Abstract

The detailed design process and experimental results of stacked multilayer diffractive optical elements are reported for an optical network unit used in optical subscriber-network applications. The optical network unit accepts two incoming light beams of 1.3- and 1.55-μm wavelengths through a single-mode optical fiber. A laser diode is also placed for bidirectional communications. The optical network unit consists of five diffractive optical elements that perform the following functions: collimation of incoming beams, focusing of the outgoing 1.55-μm beam, 3-dB splitting of the 1.3-μm beam, focusing of the 1.3-μm beam onto the photodiode, and collimation of the light emitted from a laser diode. Possible cost reductions as a result of mass production and the ease of alignment of the stacked diffractive optical elements could be ideal for constructing low-cost optical network units.

© 1998 Optical Society of America

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References

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    [CrossRef]
  2. N. Shibata and I. Yamashita, “System and component technologies toward full access network opticalization,” IEICE Trans. Electron. E80-C, 3–8 (1997).
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    [CrossRef]
  4. Y. Wakui, “The fiber-optic subscriber network in Japan,” IEEE Commun. Mag. 32, 56–63 (February 1994).
    [CrossRef]
  5. Y. Mochida, “Technologies for local-access fibering,” IEEE Commun. Mag. 32, 64–73 (February 1994).
    [CrossRef]
  6. J. Yoshida, S. Sekine, H. Terui, T. Kominato, K. Yoshino, N. Tsuzuki, M. Kobayashi, and K. Okada, “A compact optical module with a 1.3-μm/1.5-μm WDM circuit for fiber optic subscriber systems,” IEICE Trans. Electron. E75-B, 880–885 (1992).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1997 (3)

N. Shibata and I. Yamashita, “System and component technologies toward full access network opticalization,” IEICE Trans. Electron. E80-C, 3–8 (1997).

H. Sasaki, I. Fukuzaki, Y. Katsuki, and T. Kamijoh, “Miniaturized free-space wavelength-division multiplexing photonic circuit for passive double star network systems,” Electron. Lett. 33, 1577–1579 (1997).
[CrossRef]

N. Uchida, Y. Yamada, Y. Hibino, Y. Suzuki, and N. Ishihara, “Low-cost hybrid WDM module consisting of a spot-size convertor integrated laser diode and a waveguide photodiode on a PLC platform for access network systems,” IEICE Trans. Electron. E80-C, 88–97 (1997).

1996 (1)

1995 (2)

1994 (4)

D. A. Pommet, M. G. Moharam, and E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
[CrossRef]

J. Fan, B. Catanzaro, F. Kiamilev, S. C. Esener, and S. H. Lee, “Architecture of an integrated computer-aided design system for optoelectronics,” Opt. Eng. 33, 1571–1580 (1994).
[CrossRef]

Y. Wakui, “The fiber-optic subscriber network in Japan,” IEEE Commun. Mag. 32, 56–63 (February 1994).
[CrossRef]

Y. Mochida, “Technologies for local-access fibering,” IEEE Commun. Mag. 32, 64–73 (February 1994).
[CrossRef]

1993 (1)

1992 (2)

J. Yoshida, S. Sekine, H. Terui, T. Kominato, K. Yoshino, N. Tsuzuki, M. Kobayashi, and K. Okada, “A compact optical module with a 1.3-μm/1.5-μm WDM circuit for fiber optic subscriber systems,” IEICE Trans. Electron. E75-B, 880–885 (1992).

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, and H. S. Hinton, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, 465–477 (1992).
[CrossRef]

1989 (2)

1987 (2)

J. R. Stern, “Passive optical local networks for telephony applications and beyond,” Electron. Lett. 23, 1255–1257 (1987).
[CrossRef]

S. Shimada, K. Hashimoto, and K. Okada, “Fiber-optic subscriber loop system for integrated services: the strategy for introducing fibers into the subscriber network,” J. Lightwave Technol. LT-5, 1667–1675 (1987).
[CrossRef]

1982 (2)

1978 (1)

1972 (1)

L. d’Auria, J. P. Huignard, A. M. Roy, and E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

1965 (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Sys. Tech. J. 44, 455–494 (1965).

