Abstract

Far-field multicolor patterns and characters are emitted effectively in a relatively wide and deep spatial region by plastic diffractive micro-optics elements (DMOEs), which are illuminated directly by common Gaussian lasers in the visible range. Phase-only DMOEs are composed of a large number of fine step-shaped phase microstructures distributed sequentially over the plastic wafer selected. The initial DMOEs in silicon wafer are fabricated by an innovative technique with a combination of a single-mask ultraviolet photolithography and low-cost and rapid wet KOH etching. The fabricated silicon DMOEs are further converted into a nickel mask by the conventional electrochemical method, and they are finally transferred onto the surface of the plastic wafer through mature hot embossing. Morphological measurements show that the surface roughness of the plastic DMOEs is in the nanometer range, and the feature height of the phase steps in diffractive elements is in the submicrometer scale, which can be designed and adjusted flexibly according to requirements. The dimensions of the DMOEs can be changed from the order of millimeters to centimeters. A large number of pixel phase microstructures with a square microappearance employed to construct the phase-only DMOEs are created by the Gerchberg–Saxton algorithm, according to the target patterns and characters and common Gaussian lasers manipulated by the DMOEs fabricated.

© 2011 Optical Society of America

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2010 (1)

H. M. Leung, G. Zhou, H. Yu, F. S. Chau, and A. S. Kumar, “Diamond turning and soft lithography processes for liquid tunable lenses,” J. Micromech. Microeng. 20, 025021 (2010).
[CrossRef]

2008 (2)

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

J. Tan, M. Shan, C. Zhao, and J. Liu, “Design and fabrication of diffractive microlens arrays with continuous relief for parallel laser direct writing,” Appl. Opt. 47, 1430–1433 (2008).
[CrossRef] [PubMed]

2007 (1)

M. Lee and K. Kuo, “Single-step fabrication of Fresnel microlens array on sapphire substrate of flip-chip gallium nitride light emitting diode by focused ion beam,” Appl. Phys. Lett. 91, 051111(2007).
[CrossRef]

2006 (2)

2005 (3)

N. Sanner, N. Huot, E. Audouard, C. Larat, J. Huignard, and B. Loiseaux, “Programmable focal spot shaping of amplified femtosecond laser pulses,” Opt. Lett. 30, 1479–1481 (2005).
[CrossRef] [PubMed]

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far-field pattern formation,” J. Opt. A: Pure Appl. Opt. 7, S276–S279 (2005).
[CrossRef]

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far field pattern formation,” J. Opt. A: Pure Appl. Opt. 7, S276–S279 (2005).
[CrossRef]

2004 (6)

2003 (1)

C. Chen, M. Li, C. Chang, J. Sheu, G. Chi, W. Cheng, J. Yeh, J. Chang, and T. Ito, “GaN diffractive microlenses fabricated with gray-level mask,” Opt. Commun. 215, 75–78 (2003).
[CrossRef]

2002 (3)

R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications, Proc. IEEE 90, 222–244 (2002) .
[CrossRef]

J. Sze and M. Lu, “Design and fabrication of the diffractive phase element that synthesizes three-color pseudo-nondiffracting beams,” Opt. Eng. 41, 3127–3135 (2002).
[CrossRef]

J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. 27, 1463–1465 (2002).
[CrossRef]

2001 (2)

1997 (1)

1994 (1)

D. L. Kendall, W. P. Eaton, R. Manginell, T. G. Digges, Jr., “Micromirror arrays using KOH:H2O micromachining of silicon for lens templates, geodesic lenses, and other applications,” Opt. Eng. 33, 3578–3588 (1994).
[CrossRef]

Andersson, G.

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Audouard, E.

Barton, I. M.

Bengtsson, J.

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Blair, P.

Caley, A. J.

A. J. Caley and M. R. Taghizadeh, “Analysis of the effects of bias phase and wavelength choice on the design of dual-wavelength diffractive optical elements,” J. Opt. Soc. Am. A 23, 193–198(2006).
[CrossRef]

A. J. Caley and M. R. Taghizadeh, “Analysis of the effects of bias phase and wavelength choice on the design of dual-wavelength diffractive optical elements,” J. Opt. Soc. Am. A 23, 193–198(2006).
[CrossRef]

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far field pattern formation,” J. Opt. A: Pure Appl. Opt. 7, S276–S279 (2005).
[CrossRef]

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far-field pattern formation,” J. Opt. A: Pure Appl. Opt. 7, S276–S279 (2005).
[CrossRef]

Chang, C.

C. Chen, M. Li, C. Chang, J. Sheu, G. Chi, W. Cheng, J. Yeh, J. Chang, and T. Ito, “GaN diffractive microlenses fabricated with gray-level mask,” Opt. Commun. 215, 75–78 (2003).
[CrossRef]

Chang, J.

