Abstract

The possibility of realizing the fundamental logic gates and, or, and not, which operate in parallel through two interlinked χ(2) interactions of sum- and difference-frequency generations in a single type I β-BaB2O4 crystal, is presented both theoretically and experimentally. Optical bits are encoded and read as amplitude modulation of the harmonics of a Nd:YAG laser. Owing to the excellent optical resolution of the system, data can be encoded at high density. The implementation of an all-optical parallel half-adder is also shown. The output fields can be frequency converted to implement an all-optical looping circuit.

© 2004 Optical Society of America

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    [CrossRef]

2004

2003

M. Bondani, A. Allevi, and A. Andreoni, “Holography by nondegenerate χ(2) interactions,” J. Opt. Soc. Am. B 20, 1–13 (2003).
[CrossRef]

S. Pereira, P. Chak, and J. E. Sipe, “All-optical AND gate by use of a Kerr nonlinear microresonator structure,” Opt. Lett. 28, 444–446 (2003).
[CrossRef] [PubMed]

C. P. Singh and S. Roy, “All-optical switching in bacteriorhodopsin based on M state dynamics and its application to photonic logic gates,” Opt. Commun. 218, 55–66 (2003).
[CrossRef]

P. Wen, M. Sanchez, M. Gross, and S. Esener, “Vertical-cavity optical AND gate,” Opt. Commun. 219, 383–387 (2003).
[CrossRef]

J. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, and S. H. Kim, “All-optical half adder using semiconductor optical amplifier based devices,” Opt. Commun. 218, 345–349 (2003).
[CrossRef]

2002

2001

G. S. Kanter, P. Kumar, K. R. Parameswaran, and M. M. Fejer, “Wavelength-selective pulsed all-optical switching based on cascaded second-order nonlinearity in a periodically poled lithium-niobate waveguide,” IEEE Photonics Technol. Lett. 13, 341–343 (2001).
[CrossRef]

A. Chowdhury, C. Staus, B. F. Boland, T. F. Kuech, and L. McCaughan, “Experimental demonstration of 1535–1555-nm simultaneous optical wavelength interchange with a nonlinear photonic crystal,” Opt. Lett. 26, 1353–1355 (2001).
[CrossRef]

L. Brzozowski and E. H. Sargent, “All-optical analog-to-digital converters, hardlimiters, and logic gates,” J. Lightwave Technol. 19, 114–119 (2001).
[CrossRef]

A. Andreoni, M. Bondani, M. C. A. Potenza, Yu. N. Denisyuk, and E. Puddu, “Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions,” Rev. Sci. Instrum. 72, 2525–2531 (2001).
[CrossRef]

P. O. Hedekvist, A. Bhardwaj, K. Vahala, and H. Andersson, “Advanced all-optical logic gates on a spectral bus,” Appl. Opt. 40, 1761–1766 (2001).
[CrossRef]

2000

I. S. Nefedov, V. N. Gusyatnikov, P. K. Kashkarov, and A. M. Zheltikov, “Low threshold photonic band-gap optical logic gates,” Laser Phys. 10, 640–643 (2000).

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

A. Andreoni, M. Bondani, and M. C. A. Potenza, “Combination tasks performed by second-harmonic-generated holograms,” Opt. Lett. 25, 1570–1572 (2000).
[CrossRef]

K. R. Parameswaran, M. Fujimura, M. H. Chou, and M. M. Fejer, “Low-power all-optical gate based on sum frequency mixing in APE waveguides in PPLN,” IEEE Photonics Technol. Lett. 12, 654–656 (2000).
[CrossRef]

1999

1997

1996

M. Shirakawa, T. Takemori, and J. Ohtsubo, “Optical computing based on a selector logic,” Opt. Commun. 124, 333–344 (1996).
[CrossRef]

1990

1989

S. A. Collins, Jr., “Application of liquid crystals to optical logic gates,” in Liquid Crystal Chemistry, Physics, and Applications, J. W. Doane and Z. Yaniv, eds., Proc. SPIE 1080, 72–82 (1989).
[CrossRef]

1987

Albert, O.

Allevi, A.

Andersson, H.

Andreoni, A.

Assanto, G.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002).
[CrossRef]

Bhardwaj, A.