1961 (1)

K. Miyamoto, “The phase Fresnel lens,” J. Opt. Soc. of Am. 51, 17–20 (1961).
[CrossRef]

Belland, P.

Catanzaro, B.

J. Fan, B. Catanzaro, F. Kiamilev, S. C. Esener, and S. H. Lee, “Architecture of an integrated computer-aided design system for optoelectronics,” Opt. Eng. 33, 1571–1580 (1994).
[CrossRef]

Cloonan, T. J.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, and H. S. Hinton, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, 465–477 (1992).
[CrossRef]

Crenn, J. P.

d’Auria, L.

L. d’Auria, J. P. Huignard, A. M. Roy, and E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

Esener, S. C.

J. Fan, B. Catanzaro, F. Kiamilev, S. C. Esener, and S. H. Lee, “Architecture of an integrated computer-aided design system for optoelectronics,” Opt. Eng. 33, 1571–1580 (1994).
[CrossRef]

Fan, J.

J. Fan, D. Zaleta, K. S. Urquhart, and S. H. Lee, “Efficient encoding algorithms for computer-aided design of diffractive optical elements by the use of electron-beam fabrication,” Appl. Opt. 34, 2522–2533 (1995).
[CrossRef] [PubMed]

J. Fan, B. Catanzaro, F. Kiamilev, S. C. Esener, and S. H. Lee, “Architecture of an integrated computer-aided design system for optoelectronics,” Opt. Eng. 33, 1571–1580 (1994).
[CrossRef]

Farhoosh, H.

Feldman, M. R.

Fukuzaki, I.

H. Sasaki, I. Fukuzaki, Y. Katsuki, and T. Kamijoh, “Miniaturized free-space wavelength-division multiplexing photonic circuit for passive double star network systems,” Electron. Lett. 33, 1577–1579 (1997).
[CrossRef]

Gaylord, T. K.

Grann, E. B.

Guest, C. C.

Hashimoto, K.

S. Shimada, K. Hashimoto, and K. Okada, “Fiber-optic subscriber loop system for integrated services: the strategy for introducing fibers into the subscriber network,” J. Lightwave Technol. LT-5, 1667–1675 (1987).
[CrossRef]

Herzig, H. P.

Hibino, Y.

N. Uchida, Y. Yamada, Y. Hibino, Y. Suzuki, and N. Ishihara, “Low-cost hybrid WDM module consisting of a spot-size convertor integrated laser diode and a waveguide photodiode on a PLC platform for access network systems,” IEICE Trans. Electron. E80-C, 88–97 (1997).

Hinton, H. S.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, and H. S. Hinton, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, 465–477 (1992).
[CrossRef]

Huignard, J. P.

L. d’Auria, J. P. Huignard, A. M. Roy, and E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

Ishihara, N.

N. Uchida, Y. Yamada, Y. Hibino, Y. Suzuki, and N. Ishihara, “Low-cost hybrid WDM module consisting of a spot-size convertor integrated laser diode and a waveguide photodiode on a PLC platform for access network systems,” IEICE Trans. Electron. E80-C, 88–97 (1997).

Kamijoh, T.

H. Sasaki, I. Fukuzaki, Y. Katsuki, and T. Kamijoh, “Miniaturized free-space wavelength-division multiplexing photonic circuit for passive double star network systems,” Electron. Lett. 33, 1577–1579 (1997).
[CrossRef]

Katsuki, Y.

H. Sasaki, I. Fukuzaki, Y. Katsuki, and T. Kamijoh, “Miniaturized free-space wavelength-division multiplexing photonic circuit for passive double star network systems,” Electron. Lett. 33, 1577–1579 (1997).
[CrossRef]

Kiamilev, F.

J. Fan, B. Catanzaro, F. Kiamilev, S. C. Esener, and S. H. Lee, “Architecture of an integrated computer-aided design system for optoelectronics,” Opt. Eng. 33, 1571–1580 (1994).
[CrossRef]

Kobayashi, M.