C. Chen, M. Li, C. Chang, J. Sheu, G. Chi, W. Cheng, J. Yeh, J. Chang, and T. Ito, “GaN diffractive microlenses fabricated with gray-level mask,” Opt. Commun. 215, 75–78 (2003).
[CrossRef]

Chau, F. S.

H. M. Leung, G. Zhou, H. Yu, F. S. Chau, and A. S. Kumar, “Diamond turning and soft lithography processes for liquid tunable lenses,” J. Micromech. Microeng. 20, 025021 (2010).
[CrossRef]

Chen, C.

C. Chen, M. Li, C. Chang, J. Sheu, G. Chi, W. Cheng, J. Yeh, J. Chang, and T. Ito, “GaN diffractive microlenses fabricated with gray-level mask,” Opt. Commun. 215, 75–78 (2003).
[CrossRef]

Cheng, W.

C. Chen, M. Li, C. Chang, J. Sheu, G. Chi, W. Cheng, J. Yeh, J. Chang, and T. Ito, “GaN diffractive microlenses fabricated with gray-level mask,” Opt. Commun. 215, 75–78 (2003).
[CrossRef]

Chi, G.

C. Chen, M. Li, C. Chang, J. Sheu, G. Chi, W. Cheng, J. Yeh, J. Chang, and T. Ito, “GaN diffractive microlenses fabricated with gray-level mask,” Opt. Commun. 215, 75–78 (2003).
[CrossRef]

Digges, T. G.

D. L. Kendall, W. P. Eaton, R. Manginell, T. G. Digges, Jr., “Micromirror arrays using KOH:H2O micromachining of silicon for lens templates, geodesic lenses, and other applications,” Opt. Eng. 33, 3578–3588 (1994).
[CrossRef]

Eaton, W. P.

D. L. Kendall, W. P. Eaton, R. Manginell, T. G. Digges, Jr., “Micromirror arrays using KOH:H2O micromachining of silicon for lens templates, geodesic lenses, and other applications,” Opt. Eng. 33, 3578–3588 (1994).
[CrossRef]

Enoksson, P.

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Hedsten, K.

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Herzig, H. P.

Huignard, J.

Huot, N.

Ichioka, Y.

Ito, T.

C. Chen, M. Li, C. Chang, J. Sheu, G. Chi, W. Cheng, J. Yeh, J. Chang, and T. Ito, “GaN diffractive microlenses fabricated with gray-level mask,” Opt. Commun. 215, 75–78 (2003).
[CrossRef]

Karlén, D.

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Kendall, D. L.

D. L. Kendall, W. P. Eaton, R. Manginell, T. G. Digges, Jr., “Micromirror arrays using KOH:H2O micromachining of silicon for lens templates, geodesic lenses, and other applications,” Opt. Eng. 33, 3578–3588 (1994).
[CrossRef]

Kumar, A. S.

H. M. Leung, G. Zhou, H. Yu, F. S. Chau, and A. S. Kumar, “Diamond turning and soft lithography processes for liquid tunable lenses,” J. Micromech. Microeng. 20, 025021 (2010).
[CrossRef]

Kuo, K.

M. Lee and K. Kuo, “Single-step fabrication of Fresnel microlens array on sapphire substrate of flip-chip gallium nitride light emitting diode by focused ion beam,” Appl. Phys. Lett. 91, 051111(2007).
[CrossRef]

Larat, C.

Lee, M.

M. Lee and K. Kuo, “Single-step fabrication of Fresnel microlens array on sapphire substrate of flip-chip gallium nitride light emitting diode by focused ion beam,” Appl. Phys. Lett. 91, 051111(2007).
[CrossRef]

Leung, H. M.

H. M. Leung, G. Zhou, H. Yu, F. S. Chau, and A. S. Kumar, “Diamond turning and soft lithography processes for liquid tunable lenses,” J. Micromech. Microeng. 20, 025021 (2010).
[CrossRef]

Li, M.

C. Chen, M. Li, C. Chang, J. Sheu, G. Chi, W. Cheng, J. Yeh, J. Chang, and T. Ito, “GaN diffractive microlenses fabricated with gray-level mask,” Opt. Commun. 215, 75–78 (2003).
[CrossRef]

Li, Y.

Liu, J.

Liu, J. S.

Löfving, B.

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Loiseaux, B.

Lu, M.

J. Sze and M. Lu, “Design and fabrication of the diffractive phase element that synthesizes three-color pseudo-nondiffracting beams,” Opt. Eng. 41, 3127–3135 (2002).
[CrossRef]

Magath, T.

T. Magath, “Synthesis of three-dimensional light fields and of a quasi-optical power splitter at 150 GHz,” IEEE Trans. Microw. Theory Tech. 52, 2385–2389 (2004).
[CrossRef]

Manginell, R.

