Abstract

The sensitivity of optical waveguide chemical sensors for TM modes is investigated by use of the group-index method [J. Lightwave Technol. 14, 1907 (1996)]. We compare the absorption coefficients of TM modes with those of TE modes to determine in which mode the sensor should work. For open-clad-type evanescent-wave sensors, which mode is more favorable depends on whether the refractive index of the measurand is smaller or larger than that of the substrate; for buffered-clad-type evanescent-wave sensors, the choice depends on the function of the buffer layer; for sensing-layer-type evanescent-wave sensors, the TE mode has greater sensitivity than the TM mode; and for guided-wave sensors, both TE and TM modes are suitable for use in highly sensitive sensors.

© 1999 Optical Society of America

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References

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  1. A. Brecht and G. Gauglitz, “Recent developments in optical transducers for chemical or biochemical applications,” Sens. Actuators B38–39, 1–7 (1997).
    [CrossRef]
  2. L. Yang and S. S. Saavedra, “Chemical sensing using sol-gel derived planar waveguides and indicator phases,” Anal. Chem. 67, 1307–1314 (1995).
    [CrossRef]
  3. Z. A. Ansari, R. N. Karekar, and R. C. Aiyer, “Planar optical waveguide with PbCl2 cladding: a chlorine sensor,” J. Mater. Sci.: Mater. Electron. 7, 255–259 (1996).
  4. S. M. Klainer, S. L. Coulter, R. J. Pollina, and D. Saini, “Advances in miniature optical waveguide sensors,” Sens. Actuators B38–39, 176–182 (1997).
    [CrossRef]
  5. D.-K. Qing, I. Yamaguchi, K. Itoh, and M. Murabayashi, “Coupling-wave structures for optical waveguide sensors,” Opt. Rev. 4, 578–583 (1997).
    [CrossRef]
  6. R. E. Kunz, “Miniature integrated optical modules for chemical and biochemical sensing,” Sens. Actuators B38–39, 13–28 (1997).
    [CrossRef]
  7. K. A. Remley and A. Weisshaar, “Design and analysis of a silicon-based antiresonant reflecting optical waveguide chemical sensor,” Opt. Lett. 21, 1241–1243 (1996).
    [CrossRef] [PubMed]
  8. Z. Weissman, “Evanescent field sensors with periodically segmented waveguides,” Appl. Opt. 36, 1218–1222 (1997).
    [CrossRef] [PubMed]
  9. K. Benaissa and A. Nathan, “Silicon anti-resonant reflecting optical waveguides for sensor applications,” Sens. Actuators A65, 33–44 (1998).
    [CrossRef]
  10. J. D. Swalen, M. Tacke, R. Santo, K. E. Rieckhoff, and J. Fischer, “Spectra of organic molecules in thin film,” Helv. Chim Acta 61, 960–977 (1978).
    [CrossRef]
  11. G. Stewart, J. Norris, D. F. Clark, and B. Culshaw, “Evanescent-wave chemical sensors—a theoretical evaluation,” Int. J. Optoelectron. 6, 227–238 (1991).
  12. G. Stewart and B. Culshaw, “Optical waveguide modeling and design for evanescent field chemical sensors,” Opt. Quantum Electron. 26, S249–S259 (1994).
    [CrossRef]
  13. O. Parriaux and P. Dierauer, “Normalized expressions for the optical sensitivity of evanescent wave sensors,” Opt. Lett. 19, 508–510 (1994).
    [CrossRef] [PubMed]
  14. H. Gnewuch and H. Renner, “Mode-independent attenuation in evanescent-field sensors,” Appl. Opt. 34, 1473–1483 (1995).
    [CrossRef] [PubMed]
  15. D.-K. Qing, X.-M. Chen, K. Itoh, and M. Murabayashi, “A theoretical evaluation of the absorption coefficient of optical waveguide chemical or biological sensors by group index method,” J. Lightwave Technol. 14, 1907–1917 (1996).
    [CrossRef]
  16. M. J. Sun and M. W. Muller, “Measurements on four-layer isotropic waveguides,” Appl. Opt. 16, 814–815 (1977).
    [PubMed]