Boland, B. F.

Bondani, M.

Brega, A.

Brzozowski, L.

L. Brzozowski and E. H. Sargent, “All-optical analog-to-digital converters, hardlimiters, and logic gates,” J. Lightwave Technol. 19, 114–119 (2001).
[CrossRef]

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

Byun, Y. T.

J. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, and S. H. Kim, “All-optical half adder using semiconductor optical amplifier based devices,” Opt. Commun. 218, 345–349 (2003).
[CrossRef]

Castro, M.

H. Soto, J. Topomonzo, D. Erasme, G. Guekos, and M. Castro, “Experimental demonstration of all-optical logic gates using cross-polarization modulation in a semiconductor optical amplifier,” in Smart Structure, Devices, and Systems, E. C. Harvey, D. Abbott, and V. K. Varadan, eds., Proc. SPIE 4935, 495–502 (2002).
[CrossRef]

Chak, P.

Chou, M. H.

K. R. Parameswaran, M. Fujimura, M. H. Chou, and M. M. Fejer, “Low-power all-optical gate based on sum frequency mixing in APE waveguides in PPLN,” IEEE Photonics Technol. Lett. 12, 654–656 (2000).
[CrossRef]

Chou, M.-H.

Chowdhury, A.

Collins Jr., S. A.

S. A. Collins, Jr., “Application of liquid crystals to optical logic gates,” in Liquid Crystal Chemistry, Physics, and Applications, J. W. Doane and Z. Yaniv, eds., Proc. SPIE 1080, 72–82 (1989).
[CrossRef]

Conti, C.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002).
[CrossRef]

De Luca, A.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002).
[CrossRef]

DeLong, K. W.

Denisyuk, Yu. N.

A. Andreoni, M. Bondani, M. C. A. Potenza, Yu. N. Denisyuk, and E. Puddu, “Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions,” Rev. Sci. Instrum. 72, 2525–2531 (2001).
[CrossRef]

Erasme, D.

H. Soto, J. Topomonzo, D. Erasme, G. Guekos, and M. Castro, “Experimental demonstration of all-optical logic gates using cross-polarization modulation in a semiconductor optical amplifier,” in Smart Structure, Devices, and Systems, E. C. Harvey, D. Abbott, and V. K. Varadan, eds., Proc. SPIE 4935, 495–502 (2002).
[CrossRef]

Esener, S.

P. Wen, M. Sanchez, M. Gross, and S. Esener, “Vertical-cavity optical AND gate,” Opt. Commun. 219, 383–387 (2003).
[CrossRef]

Etchepare, J.

Fan, S.

Fang, H.

Fejer, M. M.

G. S. Kanter, P. Kumar, K. R. Parameswaran, and M. M. Fejer, “Wavelength-selective pulsed all-optical switching based on cascaded second-order nonlinearity in a periodically poled lithium-niobate waveguide,” IEEE Photonics Technol. Lett. 13, 341–343 (2001).
[CrossRef]

K. R. Parameswaran, M. Fujimura, M. H. Chou, and M. M. Fejer, “Low-power all-optical gate based on sum frequency mixing in APE waveguides in PPLN,” IEEE Photonics Technol. Lett. 12, 654–656 (2000).
[CrossRef]

R. Schiek, L. Friedrich, H. Fang, G. I. Stegeman, K. R. Parameswaran, M.-H. Chou, and M. M. Fejer, “Nonlinear directional coupler in periodically poled lithium niobate,” Opt. Lett. 24, 1617–1619 (1999).
[CrossRef]

Ferraro, A.

A. Allevi, A. Andreoni, M. Bondani, A. Ferraro, M. G. A. Paris, and E. Puddu, “Quantum and classical properties of the fields generated by two interlinked second-order non-linear interactions,” J. Mod. Opt. 51, 1031–1036 (2004).
[CrossRef]

M. Bondani, A. Allevi, E. Puddu, A. Andreoni, A. Ferraro, and M. G. A. Paris, “Properties of two interlinked χ(2) interactions in noncollinear phase matching,” Opt. Lett. 29, 180–182 (2004).
[CrossRef] [PubMed]

Fittinghoff, D. N.

Friedrich, L.

Fujimura, M.