D. L. Kendall, W. P. Eaton, R. Manginell, T. G. Digges, Jr., “Micromirror arrays using KOH:H2O micromachining of silicon for lens templates, geodesic lenses, and other applications,” Opt. Eng. 33, 3578–3588 (1994).
[CrossRef]

Melin, J.

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Modh, P.

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Nikolajeff, F.

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Nilsson, R.

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Ogura, Y.

Persson, K.

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Piestun, R.

R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications, Proc. IEEE 90, 222–244 (2002) .
[CrossRef]

Rödjegård, H.

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Rossi, M.

Sanner, N.

Schilling, A.

Shamir, J.

R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications, Proc. IEEE 90, 222–244 (2002) .
[CrossRef]

Shan, M.

Sheu, J.

C. Chen, M. Li, C. Chang, J. Sheu, G. Chi, W. Cheng, J. Yeh, J. Chang, and T. Ito, “GaN diffractive microlenses fabricated with gray-level mask,” Opt. Commun. 215, 75–78 (2003).
[CrossRef]

Shirai, N.

Soifer, V. A.

V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

Stauffer, L.

Sze, J.

J. Sze and M. Lu, “Design and fabrication of the diffractive phase element that synthesizes three-color pseudo-nondiffracting beams,” Opt. Eng. 41, 3127–3135 (2002).
[CrossRef]

Taghizadeh, M. R.

A. J. Caley and M. R. Taghizadeh, “Analysis of the effects of bias phase and wavelength choice on the design of dual-wavelength diffractive optical elements,” J. Opt. Soc. Am. A 23, 193–198(2006).
[CrossRef]

A. J. Caley and M. R. Taghizadeh, “Analysis of the effects of bias phase and wavelength choice on the design of dual-wavelength diffractive optical elements,” J. Opt. Soc. Am. A 23, 193–198(2006).
[CrossRef]

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far-field pattern formation,” J. Opt. A: Pure Appl. Opt. 7, S276–S279 (2005).
[CrossRef]

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far field pattern formation,” J. Opt. A: Pure Appl. Opt. 7, S276–S279 (2005).
[CrossRef]

M. J. Thomson, J. Liu, and M. R. Taghizadeh, “Iterative algorithm for the design of free-space diffractive optical elements for fiber coupling,” Appl. Opt. 43, 1996–1999 (2004).
[CrossRef] [PubMed]

M. J. Thomson, J. Liu, and M. R. Taghizadeh, “Iterative algorithm for the design of free-space diffractive optical elements for fiber coupling,” Appl. Opt. 43, 1996–1999 (2004).
[CrossRef] [PubMed]

J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. 27, 1463–1465 (2002).
[CrossRef]

I. M. Barton, P. Blair, and M. R. Taghizadeh, “Dual-wavelength operation diffractive phase elements for pattern formation,” Opt. Express 1, 54–59 (1997).
[CrossRef] [PubMed]

Tan, J.

Tanida, J.

Tao, S. H.

Thomson, M. J.

Vokinger, U.

Waddie, A. J.

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far field pattern formation,” J. Opt. A: Pure Appl. Opt. 7, S276–S279 (2005).
[CrossRef]

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far-field pattern formation,” J. Opt. A: Pure Appl. Opt. 7, S276–S279 (2005).
[CrossRef]

Yeh, J.

C. Chen, M. Li, C. Chang, J. Sheu, G. Chi, W. Cheng, J. Yeh, J. Chang, and T. Ito, “GaN diffractive microlenses fabricated with gray-level mask,” Opt. Commun. 215, 75–78 (2003).
[CrossRef]

Yu, H.

H. M. Leung, G. Zhou, H. Yu, F. S. Chau, and A. S. Kumar, “Diamond turning and soft lithography processes for liquid tunable lenses,” J. Micromech. Microeng. 20, 025021 (2010).
[CrossRef]

Yuan, X.

Zhao, C.

Zhao, Y.

Zhou, G.

H. M. Leung, G. Zhou, H. Yu, F. S. Chau, and A. S. Kumar, “Diamond turning and soft lithography processes for liquid tunable lenses,” J. Micromech. Microeng. 20, 025021 (2010).
[CrossRef]

Zhou, Q.

