Abstract

We derive a set of concise formulas to characterize the temperature sensitivity of holographic wavelength-division multiplexers–demultiplexers (H-MUX’s–H-DMUX’s). The normalized parameters such as dispersion abilities, central wavelength shift rate, and variations of insertion loss hold for general grating-based wavelength-division multiplexing–demultiplexing (WDM–WDDM) structures. The results are applicable to both wide-WDM–WDDM and dense ones working in 800-, 1300-, and 1550-nm optical wavelength windows, regardless of whether their input–output ports are single-mode or multimode fibers. Detailed analysis and experiments are carried out on a fully packaged four-channel H-MUX–H-DMUX. The experimental results at temperatures from 25 to 80 °C fit nicely with the theoretical prediction. We conclude that passive grating-based H-MUX’s–H-DMUX’s are promising for meeting the requirements on temperature sensitivity in optical data communications and telecommunications. Most of the analysis can be applied to other types of Bragg-grating-based WDM–WDDM.

© 2000 Optical Society of America

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1999

1998

P. Green, “Optical networking has arrived,” IEEE Commun. Mag. 36, 38 (1998).
[CrossRef]

J. P. Ryan, “WDM: North American deployment trends,” IEEE Commun. Mag. 36, 40–44 (1998).
[CrossRef]

C. DeCusatis, “Optical data communication: fundamentals and future directions,” Opt. Eng. 37, 3082–3099 (1998).
[CrossRef]

R. K. Butler, D. R. Polson, “Wave-division multiplexing in the Sprint long distance network,” IEEE Commun. Mag. 36, 52–55 (1998).
[CrossRef]

J. Yeh, A. Harton, K. Wyatt, “Realibility study of holographic optical elements made with DuPont photopolymer,” Appl. Opt. 37, 6270–6274 (1998).
[CrossRef]

1997

C. Zhao, J. Liu, Z. Fu, R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

1996

1995

1994

1993

1992

1990

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-waveguide grating for wavelength division multi-/demultiplexer with nanometer resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

1985

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE, 73, 894–937 (1985).
[CrossRef]

1982

J. E. Ludman, “Approximate bandwidth and the diffraction efficiency in thick holograms,” Am. J. Phys. 50, 244–246 (1982).
[CrossRef]

1981

T. H. Jamieson, “Thermal effects in optical systems,” Opt. Eng. 20, 156–160 (1981).
[CrossRef]

1969

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

1966

Baba, T.

Burckhardt, C. B.

Butler, R. K.

R. K. Butler, D. R. Polson, “Wave-division multiplexing in the Sprint long distance network,” IEEE Commun. Mag. 36, 52–55 (1998).
[CrossRef]

Chang, J.-T.

Chen, A.

S. X. Wu, C. S. Cheng, T. Huang, S. Qin, J. Yeh, Q. Gao, A. Chen, C. P. Yeh, A. Harton, K. Wyatt, “An experimental study on mechanical, thermomechanical, and optomechanical behaviors of holographic materials,” in Holographic Materials IV, T. J. Trout, ed., Proc. SPIE3294, 145–151 (1998).
[CrossRef]

Chen, R. T.

J. Liu, R. T. Chen, “Path-reversed substrate-guided-wave optical interconnects for wavelength-division demultiplexing,” Appl. Opt. 38, 3046–3052 (1999).
[CrossRef]

C. Zhao, J. Liu, Z. Fu, R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

C. C. Zhou, S. Sutton, R. T. Chen, B. V. Hunter, P. Dempewolf, “Four channel multimode wavelength division, multiplexer and demultiplexer based on photopolymer volume holographic gratings and substrate-guided waves,” in Design and Manufacturing of WDM Devices, R. T. Chen, L. S. Lome, eds., Proc. SPIE3234, 136–139 (1997).
[CrossRef]

R. T. Chen, J. Liu, X. Deng, “Multimode-fiber-compatible WDM/WDDM with an ultra-large wavelength range,” in Wavelength Division Multiplexing, R. T. Chen, L. S. Lome, eds., Vol. CR71 of SPIE Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash.1999), pp. 50–71.