J. Yoshida, S. Sekine, H. Terui, T. Kominato, K. Yoshino, N. Tsuzuki, M. Kobayashi, and K. Okada, “A compact optical module with a 1.3-μm/1.5-μm WDM circuit for fiber optic subscriber systems,” IEICE Trans. Electron. E75-B, 880–885 (1992).

Kogelnik, H.

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Sys. Tech. J. 44, 455–494 (1965).

Kominato, T.

J. Yoshida, S. Sekine, H. Terui, T. Kominato, K. Yoshino, N. Tsuzuki, M. Kobayashi, and K. Okada, “A compact optical module with a 1.3-μm/1.5-μm WDM circuit for fiber optic subscriber systems,” IEICE Trans. Electron. E75-B, 880–885 (1992).

Kuittinen, M.

Lee, S. H.

Li, Y.

Y. Li and E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Ma, J.

Magnusson, R.

McCormick, F. B.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, and H. S. Hinton, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, 465–477 (1992).
[CrossRef]

Miyamoto, K.

K. Miyamoto, “The phase Fresnel lens,” J. Opt. Soc. of Am. 51, 17–20 (1961).
[CrossRef]

Mochida, Y.

Y. Mochida, “Technologies for local-access fibering,” IEEE Commun. Mag. 32, 64–73 (February 1994).
[CrossRef]

Moharam, M. G.

Morris, J. E.

Okada, K.

J. Yoshida, S. Sekine, H. Terui, T. Kominato, K. Yoshino, N. Tsuzuki, M. Kobayashi, and K. Okada, “A compact optical module with a 1.3-μm/1.5-μm WDM circuit for fiber optic subscriber systems,” IEICE Trans. Electron. E75-B, 880–885 (1992).

S. Shimada, K. Hashimoto, and K. Okada, “Fiber-optic subscriber loop system for integrated services: the strategy for introducing fibers into the subscriber network,” J. Lightwave Technol. LT-5, 1667–1675 (1987).
[CrossRef]

Ozguz, V.

Patra, S.

Pommet, D. A.

Roy, A. M.

L. d’Auria, J. P. Huignard, A. M. Roy, and E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

Sasaki, H.

H. Sasaki, I. Fukuzaki, Y. Katsuki, and T. Kamijoh, “Miniaturized free-space wavelength-division multiplexing photonic circuit for passive double star network systems,” Electron. Lett. 33, 1577–1579 (1997).
[CrossRef]

Sasian, J. M.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, and H. S. Hinton, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, 465–477 (1992).
[CrossRef]

Sekine, S.

J. Yoshida, S. Sekine, H. Terui, T. Kominato, K. Yoshino, N. Tsuzuki, M. Kobayashi, and K. Okada, “A compact optical module with a 1.3-μm/1.5-μm WDM circuit for fiber optic subscriber systems,” IEICE Trans. Electron. E75-B, 880–885 (1992).

Shibata, N.

N. Shibata and I. Yamashita, “System and component technologies toward full access network opticalization,” IEICE Trans. Electron. E80-C, 3–8 (1997).

Shimada, S.

S. Shimada, K. Hashimoto, and K. Okada, “Fiber-optic subscriber loop system for integrated services: the strategy for introducing fibers into the subscriber network,” J. Lightwave Technol. LT-5, 1667–1675 (1987).
[CrossRef]

Spitz, E.

L. d’Auria, J. P. Huignard, A. M. Roy, and E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

Stern, J. R.

J. R. Stern, “Passive optical local networks for telephony applications and beyond,” Electron. Lett. 23, 1255–1257 (1987).
[CrossRef]

Suzuki, Y.

N. Uchida, Y. Yamada, Y. Hibino, Y. Suzuki, and N. Ishihara, “Low-cost hybrid WDM module consisting of a spot-size convertor integrated laser diode and a waveguide photodiode on a PLC platform for access network systems,” IEICE Trans. Electron. E80-C, 88–97 (1997).