1998 (1)

K. Benaissa and A. Nathan, “Silicon anti-resonant reflecting optical waveguides for sensor applications,” Sens. Actuators A65, 33–44 (1998).
[CrossRef]

1997 (5)

A. Brecht and G. Gauglitz, “Recent developments in optical transducers for chemical or biochemical applications,” Sens. Actuators B38–39, 1–7 (1997).
[CrossRef]

S. M. Klainer, S. L. Coulter, R. J. Pollina, and D. Saini, “Advances in miniature optical waveguide sensors,” Sens. Actuators B38–39, 176–182 (1997).
[CrossRef]

D.-K. Qing, I. Yamaguchi, K. Itoh, and M. Murabayashi, “Coupling-wave structures for optical waveguide sensors,” Opt. Rev. 4, 578–583 (1997).
[CrossRef]

R. E. Kunz, “Miniature integrated optical modules for chemical and biochemical sensing,” Sens. Actuators B38–39, 13–28 (1997).
[CrossRef]

Z. Weissman, “Evanescent field sensors with periodically segmented waveguides,” Appl. Opt. 36, 1218–1222 (1997).
[CrossRef] [PubMed]

1996 (3)

K. A. Remley and A. Weisshaar, “Design and analysis of a silicon-based antiresonant reflecting optical waveguide chemical sensor,” Opt. Lett. 21, 1241–1243 (1996).
[CrossRef] [PubMed]

D.-K. Qing, X.-M. Chen, K. Itoh, and M. Murabayashi, “A theoretical evaluation of the absorption coefficient of optical waveguide chemical or biological sensors by group index method,” J. Lightwave Technol. 14, 1907–1917 (1996).
[CrossRef]

Z. A. Ansari, R. N. Karekar, and R. C. Aiyer, “Planar optical waveguide with PbCl2 cladding: a chlorine sensor,” J. Mater. Sci.: Mater. Electron. 7, 255–259 (1996).

1995 (2)

L. Yang and S. S. Saavedra, “Chemical sensing using sol-gel derived planar waveguides and indicator phases,” Anal. Chem. 67, 1307–1314 (1995).
[CrossRef]

H. Gnewuch and H. Renner, “Mode-independent attenuation in evanescent-field sensors,” Appl. Opt. 34, 1473–1483 (1995).
[CrossRef] [PubMed]

1994 (2)

O. Parriaux and P. Dierauer, “Normalized expressions for the optical sensitivity of evanescent wave sensors,” Opt. Lett. 19, 508–510 (1994).
[CrossRef] [PubMed]

G. Stewart and B. Culshaw, “Optical waveguide modeling and design for evanescent field chemical sensors,” Opt. Quantum Electron. 26, S249–S259 (1994).
[CrossRef]

1991 (1)

G. Stewart, J. Norris, D. F. Clark, and B. Culshaw, “Evanescent-wave chemical sensors—a theoretical evaluation,” Int. J. Optoelectron. 6, 227–238 (1991).

1978 (1)

J. D. Swalen, M. Tacke, R. Santo, K. E. Rieckhoff, and J. Fischer, “Spectra of organic molecules in thin film,” Helv. Chim Acta 61, 960–977 (1978).
[CrossRef]

1977 (1)

Aiyer, R. C.

Z. A. Ansari, R. N. Karekar, and R. C. Aiyer, “Planar optical waveguide with PbCl2 cladding: a chlorine sensor,” J. Mater. Sci.: Mater. Electron. 7, 255–259 (1996).

Ansari, Z. A.

Z. A. Ansari, R. N. Karekar, and R. C. Aiyer, “Planar optical waveguide with PbCl2 cladding: a chlorine sensor,” J. Mater. Sci.: Mater. Electron. 7, 255–259 (1996).