X. Deng, F. Zhao, Z. Fu, J. Zou, J. Qiao, G. Kim, R. T. Chen, “Linearity of volume hologram out-coupling for wavelength-division demultiplexing,” in WDM and Photonic Switching Devices for Network Applications, R. T. Chen, G. F. Lipscomb, eds., Proc. SPIE3949, 109–119 (2000).
[CrossRef]

X. Deng, F. Zhao, R. T. Chen, “Optimal design of substrate-mode volume holographic wavelength divisiondemultiplexers,” in WDM and Photonic Switching Devices for Network Applications, R. T. Chen, G. F. Lipscomb, eds., Proc. SPIE3949, 120–136 (2000).
[CrossRef]

Cheng, C. S.

S. X. Wu, C. S. Cheng, T. Huang, S. Qin, J. Yeh, Q. Gao, A. Chen, C. P. Yeh, A. Harton, K. Wyatt, “An experimental study on mechanical, thermomechanical, and optomechanical behaviors of holographic materials,” in Holographic Materials IV, T. J. Trout, ed., Proc. SPIE3294, 145–151 (1998).
[CrossRef]

DeCusatis, C.

C. DeCusatis, “Optical data communication: fundamentals and future directions,” Opt. Eng. 37, 3082–3099 (1998).
[CrossRef]

Dempewolf, P.

C. C. Zhou, S. Sutton, R. T. Chen, B. V. Hunter, P. Dempewolf, “Four channel multimode wavelength division, multiplexer and demultiplexer based on photopolymer volume holographic gratings and substrate-guided waves,” in Design and Manufacturing of WDM Devices, R. T. Chen, L. S. Lome, eds., Proc. SPIE3234, 136–139 (1997).
[CrossRef]

Deng, X.

X. Deng, F. Zhao, R. T. Chen, “Optimal design of substrate-mode volume holographic wavelength divisiondemultiplexers,” in WDM and Photonic Switching Devices for Network Applications, R. T. Chen, G. F. Lipscomb, eds., Proc. SPIE3949, 120–136 (2000).
[CrossRef]

R. T. Chen, J. Liu, X. Deng, “Multimode-fiber-compatible WDM/WDDM with an ultra-large wavelength range,” in Wavelength Division Multiplexing, R. T. Chen, L. S. Lome, eds., Vol. CR71 of SPIE Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash.1999), pp. 50–71.

X. Deng, F. Zhao, Z. Fu, J. Zou, J. Qiao, G. Kim, R. T. Chen, “Linearity of volume hologram out-coupling for wavelength-division demultiplexing,” in WDM and Photonic Switching Devices for Network Applications, R. T. Chen, G. F. Lipscomb, eds., Proc. SPIE3949, 109–119 (2000).
[CrossRef]

Duzick, T.

W. J. Gambogi, K. Steijin, S. Mackara, T. Duzick, B. Hamzavy, J. Kelly, “HOE imaging in DuPont holographic photopolymers,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2152, 282–293 (1998).
[CrossRef]

Ferstl, M.

Fu, Z.

C. Zhao, J. Liu, Z. Fu, R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

X. Deng, F. Zhao, Z. Fu, J. Zou, J. Qiao, G. Kim, R. T. Chen, “Linearity of volume hologram out-coupling for wavelength-division demultiplexing,” in WDM and Photonic Switching Devices for Network Applications, R. T. Chen, G. F. Lipscomb, eds., Proc. SPIE3949, 109–119 (2000).
[CrossRef]

Gambogi, W. J.

W. J. Gambogi, W. A. Gerstadt, S. R. Mackara, A. W. Weber, “Holographic transmission elements using improved photopolymer films,” in Computer and Optically Generated Holographic Optics: IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 256–267 (1991).