Swanson, G. J.

G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

G. J. Swanson, “Binary optics technology: the theory and design of multilevel diffractive optical elements,” MIT Tech. Rep. 854 (Massachusetts Institute of Technology, Cambridge, Mass., 1989).

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” MIT Tech. Rep. 914 (Massachusetts Institute of Technology, Cambridge, Mass., 1991).

Terui, H.

J. Yoshida, S. Sekine, H. Terui, T. Kominato, K. Yoshino, N. Tsuzuki, M. Kobayashi, and K. Okada, “A compact optical module with a 1.3-μm/1.5-μm WDM circuit for fiber optic subscriber systems,” IEICE Trans. Electron. E75-B, 880–885 (1992).

Tooley, F. A. P.

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, and H. S. Hinton, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, 465–477 (1992).
[CrossRef]

Tsuzuki, N.

J. Yoshida, S. Sekine, H. Terui, T. Kominato, K. Yoshino, N. Tsuzuki, M. Kobayashi, and K. Okada, “A compact optical module with a 1.3-μm/1.5-μm WDM circuit for fiber optic subscriber systems,” IEICE Trans. Electron. E75-B, 880–885 (1992).

Uchida, N.

N. Uchida, Y. Yamada, Y. Hibino, Y. Suzuki, and N. Ishihara, “Low-cost hybrid WDM module consisting of a spot-size convertor integrated laser diode and a waveguide photodiode on a PLC platform for access network systems,” IEICE Trans. Electron. E80-C, 88–97 (1997).

Urquhart, K. S.

Veldkamp, W. B.

G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

Wakui, Y.

Y. Wakui, “The fiber-optic subscriber network in Japan,” IEEE Commun. Mag. 32, 56–63 (February 1994).
[CrossRef]

Welch, W. H.

Wolf, E.

Y. Li and E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Yamada, Y.

N. Uchida, Y. Yamada, Y. Hibino, Y. Suzuki, and N. Ishihara, “Low-cost hybrid WDM module consisting of a spot-size convertor integrated laser diode and a waveguide photodiode on a PLC platform for access network systems,” IEICE Trans. Electron. E80-C, 88–97 (1997).

Yamashita, I.

N. Shibata and I. Yamashita, “System and component technologies toward full access network opticalization,” IEICE Trans. Electron. E80-C, 3–8 (1997).

Yoshida, J.

J. Yoshida, S. Sekine, H. Terui, T. Kominato, K. Yoshino, N. Tsuzuki, M. Kobayashi, and K. Okada, “A compact optical module with a 1.3-μm/1.5-μm WDM circuit for fiber optic subscriber systems,” IEICE Trans. Electron. E75-B, 880–885 (1992).

Yoshino, K.

J. Yoshida, S. Sekine, H. Terui, T. Kominato, K. Yoshino, N. Tsuzuki, M. Kobayashi, and K. Okada, “A compact optical module with a 1.3-μm/1.5-μm WDM circuit for fiber optic subscriber systems,” IEICE Trans. Electron. E75-B, 880–885 (1992).

Zaleta, D.

Appl. Opt. (4)

Bell Sys. Tech. J. (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Sys. Tech. J. 44, 455–494 (1965).

Electron. Lett. (2)

J. R. Stern, “Passive optical local networks for telephony applications and beyond,” Electron. Lett. 23, 1255–1257 (1987).
[CrossRef]

H. Sasaki, I. Fukuzaki, Y. Katsuki, and T. Kamijoh, “Miniaturized free-space wavelength-division multiplexing photonic circuit for passive double star network systems,” Electron. Lett. 33, 1577–1579 (1997).
[CrossRef]

IEEE Commun. Mag. (2)

Y. Wakui, “The fiber-optic subscriber network in Japan,” IEEE Commun. Mag. 32, 56–63 (February 1994).
[CrossRef]

Y. Mochida, “Technologies for local-access fibering,” IEEE Commun. Mag. 32, 64–73 (February 1994).
[CrossRef]

IEICE Trans. Electron. (3)

J. Yoshida, S. Sekine, H. Terui, T. Kominato, K. Yoshino, N. Tsuzuki, M. Kobayashi, and K. Okada, “A compact optical module with a 1.3-μm/1.5-μm WDM circuit for fiber optic subscriber systems,” IEICE Trans. Electron. E75-B, 880–885 (1992).