Benaissa, K.

K. Benaissa and A. Nathan, “Silicon anti-resonant reflecting optical waveguides for sensor applications,” Sens. Actuators A65, 33–44 (1998).
[CrossRef]

Brecht, A.

A. Brecht and G. Gauglitz, “Recent developments in optical transducers for chemical or biochemical applications,” Sens. Actuators B38–39, 1–7 (1997).
[CrossRef]

Chen, X.-M.

D.-K. Qing, X.-M. Chen, K. Itoh, and M. Murabayashi, “A theoretical evaluation of the absorption coefficient of optical waveguide chemical or biological sensors by group index method,” J. Lightwave Technol. 14, 1907–1917 (1996).
[CrossRef]

Clark, D. F.

G. Stewart, J. Norris, D. F. Clark, and B. Culshaw, “Evanescent-wave chemical sensors—a theoretical evaluation,” Int. J. Optoelectron. 6, 227–238 (1991).

Coulter, S. L.

S. M. Klainer, S. L. Coulter, R. J. Pollina, and D. Saini, “Advances in miniature optical waveguide sensors,” Sens. Actuators B38–39, 176–182 (1997).
[CrossRef]

Culshaw, B.

G. Stewart and B. Culshaw, “Optical waveguide modeling and design for evanescent field chemical sensors,” Opt. Quantum Electron. 26, S249–S259 (1994).
[CrossRef]

G. Stewart, J. Norris, D. F. Clark, and B. Culshaw, “Evanescent-wave chemical sensors—a theoretical evaluation,” Int. J. Optoelectron. 6, 227–238 (1991).

Dierauer, P.

Fischer, J.

J. D. Swalen, M. Tacke, R. Santo, K. E. Rieckhoff, and J. Fischer, “Spectra of organic molecules in thin film,” Helv. Chim Acta 61, 960–977 (1978).
[CrossRef]

Gauglitz, G.

A. Brecht and G. Gauglitz, “Recent developments in optical transducers for chemical or biochemical applications,” Sens. Actuators B38–39, 1–7 (1997).
[CrossRef]

Gnewuch, H.

Itoh, K.

D.-K. Qing, I. Yamaguchi, K. Itoh, and M. Murabayashi, “Coupling-wave structures for optical waveguide sensors,” Opt. Rev. 4, 578–583 (1997).
[CrossRef]

D.-K. Qing, X.-M. Chen, K. Itoh, and M. Murabayashi, “A theoretical evaluation of the absorption coefficient of optical waveguide chemical or biological sensors by group index method,” J. Lightwave Technol. 14, 1907–1917 (1996).
[CrossRef]

Karekar, R. N.

Z. A. Ansari, R. N. Karekar, and R. C. Aiyer, “Planar optical waveguide with PbCl2 cladding: a chlorine sensor,” J. Mater. Sci.: Mater. Electron. 7, 255–259 (1996).

Klainer, S. M.

S. M. Klainer, S. L. Coulter, R. J. Pollina, and D. Saini, “Advances in miniature optical waveguide sensors,” Sens. Actuators B38–39, 176–182 (1997).
[CrossRef]

Kunz, R. E.

R. E. Kunz, “Miniature integrated optical modules for chemical and biochemical sensing,” Sens. Actuators B38–39, 13–28 (1997).
[CrossRef]

Muller, M. W.

Murabayashi, M.

D.-K. Qing, I. Yamaguchi, K. Itoh, and M. Murabayashi, “Coupling-wave structures for optical waveguide sensors,” Opt. Rev. 4, 578–583 (1997).
[CrossRef]

D.-K. Qing, X.-M. Chen, K. Itoh, and M. Murabayashi, “A theoretical evaluation of the absorption coefficient of optical waveguide chemical or biological sensors by group index method,” J. Lightwave Technol. 14, 1907–1917 (1996).
[CrossRef]

Nathan, A.