K. R. Parameswaran, M. Fujimura, M. H. Chou, and M. M. Fejer, “Low-power all-optical gate based on sum frequency mixing in APE waveguides in PPLN,” IEEE Photonics Technol. Lett. 12, 654–656 (2000).
[CrossRef]

Gregory, D. A.

Gross, M.

P. Wen, M. Sanchez, M. Gross, and S. Esener, “Vertical-cavity optical AND gate,” Opt. Commun. 219, 383–387 (2003).
[CrossRef]

Guekos, G.

H. Soto, J. Topomonzo, D. Erasme, G. Guekos, and M. Castro, “Experimental demonstration of all-optical logic gates using cross-polarization modulation in a semiconductor optical amplifier,” in Smart Structure, Devices, and Systems, E. C. Harvey, D. Abbott, and V. K. Varadan, eds., Proc. SPIE 4935, 495–502 (2002).
[CrossRef]

Gusyatnikov, V. N.

I. S. Nefedov, V. N. Gusyatnikov, P. K. Kashkarov, and A. M. Zheltikov, “Low threshold photonic band-gap optical logic gates,” Laser Phys. 10, 640–643 (2000).

Hedekvist, P. O.

Huang, D.

Huang, Y.

Ibanescu, M.

Ippen, E.

Jhon, Y. M.

J. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, and S. H. Kim, “All-optical half adder using semiconductor optical amplifier based devices,” Opt. Commun. 218, 345–349 (2003).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Jutamulia, S.

Kanter, G. S.

G. S. Kanter, P. Kumar, K. R. Parameswaran, and M. M. Fejer, “Wavelength-selective pulsed all-optical switching based on cascaded second-order nonlinearity in a periodically poled lithium-niobate waveguide,” IEEE Photonics Technol. Lett. 13, 341–343 (2001).
[CrossRef]

Kashkarov, P. K.

I. S. Nefedov, V. N. Gusyatnikov, P. K. Kashkarov, and A. M. Zheltikov, “Low threshold photonic band-gap optical logic gates,” Laser Phys. 10, 640–643 (2000).

Khan, A. H.

Kim, J. H.

J. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, and S. H. Kim, “All-optical half adder using semiconductor optical amplifier based devices,” Opt. Commun. 218, 345–349 (2003).
[CrossRef]

Kim, S. H.

J. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, and S. H. Kim, “All-optical half adder using semiconductor optical amplifier based devices,” Opt. Commun. 218, 345–349 (2003).
[CrossRef]

Kivshar, Y. S.

Krumbügel, M. A.

Kuech, T. F.

Kumar, P.

G. S. Kanter, P. Kumar, K. R. Parameswaran, and M. M. Fejer, “Wavelength-selective pulsed all-optical switching based on cascaded second-order nonlinearity in a periodically poled lithium-niobate waveguide,” IEEE Photonics Technol. Lett. 13, 341–343 (2001).
[CrossRef]

Lee, S.

J. H. Kim, Y. T. Byun, Y. M. Jhon, S. Lee, D. H. Woo, and S. H. Kim, “All-optical half adder using semiconductor optical amplifier based devices,” Opt. Commun. 218, 345–349 (2003).
[CrossRef]

Liu, D.

McCaughan, L.

Minkovski, N.

Nefedov, I. S.

I. S. Nefedov, V. N. Gusyatnikov, P. K. Kashkarov, and A. M. Zheltikov, “Low threshold photonic band-gap optical logic gates,” Laser Phys. 10, 640–643 (2000).

Nejib, U. R.

Ohtsubo, J.

M. Shirakawa, T. Takemori, and J. Ohtsubo, “Optical computing based on a selector logic,” Opt. Commun. 124, 333–344 (1996).
[CrossRef]

Ohzu, H.

Y. Takaki and H. Ohzu, “Optical half-adder using wavefront superposition,” Appl. Opt. 29, 4351–4358 (1990).
[CrossRef] [PubMed]

Y. Takaki and H. Ohzu, “Optical logic operations by holographic filters,” Jpn. J. Opt. 16, 345–351 (1987).

Parameswaran, K. R.