N. Shibata and I. Yamashita, “System and component technologies toward full access network opticalization,” IEICE Trans. Electron. E80-C, 3–8 (1997).

N. Uchida, Y. Yamada, Y. Hibino, Y. Suzuki, and N. Ishihara, “Low-cost hybrid WDM module consisting of a spot-size convertor integrated laser diode and a waveguide photodiode on a PLC platform for access network systems,” IEICE Trans. Electron. E80-C, 88–97 (1997).

J. Lightwave Technol. (1)

S. Shimada, K. Hashimoto, and K. Okada, “Fiber-optic subscriber loop system for integrated services: the strategy for introducing fibers into the subscriber network,” J. Lightwave Technol. LT-5, 1667–1675 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. of Am. (1)

K. Miyamoto, “The phase Fresnel lens,” J. Opt. Soc. of Am. 51, 17–20 (1961).
[CrossRef]

Opt. Commun. (2)

L. d’Auria, J. P. Huignard, A. M. Roy, and E. Spitz, “Photolithographic fabrication of thin film lenses,” Opt. Commun. 5, 232–235 (1972).
[CrossRef]

Y. Li and E. Wolf, “Focal shift in focused truncated Gaussian beams,” Opt. Commun. 42, 151–156 (1982).
[CrossRef]

Opt. Eng. (2)

G. J. Swanson and W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

J. Fan, B. Catanzaro, F. Kiamilev, S. C. Esener, and S. H. Lee, “Architecture of an integrated computer-aided design system for optoelectronics,” Opt. Eng. 33, 1571–1580 (1994).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

F. B. McCormick, F. A. P. Tooley, T. J. Cloonan, J. M. Sasian, and H. S. Hinton, “Optical interconnections using microlens arrays,” Opt. Quantum Electron. 24, 465–477 (1992).
[CrossRef]

Other (3)

G. J. Swanson, “Binary optics technology: the theory and design of multilevel diffractive optical elements,” MIT Tech. Rep. 854 (Massachusetts Institute of Technology, Cambridge, Mass., 1989).

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” MIT Tech. Rep. 914 (Massachusetts Institute of Technology, Cambridge, Mass., 1991).

code v is a registered trademark of Optical Research Associates, 3280 East Foothill Boulevard, Pasadena, Calif. 91107.

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Figures (15)

Fig. 1
Fig. 1

Generalized diagram of the optical subscriber-network unit.

Fig. 2
Fig. 2

Schematic diagram of the photonic circuit of the optical subscriber-network unit consisting of stacked multilayer DOE’s.

Fig. 3
Fig. 3

Wafer-scale mass production of the proposed photonic circuits for ONU applications.

Fig. 4
Fig. 4

OPD function and its corresponding cross section of the DOE. P denotes the minimum feature size of the DOE.

Fig. 5
Fig. 5

Beam deflection by a linear grating.

Fig. 6
Fig. 6

Focal length of the diffractive lens.

Fig. 7
Fig. 7

Detailed photonic-circuit diagram of the relay-lens system for the 1.55-μm beam.

Fig. 8
Fig. 8

Detailed photonic-circuit diagram of the photodetector focusing lens.

Fig. 9
Fig. 9

Detailed photonic-circuit diagram of the laser-diode collimator.

Fig. 10
Fig. 10

Scanning electron micrograph photographs of (a) the optical fiber collimator DOE 1 and (b) the closeup view of the intersecting region between DOE’s 1 and 2.

Fig. 11
Fig. 11

Lateral-alignment tolerance of the input and the output optical fiber pair. The input and the output fiber spacing was fixed at 250 μm, and the pair moved together along the x axis.