K. Benaissa and A. Nathan, “Silicon anti-resonant reflecting optical waveguides for sensor applications,” Sens. Actuators A65, 33–44 (1998).
[CrossRef]

Norris, J.

G. Stewart, J. Norris, D. F. Clark, and B. Culshaw, “Evanescent-wave chemical sensors—a theoretical evaluation,” Int. J. Optoelectron. 6, 227–238 (1991).

Parriaux, O.

Pollina, R. J.

S. M. Klainer, S. L. Coulter, R. J. Pollina, and D. Saini, “Advances in miniature optical waveguide sensors,” Sens. Actuators B38–39, 176–182 (1997).
[CrossRef]

Qing, D.-K.

D.-K. Qing, I. Yamaguchi, K. Itoh, and M. Murabayashi, “Coupling-wave structures for optical waveguide sensors,” Opt. Rev. 4, 578–583 (1997).
[CrossRef]

D.-K. Qing, X.-M. Chen, K. Itoh, and M. Murabayashi, “A theoretical evaluation of the absorption coefficient of optical waveguide chemical or biological sensors by group index method,” J. Lightwave Technol. 14, 1907–1917 (1996).
[CrossRef]

Remley, K. A.

Renner, H.

Rieckhoff, K. E.

J. D. Swalen, M. Tacke, R. Santo, K. E. Rieckhoff, and J. Fischer, “Spectra of organic molecules in thin film,” Helv. Chim Acta 61, 960–977 (1978).
[CrossRef]

Saavedra, S. S.

L. Yang and S. S. Saavedra, “Chemical sensing using sol-gel derived planar waveguides and indicator phases,” Anal. Chem. 67, 1307–1314 (1995).
[CrossRef]

Saini, D.

S. M. Klainer, S. L. Coulter, R. J. Pollina, and D. Saini, “Advances in miniature optical waveguide sensors,” Sens. Actuators B38–39, 176–182 (1997).
[CrossRef]

Santo, R.

J. D. Swalen, M. Tacke, R. Santo, K. E. Rieckhoff, and J. Fischer, “Spectra of organic molecules in thin film,” Helv. Chim Acta 61, 960–977 (1978).
[CrossRef]

Stewart, G.

G. Stewart and B. Culshaw, “Optical waveguide modeling and design for evanescent field chemical sensors,” Opt. Quantum Electron. 26, S249–S259 (1994).
[CrossRef]

G. Stewart, J. Norris, D. F. Clark, and B. Culshaw, “Evanescent-wave chemical sensors—a theoretical evaluation,” Int. J. Optoelectron. 6, 227–238 (1991).

Sun, M. J.

Swalen, J. D.

J. D. Swalen, M. Tacke, R. Santo, K. E. Rieckhoff, and J. Fischer, “Spectra of organic molecules in thin film,” Helv. Chim Acta 61, 960–977 (1978).
[CrossRef]

Tacke, M.

J. D. Swalen, M. Tacke, R. Santo, K. E. Rieckhoff, and J. Fischer, “Spectra of organic molecules in thin film,” Helv. Chim Acta 61, 960–977 (1978).
[CrossRef]

Weisshaar, A.

Weissman, Z.

Yamaguchi, I.

D.-K. Qing, I. Yamaguchi, K. Itoh, and M. Murabayashi, “Coupling-wave structures for optical waveguide sensors,” Opt. Rev. 4, 578–583 (1997).
[CrossRef]

Yang, L.

L. Yang and S. S. Saavedra, “Chemical sensing using sol-gel derived planar waveguides and indicator phases,” Anal. Chem. 67, 1307–1314 (1995).
[CrossRef]

Anal. Chem. (1)

L. Yang and S. S. Saavedra, “Chemical sensing using sol-gel derived planar waveguides and indicator phases,” Anal. Chem. 67, 1307–1314 (1995).
[CrossRef]

Appl. Opt. (3)

Helv. Chim Acta (1)

J. D. Swalen, M. Tacke, R. Santo, K. E. Rieckhoff, and J. Fischer, “Spectra of organic molecules in thin film,” Helv. Chim Acta 61, 960–977 (1978).
[CrossRef]

Int. J. Optoelectron. (1)

G. Stewart, J. Norris, D. F. Clark, and B. Culshaw, “Evanescent-wave chemical sensors—a theoretical evaluation,” Int. J. Optoelectron. 6, 227–238 (1991).