W. J. Gambogi, A. M. Weber, T. J. Trout, “Advances and applications of DuPont holographic photopolymers,” in Holographic Imaging and Materials, T. H. Jeong, ed., Proc. SPIE2043, 2–13 (1993).
[CrossRef]

W. J. Gambogi, K. Steijin, S. Mackara, T. Duzick, B. Hamzavy, J. Kelly, “HOE imaging in DuPont holographic photopolymers,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2152, 282–293 (1998).
[CrossRef]

Gao, Q.

S. X. Wu, C. S. Cheng, T. Huang, S. Qin, J. Yeh, Q. Gao, A. Chen, C. P. Yeh, A. Harton, K. Wyatt, “An experimental study on mechanical, thermomechanical, and optomechanical behaviors of holographic materials,” in Holographic Materials IV, T. J. Trout, ed., Proc. SPIE3294, 145–151 (1998).
[CrossRef]

Gaylord, T. K.

Gerstadt, W. A.

W. J. Gambogi, W. A. Gerstadt, S. R. Mackara, A. W. Weber, “Holographic transmission elements using improved photopolymer films,” in Computer and Optically Generated Holographic Optics: IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 256–267 (1991).

Grann, E. B.

Green, P.

P. Green, “Optical networking has arrived,” IEEE Commun. Mag. 36, 38 (1998).
[CrossRef]

Hamzavy, B.

W. J. Gambogi, K. Steijin, S. Mackara, T. Duzick, B. Hamzavy, J. Kelly, “HOE imaging in DuPont holographic photopolymers,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2152, 282–293 (1998).
[CrossRef]

Harton, A.

J. Yeh, A. Harton, K. Wyatt, “Realibility study of holographic optical elements made with DuPont photopolymer,” Appl. Opt. 37, 6270–6274 (1998).
[CrossRef]

S. X. Wu, C. S. Cheng, T. Huang, S. Qin, J. Yeh, Q. Gao, A. Chen, C. P. Yeh, A. Harton, K. Wyatt, “An experimental study on mechanical, thermomechanical, and optomechanical behaviors of holographic materials,” in Holographic Materials IV, T. J. Trout, ed., Proc. SPIE3294, 145–151 (1998).
[CrossRef]

Hellmich, H.

Huang, T.

S. X. Wu, C. S. Cheng, T. Huang, S. Qin, J. Yeh, Q. Gao, A. Chen, C. P. Yeh, A. Harton, K. Wyatt, “An experimental study on mechanical, thermomechanical, and optomechanical behaviors of holographic materials,” in Holographic Materials IV, T. J. Trout, ed., Proc. SPIE3294, 145–151 (1998).
[CrossRef]

Huang, Y. T.

Huang, Y.-T.

Hunter, B. V.

C. C. Zhou, S. Sutton, R. T. Chen, B. V. Hunter, P. Dempewolf, “Four channel multimode wavelength division, multiplexer and demultiplexer based on photopolymer volume holographic gratings and substrate-guided waves,” in Design and Manufacturing of WDM Devices, R. T. Chen, L. S. Lome, eds., Proc. SPIE3234, 136–139 (1997).
[CrossRef]

Iga, K.

Intani, D.

Jamieson, T. H.

T. H. Jamieson, “Thermal effects in optical systems,” Opt. Eng. 20, 156–160 (1981).
[CrossRef]

Kato, K.

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-waveguide grating for wavelength division multi-/demultiplexer with nanometer resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

Kelly, J.

W. J. Gambogi, K. Steijin, S. Mackara, T. Duzick, B. Hamzavy, J. Kelly, “HOE imaging in DuPont holographic photopolymers,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2152, 282–293 (1998).
[CrossRef]

Kim, G.

X. Deng, F. Zhao, Z. Fu, J. Zou, J. Qiao, G. Kim, R. T. Chen, “Linearity of volume hologram out-coupling for wavelength-division demultiplexing,” in WDM and Photonic Switching Devices for Network Applications, R. T. Chen, G. F. Lipscomb, eds., Proc. SPIE3949, 109–119 (2000).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Kuhlow, B.