G. S. Kanter, P. Kumar, K. R. Parameswaran, and M. M. Fejer, “Wavelength-selective pulsed all-optical switching based on cascaded second-order nonlinearity in a periodically poled lithium-niobate waveguide,” IEEE Photonics Technol. Lett. 13, 341–343 (2001).
[CrossRef]

K. R. Parameswaran, M. Fujimura, M. H. Chou, and M. M. Fejer, “Low-power all-optical gate based on sum frequency mixing in APE waveguides in PPLN,” IEEE Photonics Technol. Lett. 12, 654–656 (2000).
[CrossRef]

R. Schiek, L. Friedrich, H. Fang, G. I. Stegeman, K. R. Parameswaran, M.-H. Chou, and M. M. Fejer, “Nonlinear directional coupler in periodically poled lithium niobate,” Opt. Lett. 24, 1617–1619 (1999).
[CrossRef]

Paris, M. G. A.

A. Allevi, A. Andreoni, M. Bondani, A. Ferraro, M. G. A. Paris, and E. Puddu, “Quantum and classical properties of the fields generated by two interlinked second-order non-linear interactions,” J. Mod. Opt. 51, 1031–1036 (2004).
[CrossRef]

M. Bondani, A. Allevi, E. Puddu, A. Andreoni, A. Ferraro, and M. G. A. Paris, “Properties of two interlinked χ(2) interactions in noncollinear phase matching,” Opt. Lett. 29, 180–182 (2004).
[CrossRef] [PubMed]

Peccianti, M.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “All-optical switching and logic gating with spatial solitons in liquid crystals,” Appl. Phys. Lett. 81, 3335–3337 (2002).
[CrossRef]

Pereira, S.

Petrov, G. I.

Potenza, M. C. A.

A. Andreoni, M. Bondani, M. C. A. Potenza, Yu. N. Denisyuk, and E. Puddu, “Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions,” Rev. Sci. Instrum. 72, 2525–2531 (2001).
[CrossRef]

A. Andreoni, M. Bondani, and M. C. A. Potenza, “Combination tasks performed by second-harmonic-generated holograms,” Opt. Lett. 25, 1570–1572 (2000).
[CrossRef]

Puddu, E.

M. Bondani, A. Allevi, A. Brega, E. Puddu, and A. Andreoni, “Difference-frequency-generated holograms of two-dimensional objects,” J. Opt. Soc. Am. B 21, 280–288 (2004).
[CrossRef]

A. Allevi, A. Andreoni, M. Bondani, A. Ferraro, M. G. A. Paris, and E. Puddu, “Quantum and classical properties of the fields generated by two interlinked second-order non-linear interactions,” J. Mod. Opt. 51, 1031–1036 (2004).
[CrossRef]

M. Bondani, A. Allevi, E. Puddu, A. Andreoni, A. Ferraro, and M. G. A. Paris, “Properties of two interlinked χ(2) interactions in noncollinear phase matching,” Opt. Lett. 29, 180–182 (2004).
[CrossRef] [PubMed]

A. Andreoni, M. Bondani, M. C. A. Potenza, Yu. N. Denisyuk, and E. Puddu, “Boolean algebra operations performed on optical bits by the generation of holographic fields through second-order nonlinear interactions,” Rev. Sci. Instrum. 72, 2525–2531 (2001).
[CrossRef]

Roy, S.

C. P. Singh and S. Roy, “All-optical switching in bacteriorhodopsin based on M state dynamics and its application to photonic logic gates,” Opt. Commun. 218, 55–66 (2003).
[CrossRef]

Saltiel, S. M.

Sanchez, M.

P. Wen, M. Sanchez, M. Gross, and S. Esener, “Vertical-cavity optical AND gate,” Opt. Commun. 219, 383–387 (2003).
[CrossRef]

Sargent, E. H.

L. Brzozowski and E. H. Sargent, “All-optical analog-to-digital converters, hardlimiters, and logic gates,” J. Lightwave Technol. 19, 114–119 (2001).
[CrossRef]

L. Brzozowski and E. H. Sargent, “Optical signal processing using nonlinear distributed feedback structures,” IEEE J. Quantum Electron. 36, 550–555 (2000).
[CrossRef]

Schiek, R.

Shirakawa, M.

M. Shirakawa, T. Takemori, and J. Ohtsubo, “Optical computing based on a selector logic,” Opt. Commun. 124, 333–344 (1996).
[CrossRef]

Singh, C. P.