J. Lightwave Technol. (1)

D.-K. Qing, X.-M. Chen, K. Itoh, and M. Murabayashi, “A theoretical evaluation of the absorption coefficient of optical waveguide chemical or biological sensors by group index method,” J. Lightwave Technol. 14, 1907–1917 (1996).
[CrossRef]

J. Mater. Sci.: Mater. Electron. (1)

Z. A. Ansari, R. N. Karekar, and R. C. Aiyer, “Planar optical waveguide with PbCl2 cladding: a chlorine sensor,” J. Mater. Sci.: Mater. Electron. 7, 255–259 (1996).

Opt. Lett. (2)

Opt. Quantum Electron. (1)

G. Stewart and B. Culshaw, “Optical waveguide modeling and design for evanescent field chemical sensors,” Opt. Quantum Electron. 26, S249–S259 (1994).
[CrossRef]

Opt. Rev. (1)

D.-K. Qing, I. Yamaguchi, K. Itoh, and M. Murabayashi, “Coupling-wave structures for optical waveguide sensors,” Opt. Rev. 4, 578–583 (1997).
[CrossRef]

Sens. Actuators (4)

R. E. Kunz, “Miniature integrated optical modules for chemical and biochemical sensing,” Sens. Actuators B38–39, 13–28 (1997).
[CrossRef]

K. Benaissa and A. Nathan, “Silicon anti-resonant reflecting optical waveguides for sensor applications,” Sens. Actuators A65, 33–44 (1998).
[CrossRef]

S. M. Klainer, S. L. Coulter, R. J. Pollina, and D. Saini, “Advances in miniature optical waveguide sensors,” Sens. Actuators B38–39, 176–182 (1997).
[CrossRef]

A. Brecht and G. Gauglitz, “Recent developments in optical transducers for chemical or biochemical applications,” Sens. Actuators B38–39, 1–7 (1997).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Model of an absorptive medium with light propagating in the z direction.

Fig. 2
Fig. 2

Schematic structures of (a) open-clad-type evanescent-wave sensors, (b) buffered-clad-type evanescent-wave sensors, (c) sensing-layer-type evanescent wave sensors, and (d) guided-wave sensors.

Fig. 3
Fig. 3

(a) Absorption coefficient, (b) group index of the cladding, (c) power fraction of the cladding, (d) power fraction of the guiding film, and (e) power fraction of the substrate as functions of the thickness of the guiding film for a step-index waveguide with nabs=1.33, nf=1.52, ns=1.51, and λ=0.6328 µm (for open-clad-type evanescent-wave sensors).

Fig. 4
Fig. 4

Power fractions of (a) the cladding, (b) the guiding film, and (c) the substrate as functions of the thickness of the guiding film. nf, ns, and λ are fixed to be 1.52, 1.51, and 0.6328 µm, respectively; nabs varies among 1.0, 1.33, and 1.51 (for open-clad-type evanescent-wave sensors).  

Fig. 5
Fig. 5

(a) Absorption coefficient, (b) group index of the cladding, and (c) power fraction of the cladding as functions of the thickness of the guiding film for step-index waveguides with nabs=1.33, nf=1.52, ns=1.51, nbuf=1.40, and λ=0.6328 µm. The refractive index of the buffer layer is smaller than that of the guiding film, and the thickness of the buffer layer varies among 0.1, 0.2, 0.3 µm (for buffered-clad-type evanescent-wave sensors).