Li, L.

Liu, J.

J. Liu, R. T. Chen, “Path-reversed substrate-guided-wave optical interconnects for wavelength-division demultiplexing,” Appl. Opt. 38, 3046–3052 (1999).
[CrossRef]

C. Zhao, J. Liu, Z. Fu, R. T. Chen, “Shrinkage-corrected volume holograms based on photopolymeric phase media for surface-normal optical interconnects,” Appl. Phys. Lett. 71, 1464–1466 (1997).
[CrossRef]

R. T. Chen, J. Liu, X. Deng, “Multimode-fiber-compatible WDM/WDDM with an ultra-large wavelength range,” in Wavelength Division Multiplexing, R. T. Chen, L. S. Lome, eds., Vol. CR71 of SPIE Critical Reviews of Optical Science and Technology (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash.1999), pp. 50–71.

Ludman, J. E.

J. E. Ludman, “Approximate bandwidth and the diffraction efficiency in thick holograms,” Am. J. Phys. 50, 244–246 (1982).
[CrossRef]

Mackara, S.

W. J. Gambogi, K. Steijin, S. Mackara, T. Duzick, B. Hamzavy, J. Kelly, “HOE imaging in DuPont holographic photopolymers,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2152, 282–293 (1998).
[CrossRef]

Mackara, S. R.

W. J. Gambogi, W. A. Gerstadt, S. R. Mackara, A. W. Weber, “Holographic transmission elements using improved photopolymer films,” in Computer and Optically Generated Holographic Optics: IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 256–267 (1991).

Moharam, M. G.

Nishi, I.

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-waveguide grating for wavelength division multi-/demultiplexer with nanometer resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

Orwoll, R. A.

R. A. Orwoll, “Densities, coefficients of thermal expansion, and compressibilities of amorphous polymers,” in Physical Properties of Polymers Handbook, J. E. Mark, ed. (American Institute of Physics, Woodbury, N.Y., 1996).

Pawlowski, E.

Polson, D. R.

R. K. Butler, D. R. Polson, “Wave-division multiplexing in the Sprint long distance network,” IEEE Commun. Mag. 36, 52–55 (1998).
[CrossRef]

Pommet, D. A.

Qiao, J.

X. Deng, F. Zhao, Z. Fu, J. Zou, J. Qiao, G. Kim, R. T. Chen, “Linearity of volume hologram out-coupling for wavelength-division demultiplexing,” in WDM and Photonic Switching Devices for Network Applications, R. T. Chen, G. F. Lipscomb, eds., Proc. SPIE3949, 109–119 (2000).
[CrossRef]

Qin, S.

S. X. Wu, C. S. Cheng, T. Huang, S. Qin, J. Yeh, Q. Gao, A. Chen, C. P. Yeh, A. Harton, K. Wyatt, “An experimental study on mechanical, thermomechanical, and optomechanical behaviors of holographic materials,” in Holographic Materials IV, T. J. Trout, ed., Proc. SPIE3294, 145–151 (1998).
[CrossRef]

Ryan, J. P.

J. P. Ryan, “WDM: North American deployment trends,” IEEE Commun. Mag. 36, 40–44 (1998).
[CrossRef]

Salgueiro, J. R.

Steijin, K.

W. J. Gambogi, K. Steijin, S. Mackara, T. Duzick, B. Hamzavy, J. Kelly, “HOE imaging in DuPont holographic photopolymers,” in Diffractive and Holographic Optics Technology, I. Cindrich, S. H. Lee, eds., Proc. SPIE2152, 282–293 (1998).
[CrossRef]

Su, D. C.

Su, D.-C.

Sutton, S.

C. C. Zhou, S. Sutton, R. T. Chen, B. V. Hunter, P. Dempewolf, “Four channel multimode wavelength division, multiplexer and demultiplexer based on photopolymer volume holographic gratings and substrate-guided waves,” in Design and Manufacturing of WDM Devices, R. T. Chen, L. S. Lome, eds., Proc. SPIE3234, 136–139 (1997).
[CrossRef]

Suzuki, S.