C. P. Singh and S. Roy, “All-optical switching in bacteriorhodopsin based on M state dynamics and its application to photonic logic gates,” Opt. Commun. 218, 55–66 (2003).
[CrossRef]

Sipe, J. E.

Soljaceic, M.

Soto, H.

H. Soto, J. Topomonzo, D. Erasme, G. Guekos, and M. Castro, “Experimental demonstration of all-optical logic gates using cross-polarization modulation in a semiconductor optical amplifier,” in Smart Structure, Devices, and Systems, E. C. Harvey, D. Abbott, and V. K. Varadan, eds., Proc. SPIE 4935, 495–502 (2002).
[CrossRef]

Staus, C.

Stegeman, G. I.

Sun, J.

Sweetser, J. N.

Takaki, Y.

Y. Takaki and H. Ohzu, “Optical half-adder using wavefront superposition,” Appl. Opt. 29, 4351–4358 (1990).
[CrossRef] [PubMed]

Y. Takaki and H. Ohzu, “Optical logic operations by holographic filters,” Jpn. J. Opt. 16, 345–351 (1987).

Takemori, T.

M. Shirakawa, T. Takemori, and J. Ohtsubo, “Optical computing based on a selector logic,” Opt. Commun. 124, 333–344 (1996).
[CrossRef]

Topomonzo, J.

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Figures (14)

Fig. 1
Fig. 1

Scheme of the interaction inside the crystal, with the y axis lying on the entrance face of the crystal, ϑj’s internal phase-matching angles, ϑ cut angle, and kjo,e wave vectors of the interacting fields.

Fig. 2
Fig. 2

Experimental setup for the implementation of the and and or gates. HS1,2, harmonic separators; P1,2, thin-film plate polarizers; BBO, β-BaB2O4 crystal cut at 32 deg. The inputs are encoded on fields a4 and a5 as binary digits A4 and A5, and the outputs are read on fields a2(z) and a3(z), respectively.

Fig. 3
Fig. 3

(a) and (b) Schemes of the masks used to encode the data on the pump fields a4 and a5, respectively, to implement the and and or gates; (c) scheme of the output of the and gate expected on the generated field a2(z); (d) scheme of the output of the or gate expected on the generated field a3(z). The filled circles, the logic value 1; the dashed circles, the logic value 0.

Fig. 4
Fig. 4

Images (four gray levels) detected by the CCD camera as averages over 50 pulses relative to the and and or gates. (a) and (b) Intensity maps of fields a4 and a5, respectively, as detected at the BBO crystal entrance; (c) intensity map of field a2(z), which encodes the output of the and gate; (d) intensity map of field a3(z), which encodes the output of the or gate.

Fig. 5
Fig. 5

Experimental setup for the implementation of the not gate. HS, P1,2, and BBO as in Fig. 2; BS, beam splitter with T=0.67, R=0.33 at 45 deg. The binary number A5 is encoded on the pump field a5; the depleted field a3(z) represents the negation of A5, and field a2(z) carries its copy.

Fig. 6
Fig. 6

(a) Scheme of the mask used to encode the data on the pump field a5 to implement the not gate; (b) scheme of the output of the not gate expected on the depleted field a3(z), where black circles represent logic value 0.

Fig. 7
Fig. 7

Intensity maps (four gray levels) of the images detected by the CCD camera. (a) Pump field a5 (average over 50 shots) at the BBO crystal entrance face; (b) depleted field a3(z) (single shot) giving the output of the not gate; (c) generated field a2(z) (average over 50 shots) giving the replica of the input field a5.

Fig. 8
Fig. 8

Experimental setup for the implementation of the half-adder both in cases 1 and 2. Abbreviations as in Fig. 2. The results of the half-adder are read on fields a2(z) and a3(z) as arrays.

Fig. 9
Fig. 9

Schemes of the masks used to encode the data as amplitude modulations on the pump and seed fields to implement the half-adder in case 1. (a) and (b) Pump fields a4 and a5, respectively; (c) and (d) seed fields a1(0) and a2(0), respectively. White zones encode logic value 1, and dashed circles and gray area in (d) encode logic value 0.