Fig. 6
Fig. 6

(a) Absorption coefficient, (b) group index of the cladding, (c) power fraction of the cladding as functions of thickness of the guiding film for step-index waveguides with nabs=1.33, nf=1.52, ns=1.51, nbuf=1.80, and λ=0.6328 µm. The refractive index of the buffer layer is larger than that of the guiding film, and the thickness of the buffer layer varies among 0.01, 0.03, 0.05 µm (for buffered-clad-type evanescent-wave sensors).

Fig. 7
Fig. 7

(a) Absorption coefficient, (b) group index of the sensing layer, and (c) power fraction of the sensing layer as functions of the thickness of the guiding film for step-index waveguides with nc=1.33, nf=1.52, ns=nabs=1.51, and λ=0.6328 µm. The refractive index of the sensing layer is less than that of the guiding film, and the thickness of the sensing layer varies among 0.2, 0.3, 0.5 µm (for sensing-layer-type evanescent-wave sensors).

Fig. 8
Fig. 8

(a) Absorption coefficient, (b) group index of the sensing layer, and (c) power fraction of the sensing layer as functions of thickness of the guiding film for step-index waveguides with nc=1.33, nf=1.52, ns=1.51, nabs=1.80, and λ=0.6328 µm. The refractive index of the sensing layer is larger than that of the guiding film, and the thickness of the sensing layer varies among 0.02, 0.03, 0.05 µm (for sensing-layer-type evanescent-wave sensors).

Fig. 9
Fig. 9

Absorption coefficient as a function of thickness of the guiding film for step-index waveguides with nc=1.33, nabs=1.52, ns=1.51, and λ=0.6328 µm (for guided-wave sensors).

Equations (57)