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-waveguide grating for wavelength division multi-/demultiplexer with nanometer resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

Syms, R. R. A.

R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, 1990).

Takahashi, H.

H. Takahashi, S. Suzuki, K. Kato, I. Nishi, “Arrayed-waveguide grating for wavelength division multi-/demultiplexer with nanometer resolution,” Electron. Lett. 26, 87–88 (1990).
[CrossRef]

Trout, T. J.

W. J. Gambogi, A. M. Weber, T. J. Trout, “Advances and applications of DuPont holographic photopolymers,” in Holographic Imaging and Materials, T. H. Jeong, ed., Proc. SPIE2043, 2–13 (1993).
[CrossRef]

Tsai, Y. K.

Warmuth, C.

Weber, A. M.

W. J. Gambogi, A. M. Weber, T. J. Trout, “Advances and applications of DuPont holographic photopolymers,” in Holographic Imaging and Materials, T. H. Jeong, ed., Proc. SPIE2043, 2–13 (1993).
[CrossRef]

Weber, A. W.

W. J. Gambogi, W. A. Gerstadt, S. R. Mackara, A. W. Weber, “Holographic transmission elements using improved photopolymer films,” in Computer and Optically Generated Holographic Optics: IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 256–267 (1991).

Wu, S. X.

S. X. Wu, C. S. Cheng, T. Huang, S. Qin, J. Yeh, Q. Gao, A. Chen, C. P. Yeh, A. Harton, K. Wyatt, “An experimental study on mechanical, thermomechanical, and optomechanical behaviors of holographic materials,” in Holographic Materials IV, T. J. Trout, ed., Proc. SPIE3294, 145–151 (1998).
[CrossRef]

Wyatt, K.

J. Yeh, A. Harton, K. Wyatt, “Realibility study of holographic optical elements made with DuPont photopolymer,” Appl. Opt. 37, 6270–6274 (1998).
[CrossRef]

S. X. Wu, C. S. Cheng, T. Huang, S. Qin, J. Yeh, Q. Gao, A. Chen, C. P. Yeh, A. Harton, K. Wyatt, “An experimental study on mechanical, thermomechanical, and optomechanical behaviors of holographic materials,” in Holographic Materials IV, T. J. Trout, ed., Proc. SPIE3294, 145–151 (1998).
[CrossRef]

Yeh, C. P.

S. X. Wu, C. S. Cheng, T. Huang, S. Qin, J. Yeh, Q. Gao, A. Chen, C. P. Yeh, A. Harton, K. Wyatt, “An experimental study on mechanical, thermomechanical, and optomechanical behaviors of holographic materials,” in Holographic Materials IV, T. J. Trout, ed., Proc. SPIE3294, 145–151 (1998).
[CrossRef]

Yeh, J.

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Figures (8)

Fig. 1
Fig. 1

Diffraction geometry of holographic gratings. The space is divided into three layers, namely, the incident space (or superstrate), the grating layer, and the transmission layer (or substrate), denoted with -1, 0, and +1, respectively.

Fig. 2
Fig. 2

Normalized material dispersion abilities of BK7 glass and synthetic fused silica.

Fig. 3
Fig. 3

Coupling focused beams from free space to V-grooved linear fiber array. The fibers are centered at (x l , y l ), whereas the beams are focused at (xk, yk), k, l = 1, 2, 3, … , N c .

Fig. 4
Fig. 4

EIB model of coupling light from free space to fibers. The z axis is the fractional energy in the fiber core while the size of the focused beam is 2σ, the misalignment between the focus center to the fiber center is x dev, and the fiber radius is R f .

Fig. 5
Fig. 5

Schematic top view (not to scale) of the fully packaged four-channel 1 H-DMUX. Both the input (before the collimator S) and the four outputs (at V) are multimode gradient-index fibers with a core size of 62.5 µm. H, hologram; M, mirror; L, coupling lens; V, four-channel fibers in a linear spaced V groove.