Fig. 10
Fig. 10

Intensity maps (four gray levels) of the images detected by the CCD camera in single shot relative to the half-adder in case 1. (a) and (b) Pump fields a4 and a5, respectively, as detected at the BBO crystal entrance; (c) seed field a1(0), as detected at the BBO crystal entrance; (d) generated field a3(z) that encodes the sum; (e) generated field a2(z) that encodes the carry.

Fig. 11
Fig. 11

Schemes of the masks used to encode the data as amplitude modulations on the pump and seed fields to implement the half-adder in case 2. (a) and (b) Pump fields a4 and a5, respectively; (c) and (d) seed fields a1(0) and a2(0), respectively. White zones encode logic value 1, dashed circles and gray area in (d) encode logic value 0.

Fig. 12
Fig. 12

Intensity maps (four gray levels) of the images detected by the CCD camera in single shot relative to the half-adder in case 2. (a) and (b) Pump fields a4 and a5, respectively, as detected at the BBO crystal entrance; (c) and (d) seed fields a1(0) and a2(0), respectively, as detected at the BBO crystal entrance; (e) generated field a3(z) that encodes the sum; (f) generated field a2(z) that encodes the carry.

Fig. 13
Fig. 13

Experimental setup for the replication of the half-adder in case 1. BS, as in Fig. 5; M1,2, high-reflectance mirrors at 1064 nm; L1, cylindrical lens with focal length f=200 mm; BBO, β-BaB2O4 crystal having cut angle=22.8 deg; F, filter to erase the useless data on the generated field a3(z); the data encoded on the fields injected into the second BBO are X·Y (on field a2) and X·Y¯+X·¯Y (on field a1).

Fig. 14
Fig. 14

Intensity maps (four gray levels) of the images detected by the CCD camera in single shot relative to the loopable half-adder described in case 1. (a) Field a4(z) generated by the second BBO, which appears as the replica of the sum; (b) field a5(z) generated by the second BBO, which appears as the replica of the carry; (c) superimposed fields a4(z) and a5(z) generated by the second BBO, which carry the sum and carry of the half-adder.

Tables (3)

Tables Icon

Table 1 Truth Table for the Basic Logic Gates in the Interaction Scheme

Tables Icon

Table 2 Truth Table for the Half-Adder in the Interaction Scheme

Tables Icon

Table 3 Logic Functions of Two Binary Variablesa

Equations (18)

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da1/dz=-ig1a3*,
da2/dz=-ig2a3,
da3/dz=-ig1a1*-ig2*a2,
g1=2ω1ω3ω4η03n1on3on4e1/2[d22 cos(ϑ-ϑ4)+d31 sin(ϑ-ϑ4)]a4(0),
g2=2ω2ω3ω5η03n2en3on5o1/2[d22 cos(ϑ-ϑ2)+d31 sin(ϑ-ϑ2)]a5(0),
a1(z)=-1Γ2[|g1|2 cos(Γz)-|g2|2]a1(0)-g1g2Γ2[cos(Γz)-1]a2*(0)-i g1Γ sin(Γz)a3*(0),
a2(z)=g1g2Γ2[cos(Γz)-1]a1*(0)-1Γ2[|g1|2-|g2|2 cos(Γz)]a2(0)-i g2Γ sin(Γz)a3(0),
a3(z)=-i g1Γ sin(Γz)a1*(0)-i g2*Γ sin(Γz)a2(0)+cos(Γz)a3(0),
|a2(z)|2=|g1|2|g2|2|a1(0)|2cos2(|g2|2-|g1|2z)-2 cos(|g2|2-|g1|2z)+1(|g2|2-|g1|2)2.
A2=A1·A4·A5,
|a3(z)|2={|g1|2|a1(0)|2+|g2|2|a2(0)|2+2 Re[g1*g2*a1(0)a2(0)]} × sin2(|g2|2-|g1|2z)|g2|2-|g1|2.
A3=A1·A4+A2·A5,
|a3(z)|2=cos2(|g2|z)|a3(0)|2.
A3=A¯5,
A2=A4·A5,
A3=A4+A5.
A1=Y¯1,A2=X¯0,A4=XX,A5=YY.
A3=X·Y¯+Y·X¯X,A2=X¯X·Y.

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