Appl. Opt. (6)

Appl. Phys. Lett. (1)

M. Lee and K. Kuo, “Single-step fabrication of Fresnel microlens array on sapphire substrate of flip-chip gallium nitride light emitting diode by focused ion beam,” Appl. Phys. Lett. 91, 051111(2007).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (1)

T. Magath, “Synthesis of three-dimensional light fields and of a quasi-optical power splitter at 150 GHz,” IEEE Trans. Microw. Theory Tech. 52, 2385–2389 (2004).
[CrossRef]

J. Micromech. Microeng. (1)

H. M. Leung, G. Zhou, H. Yu, F. S. Chau, and A. S. Kumar, “Diamond turning and soft lithography processes for liquid tunable lenses,” J. Micromech. Microeng. 20, 025021 (2010).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (2)

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far field pattern formation,” J. Opt. A: Pure Appl. Opt. 7, S276–S279 (2005).
[CrossRef]

A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far-field pattern formation,” J. Opt. A: Pure Appl. Opt. 7, S276–S279 (2005).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

C. Chen, M. Li, C. Chang, J. Sheu, G. Chi, W. Cheng, J. Yeh, J. Chang, and T. Ito, “GaN diffractive microlenses fabricated with gray-level mask,” Opt. Commun. 215, 75–78 (2003).
[CrossRef]

Opt. Eng. (2)

D. L. Kendall, W. P. Eaton, R. Manginell, T. G. Digges, Jr., “Micromirror arrays using KOH:H2O micromachining of silicon for lens templates, geodesic lenses, and other applications,” Opt. Eng. 33, 3578–3588 (1994).
[CrossRef]

J. Sze and M. Lu, “Design and fabrication of the diffractive phase element that synthesizes three-color pseudo-nondiffracting beams,” Opt. Eng. 41, 3127–3135 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Proc. IEEE (1)

R. Piestun and J. Shamir, “Synthesis of three-dimensional light fields and applications, Proc. IEEE 90, 222–244 (2002) .
[CrossRef]

Sens. Actuators A: Phys. (1)

K. Hedsten, J. Melin, J. Bengtsson, P. Modh, D. Karlén, B. Löfving, R. Nilsson, H. Rödjegård, K. Persson, P. Enoksson, F. Nikolajeff, and G. Andersson, “MEMS-based VCSEL beam steering using replicated polymer diffractive lens,” Sens. Actuators A: Phys. 142, 336–345 (2008).
[CrossRef]

Other (1)

V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

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Figures (8)

Fig. 1
Fig. 1

Typical diffractive integral transformation for forming the desired far-field beam intensity distribution or patterns by DMOEs.

Fig. 2
Fig. 2

Forming far-field head patterns through the diffractive elements with a two-level, eight-level, and arbitrary phase step, respectively.

Fig. 3
Fig. 3

Silicon diffractive microstructures with a step-shaped profile shaped by single-mask UV photolithography and dual-step KOH : H 2 O etching.

Fig. 4
Fig. 4

Shaping silicon phase microstructures with the desired phase steps and phase facets by further etching neighboring converse pyramid-shaped microholes with different aperture and depth over { 100 } -oriented silicon wafer, which are preformed by the first or common KOH : H 2 O etching.

Fig. 5
Fig. 5

The simulated phase microstructure distribution for (a) emission head patterns and (b) Chinese characters through directly illuminating them by conventional Gaussian laser beams with a circular profile and uniform beam distribution.

Fig. 6
Fig. 6

Optical photograph of the partial DMOE and several typical phase steps fabricated in a silicon wafer.

Fig. 7
Fig. 7

Emitting multicolor patterns by plastic DMOEs fabricated. The Gaussian lasers with wavelengths of approximately 450, 650, and 532 nm , respectively, are normally incident upon the patterned surface of the DMOEs and then output from the backside of the elements, and therefore they form the far-field beam intensity distributions correspondent to the Lena portrait.

Fig. 8
Fig. 8

Chinese characters emitted from a plastic DMOE fabricated, which manipulates the incident Gaussian laser beams with the wavelength of 532 nm .

Tables (1)

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Table 1 Main Properties of Diffractive Elements for Emission Far-Field Patterns

Equations (6)

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F ( x , y ) = i k 2 π z e i k z W ( u , v ) H ( u x , v y , z ) d u d v ,
I 0 ( u , v ) = | F ( x , y ) | 2 = | A 0 ( u , v ) e i φ ( u , v ) H ( u x , v y , z ) d u d v | 2 ,
W ¯ ( u , v ) = { A 0 ( u , v ) W ( u , v ) | W ( u , v ) | 1 , ( u , v ) Q 0 , ( u , v ) Q ,
δ F 2 = Ω [ | F ( x , y ) | B 0 ( x , y ) ] 2 d x d y Ω B 0 2 ( x , y ) d x d y ,
δ W 2 = Ω [ | W ( u , v ) | A 0 ( u , v ) ] 2 d u d v Ω 0 A 0 2 ( u , v ) d u d v ,
R j = m = 1 256 n = 1 256 A j ( m , n ) × B j ( m , n ) m = 1 256 n = 1 256 [ A j ( m , n ) ] 2 m = 1 256 n = 1 256 [ B j ( m , n ) ] 2 ,

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