Fig. 6
Fig. 6

Schematic side view (not to the scale) of the fully packaged four-channel H-DMUX. The details of the dimensions and the materials of the components are listed in Tables 1 and 2.

Fig. 7
Fig. 7

CWS’s of the fully packaged H-MUX–H-DMUX when working temperature changes. The temperature of device is electronially controlled with an accuracy of ±1 °C from 25 to 80 °C at a 5 °C step. (a) Linear fitting of the average CWS’s of the four channels; the slope is -1.785 × 10-2 nm/°C. (b) Boundary of the CWS with the CTE of the Mylar (on top of the hologram) for the hologram. (c) Boundary of the CWS with the CTE of the BK7 glass for the hologram.

Fig. 8
Fig. 8

ILV’s versus the working temperatures. The temperature of the device is electronially controlled with an accuracy of ±1 °C with an accuracy of ±1 °C from 25 to 80 °C at a 5 °C step.

Tables (2)

Tables Icon

Table 1 Coefficients of Thermal Expansions of the Materials Involved in the H-MUX–H-DMUXa

Tables Icon

Table 2 Dimensions of the Components Illustrated in Figs. 5 and 6 as well as the Materials Involved

Equations (32)

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

θo,m±=arcsink0+μsinθi+mKk0±μ,m=0, ±1, ±2,,
Dt±λ=tanθo,m±DM+λ-DM±λ+DG±λ1-DM+λ,
DM±λλn±λdn±λdλ,
DG±λsinθo,m±1-nr±λsinθicosθo,m±.
Dt±λ=DG±λ1-DM+λ.
DM±λDM±λc,
λl=λc+l-Nc+12λs,  l=1, 2, 3,, Nc,
Δλλc=Λxtcosθo,-1±cos ϕsinδθ,
Q=2πλct/n0Λ2>10,
ηJ=sin2νJ2+ζ21/21+ζ/νJ2,  ζ=ϑ 12CS,  J=s, p,
νs=πn1tλCRCS1/2,  νp=νs cos2θi-ϕ,
CR=cosθi,  CS=cosθi-Kg/k0cosϕ.
Ek,l=-+ WF,lx, yIkx, ydxdy-+ Ikx, ydxdy,k, l=1, 2, 3,, Nc.
Rk,l=-10 log10Ek,l/El,l,
ΓlT1λc,lλc,lT=βT+1-τγ++τγ±,l=1, 2, 3,, Nc,  m0,
βT=1ΛxΛxT,
γ±=1n±n±T.
τ=sinθo,m±sinθo,m±-nr± sinθi.
1γJηJT=2 cosνJsinνJνJγg+βT1-Kg cosϕ2k0CS,J=s, p,
θo,m±T=-ΓlTDG±λ.
δSRffRfθo,m±T ΔT,
δSRf-fDG±λλsλcRfλcΓlTΔTλs=-s0RfΔλλs
ΔαM=αMTH-αMTLαMTLβg-βB+βM2ΔT,αM1,
Δθo±=Δθo,m±-2ΔαM.
Δθo±dθo,m±dT ΔT=-ΓlnTDG±λΔT=-ΓlT+αMTL2βg-βB-βMDG±λDG±λΔT,
ΓlnT1λc,ldλc,ldT=ΓlT+αMTL2βg-βB-βMDG±λ.
αf1fdfdT=βglass-1nglass-nairdnglassdT-nglassdnairdT.
Δy=yv+βBK7RL+βsteeltM-βAltP+βNO A61tg+βBK7tG+βSitSiΔT,
Δz=fαf-βAlsB+βVsvΔT,
Ptλ=Poλ/PiλPtrλ=Porλ/Pirλ,δλ  λBW  λW,
Ptrλ=Ptλ * hλ,
λ= λPtrλdλ Ptrλdλ=PtνPtνν=0+ννν=0,

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