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

α(L)ln P(0)P(L)=0LCNABhν Ng(z)cF(z)dz,
α(z)=CNABhν nabscz.
εConv=0.438NABhν(nabs/c).
2Hy(x)x2+[n2(x)k02-β2]Hy(x)=0,
Ex(x)=βωε0n2(x)Hy(x),
Ez(x)=-jω0n2(x)Hy(x)x,
ρw(x)=14[(x)(ExEx*+EzEz*)+μ(x)HyHy*],
P=β2ω0 HyHy*n2(x)dx,
αWgd(L)=CNABhν NgArFArcL,
NgAr=Arρw(x)dxPAr=cω02β Ar[(ExEx*+EzEz*)+μHyHy*]dxAr HyHy*dxn(x)2,
FAr=Ar HyHy*n(x)2dx-+ HyHy*n(x)2dx,
εWgd=εConv NgArnabsFAr.
Hy=A cos(φc)exp(-γcx)(0x+),
Hy=A cos(γfx+φc)(-dx0),
Hy=A cos(γfd-φc)exp[γs(x+d)]
(-x-d),
γck0(Neff2-nabs2)1/2,
γfk0(nf2-Neff2)1/2,
γsk0(Neff2-ns2)1/2
γfd=mπ+tan-1nf2γsns2γf+tan-1nf2γcnabs2γf,
m=0, 1, 2,
NgC=Neff.
εEvan=εConv(Neff/nabs)FC,
FC0+(HyHy*/nabs2)dx-+[HyHy*/n2(x)]dx=nabs2γf2γc(nabs4γf2+nf4γc2)×dnf2+nabs2(γc2+γf2)(nabs4γf2+nf4γc2)1γc+ns2(γs2+γf2)(ns4γf2+nf4γs2)1γs-1.
Hy=AG sinh ϕ exp[-γc(x-d2)](d2x+),
Hy=AG sinh[-γb(x-d2)+ϕ](0xd2),
Hy=A cos[-γf(x+d1)+φs](-d1x0),
Hy=A cos φs exp[γs(x+d1)](-x-d1),
γck0(Neff2-nabs2)1/2,
γfk0(nf2-Neff2)1/2,
γsk0(Neff2-ns2)1/2,
γbk0(N2eff-n2buf)1/2,
φstan-1nf2γsns2γf,
ϕtanh-1nabs2γbnbuf2γc,
Gcos(γfd1-φs)/sinh(γbd2+ϕ).
γfd1=mπ+φs+tan-1nf2γbnbuf2γfcoth(γbd2+ϕ),
m=0, 1, 2, 3.
εEvan=εConv(Neff/nabs)FC,
FCd2+(HyHy*/nabs2)dx-+[HyHy*/n2(x)]dx=G2 sinh2 ϕnabs2γcd1nf2+ns2(γf2+γs2)γs(ns4γf2+nf4γs2)-G2d2nbuf2+nabs2(γc2-γb2)(nbuf4γc2-nabs4γb2)γc+γf2+γb22nf2γfγb2sin 2(γfd1-φs)-1.
Hy=A cos φc exp[-γc(x-d2)](d2x+),
Hy=A cos[γb(x-d2)+φc](0xd2),
Hy=AG cos[γf(x+d1)-φs](-d1x0),
Hy=AG cos φs exp[γs(x+d1)](-x-d1),
γfd1=m1π+φs-tan-1nf2γbnbuf2γftan(γbd2-φc-m2π),
m1,2=0, 1, 2, 3.
FCd2+(HyHy*/nabs2)dx-+[HyHy*/n2(x)]dx=cos2 φcnabs2γcd2nbuf2+nabs2(γb2+γc2)(nabs4γb2+nbuf4γc2)γc+G2d1nf2+ns2(γf2+γs2)(ns4γf2+nf4γs2)γs-γb2-γf22nbuf2γbγf2sin 2(γbd2-φc)-1.
NSLg=Neff+(Neff2-nabs2)d2NeffT(d2),
T(d2)20d2 sinh2[-(γslx-d2)+ϕ]dx=sinh[2(γsld2+ϕ)]-sinh(2ϕ)2γsl-d2.
εEvan=εConvnabsNeff+(Neff2-nabs2)d2NeffT(d2)FSL,
FSL0d2(HyHy*/nabs2)dx-+[HyHy*dx/n2(x)]=1nabs2G2[sinh 2(γsld2+ϕ)-sinh 2ϕ]2γsl-G2d2×d1nf2+ns2(γf2+γs2)γs(ns4γf2+nf4γs2)-G2d2nabs2+nc2(γc2-γsl2)(nabs4γc2-nc4γsl2)γc+γf2+γsl22nf2γfγsl2sin 2(γfd1-φs)-1.
NSLg=Neff+(nabs2-Neff2)d2NeffT(d2),
T(d2)20d2 cos2[γsl(x-d2)+φc]dx=sin[2(γsld2-φc)]+sin 2φc2γsl+d2.
εEvan=εConvnabsNeff+(nabs2-Neff2)d2NeffT(d2)FSL,
FSL0d2(HyHy*/nabs2)dx-+[HyHy*/n2(x)]dx=1nabs2sin(γsld2)cos(γsld2-2φc)2γsl+d2×d2nabs2+nc2(γsl2+γc2)(nc4γsl2+nsbs4γc2)γc+G2d1nf2+ns2(γf2+γs2)(ns4γf2+nf4γs2)γs-γsl2-γf22nabs2γslγf2sin 2(γsld2-φc)-1.
NGg=Neff+nabs2-Neff2Neff1+1dnabs2ns2γsns4γf2+nabs4γs2+nabs2nc2γcnc4γf2+nabs4γc2-1.
εGuid=εConvnabsNeff+nabs2-Neff2Neff×1+1dnabs2ns2γsns4γf2+nabs4γs2+nabs2nc2γcnc4γf2+nabs4γc2-1FG,
FG-d0(HyHy*/nabs2)dx-+[HyHy*/n2(x)]dx=dnabs2+ns2γsns4γf2+nabs4γs2+nc2γcnc4γf2+nabs4γc2×dnabs2+nc2(γc2+γf2)(nc4γf2+nabs4γc2)1γc+ns2(γs2+γf2)(ns4γf2+nabs4γs2)1γs-1.

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