Fig. 12
Fig. 12

Lateral-alignment tolerance of the laser diode in the prototypical ONU photonic circuit.

Fig. 13
Fig. 13

Coupling characteristics of the optical fiber and the laser diode demonstrating the validity of aligning the optical fibers and the laser diode independently. First, the input and the output optical fiber pair was aligned by monitoring of the output power at the output port. Second, the laser diode was aligned so that the maximum 1.3-μm output power is obtained at the common port. Then the optical fiber pair was again shifted for a better coupling efficiency.

Fig. 14
Fig. 14

(a) Stacked DOE’s for the prototypical ONU photonic circuits. (b) Packaged ONU module.

Fig. 15
Fig. 15

Four typical setups of DOE’s used in the optical subscriber-network unit: (a) DOE in which a prism bends the incident collimated beam. (b) An off-axis collimator. (c) An off-axis focusing lens. (d) Imaging between the point source and the point image.

Tables (1)

Tables Icon

Table 1 Optical Losses of the Proposed ONU System

Equations (44)

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

ρ x ,   y =   C M x m y n ,
M = m + n 2 + m + 3 n / 2 .
ρ x ,   y = r = 0 1 r ! x   η + y   ζ r ρ η ,   ζ | η = ζ = 0 + Δ .
P = λ N | grad   ρ x ,   y | ,
r = λ n 2 λ sin   θ 2 - n 1 λ sin   θ 1 ,
r = λ n 2 λ sin   θ 2 - n 1 λ sin   θ 1 ,
θ 2 = sin - 1 1 n 2 λ λ λ n 2 λ sin   θ 2 - n 1 λ sin   θ 1 + n 1 λ n 2 λ   sin   θ 1 .
r k 2 + f 2 = f + k λ 2 , r k 2 = 2 k λ f + k λ 2 ,
r k 2 = 2 k λ f .
f = f   λ λ .
Θ 0 = tan - 1 S 1 + S 2 2 T 2 ,
ρ DOE 1 x ,   y = n 1.55 x 2 + y 2 + T 1 2 1 / 2 - n 1.55 T 1 + n 1.55 S 1 + S 2 x S 1 + S 2 2 + 4 T 2 2 1 / 2 ,
ρ DOE 2 x ,   y = n 1.55 x 2 + y 2 + T 1 2 1 / 2 - n 1.55 T 1 - n 1.55 S 1 + S 2 x S 1 + S 2 2 + 4 T 2 2 1 / 2 .
Θ 1 = sin - 1 1.3 1.55 n 1.3 n 1.55   sin   Θ 0 .
Θ 2 = tan - 1 x 1 + x 2 T 3 ,
x 1 = S 1 - T 2   tan   Θ 1 .
ρ DOE 3 x ,   y = n 1.3 x 1 - S 1 x x 1 - S 1 2 + T 2 2 1 / 2 + n 1.3 x 1 + x 2 x x 1 + x 2 2 + T 3 2 1 / 2 .
n 1.3 T 1 - n 1.3 D 1 = n 1.3 T 1 1.3 1.55 .
D 3 = x 3 - x 1 2 + T 3 2 1 / 2 ,
Θ 3 = tan - 1 x 3 - x 1 T 3 .
