriman, “Spatial light modulator considerations for beam control in optical manipulation applications,” Proc. SPIE 6326, 63262W, 63262W-11 (2006), http://link.aip.org/link/?PSI/6326/63262W/1 .
[CrossRef]

L. Xu, L. Y. Wu, J. Zhang, and X. Liu, “Effect of phase valley on diffraction efficiency of liquid crystal optical phased array,” Proc. SPIE 7133, 71333L, 71333L-8 (2008), http://dx.doi.org/10.1117/12.821247 .
[CrossRef]

M. L. Scott, L. A. Bieber, and T. S. Kalkur, “Gray scale deformable grating spatial light modulator for high speed optical processing,” Proc. SPIE 3046, 129–136 (1997), http://dx.doi.org/10.1117/12.276600 .
[CrossRef]

Other (2)

D. C. O'Shea, T. J. Suleski, A. D. Kathman, and D. W. Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE, 2004).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Supplementary Material (2)

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Fig. 1
Fig. 1

Simulation results of linearly controlling the power of a single spot with a resolution of 0.002 by GSW. (a) Use the same weighting as Eq. (1) in Ref. [3]. and set the convergence parameter to 0.25 to linearly adjust the power at a single spot (Media 1) located in signal area 362 × 362 centering in hologram 512 × 512; during the iterations per aiming power, the generated complex amplitudes in dummy area are kept until the desired power is obtained, there are two spots containing most power of dummy area. (b) 500 equidistant powers of spot (128, 128, 0) are obtained in more than 9 hours by Matlab in PC with 2.8GHz Intel Pentium D 820 CPU and 512MB Memory, and (c) the number of iterations per aiming power is less than 500, and by average, 105 iterations.

Fig. 2
Fig. 2

Tunable binary phase grating. (a) Phase profile. (b) 2D gray image.

Fig. 3
Fig. 3

Schematic diagram of Fourier optics propagation from SLM plane (back focal plane) to imaging plane.

Fig. 4
Fig. 4

Normalized simulated power adjusted by a tunable binary phase grating.

Fig. 5
Fig. 5

Measurement setup, LCSLM is placed in the back focal plane of the lens.

Fig. 6
Fig. 6

Control the total power of four 3D spots with a projection spacing 128 by superimposing binary phase grating with a conventional phase-only hologram by sweeping the phase depth of grating from 0 to 2π with a step of 2π/1600. A 100-iteration phase-only hologram displayed by (a) a 8-bit gray image generated (b) 4 3D spots marked in turn by 1, 2, 3 and 4 (Media 2 captured nearby the front focal plane in Fig. 5 one-by-one by a CCD of PointGrey GRAS-20S4M-C). (c) Measured relative powers of these 4 spots. (d) Total spot power, a sum of the relative powers of these spots, has up-shifted the minimal total power above zero. (e) Total spot power, normalized in range of 0-1, can be controlled linearly in range of (f) phase depth [0,π] and (g) phase depth [0,2π].

Fig. 7
Fig. 7

Control the total spot power to random values by superimposing binary phase grating with the same hologram as shown in Fig. 6(a) by changing the phase depth of grating to the sequence of random values going from 0 to 2π. (a) 100 phase depths of grating chosen for the following step-by-step measurement. In order to make sure the largest power adjustment range to be measured for the normalization of total spot power, the phase depth of the first step was set to 0, the phase depth of the 50th step was set to π, and the phase depth of the last step was set to 2π; the other 97 phase depths were chosen using an uniformly distributed random function. (b) Normalized total spot power can be controlled randomly using phase depth shown in (a). (c) The standard deviation of normalized total spot powers per step can be obtained upon completion of 10 times 100-step measurement, and the arithmetic mean of all standard deviations is less than 0.0006.

Tables (2)

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Table 1 Simulation Results of Adjusting Normalized Power by Eq. (8)*

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Table 2 Measurement Results of Four Equal-Bright 3D Spots

Equations (8)

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U ( ξ ) = C F [ e i φ BG ( x ) ] = C 2 n = δ ( ξ n 2 d pixel ) sinc ( n 2 ) ( 1 + e i φ e i n π ) e i n π 2
I 0 ( φ ) = | U ( ξ ) | 2 = I 0 ( 1 + cos φ ) / 2
U 1 ( x , y ) = 1 λ f F [ e i φ ( x , y ) e i π λ f ( x 2 + y 2 ) ]
U 1 ( ξ , η , z ) = 1 λ f F [ e i φ ( x , y ) e i π z λ f 2 ( x 2 + y 2 ) ]
φ ( x , y ) = φ H ( x , y ) + φ BG ( x )
U ( ξ , η , z ) = U H ( ξ , η , z ) 1 2 n = δ ( ξ n 2 d pixel ) sinc ( n 2 ) ( 1 + e i φ e i n π ) e i n π 2
I ( ξ , η , z ) = | U H ( ξ , η , z ) | 2 ( 1 + cos φ ) / 2
P Total ( ξ , η , z ) = ( 1 + cos φ ) / 2

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