ρ DOE 4 x ,   y = n 1.3 X 41 - x 2 + Y 41 - y 2 + Z 41 2 1 / 2 - n 1.3 L 41 + X 42 - x 2 + Y 42 - y 2 + Z 42 2 1 / 2 - L 42 ,
X 41 = - D 1 + D 3 sin   Θ 3 ,     Y 41 = 0 ,     Z 41 = - D 1 + D 3 cos   Θ 3 ,
X 42 = x 4 - x 3 ,     Y 42 = 0 ,     Z 42 = T 4 ,
L 41 = X 41 2 + Y 41 2 + Z 41 2 1 / 2 = D 1 + D 3 ,
L 42 = X 42 2 + Y 42 2 + Z 42 2 1 / 2 = x 4 - x 3 2 + T 4 2 1 / 2 .
D 2 = x 2 - x 1 2 + T 3 2 1 / 2 ,
Θ 2 = tan - 1 x 1 + x 2 T 3 ,
ρ DOE 5 x ,   y = n 1.3 X 51 - x 2 + Y 51 - y 2 + Z 51 2 1 / 2 - n 1.3 L 51 + X 52 - x 2 + Y 52 - y 2 + Z 52 2 1 / 2 - L 52 ,
X 51 = D 1 + D 2 sin   Θ 2 ,     Y 51 = 0 , Z 51 = - D 1 + D 2 cos   Θ 2 ,
X 52 = 0 ,     Y 52 = 0 ,     Z 52 = T 5 ,
L 51 = X 51 2 + Y 51 2 + Z 51 2 1 / 2 = D 1 + D 2 ,
L 52 = X 52 2 + Y 52 2 + Z 52 2 1 / 2 = T 5 .
ρ 1 x ,   y = n 1 α 1 x + β 1 y α 1 2 + β 1 2 + γ 1 2 1 / 2 - n 2 α 2 x + β 2 y α 2 2 + β 2 2 + γ 2 2 1 / 2 .
ρ 2 x ,   y = n 1 X - x 2 + Y - y 2 + Z 2 1 / 2 - n 1 L - n 2 α x + β y α 2 + β 2 + γ 2 1 / 2 ,
L = X 2 + Y 2 + Z 2 1 / 2 .
ρ 3 x ,   y = - n 2 X - x 2 + Y - y 2 + Z 2 1 / 2 + n 2 L + n 1 α x + β y α 2 + β 2 + γ 2 1 / 2 ,
L = X 2 + Y 2 + Z 2 1 / 2 .
L 1 = X 1 2 + Y 1 2 + Z 1 2 1 / 2 ,
L 2 = X 2 2 + Y 2 2 + Z 2 2 1 / 2 ,
ρ 4 x ,   y = n 1 X 1 - x 2 + Y 1 - y 2 + Z 1 2 1 / 2 - n 1 L 1 + n 2 X 2 - x 2 + Y 2 - y 2 + Z 2 2 1 / 2 - n 2 L 2 .
P 1 = λ N   n 1 2 α 1 2 + β 1 2 α 1 2 + β 1 2 + γ 1 2 + n 2 2 α 2 2 + β 2 2 α 2 2 + β 2 2 + γ 2 2 - 2 n 1 n 2 α 1 α 2 + β 1 β 2 α 1 2 + β 1 2 + γ 1 2 1 / 2 α 2 2 + β 2 2 + γ 2 2 1 / 2 1 / 2 .
P 2 = λ N   n 2 2 α 2 + β 2 α 2 + β 2 + γ 2 + n 1 2 X - x 2 + Y - y 2 X - x 2 + Y - y 2 + Z 2 + 2 n 1 n 2 α X - x + β Y - y α 2 + β 2 + γ 2 1 / 2 X - x 2 + Y - y 2 + Z 2 1 / 2 1 / 2 ,
P 3 = λ N   n 1 2 α 2 + β 2 α 2 + β 2 + γ 2 + n 2 2 X - x 2 + Y - y 2 X - x 2 + Y - y 2 + Z 2 - 2 n 1 n 2 α X - x + β Y - y α 2 + β 2 + γ 2 1 / 2 X - x 2 + Y - y 2 + Z 2 1 / 2 1 / 2 ,
P 4 = λ N   n 1 2 X 1 - x 2 + Y 1 - y 2 X 1 - x 2 + Y 1 - y 2 + Z 1 2 + n 2 2 X 2 - x 2 + Y 2 - y 2 X 2 - x 2 + Y 2 - y 2 + Z 2 2 + 2 n 1 n 2 X 1 - x X 2 - x + Y 1 - y Y 2 - y X 1 - x 2 + Y 1 - y 2 + Z 1 2 1 / 2 X 2 - x 2 + Y 2 - y 2 + Z 2 2 1 / 2 1 / 2 .

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