Abstract

Grating couplers on planar waveguides can be used as integrated-optical sensors responding to (1) changes in the refractive index of a liquid sample covering the waveguide (differential refractometer) and (2) the adsorption and desorption, respectively, of molecules out of a gaseous or liquid sample on the waveguide (gas or chemical sensor). A theory of the sensor sensitivities is developed; conditions for the waveguide parameters in order to obtain high sensitivities are derived. It is shown that effects (1) and (2) can be distinguished by measurements of the effective index changes of both the TE0 and the TM0 modes. In the analysis both nonporous and microporous waveguiding films are considered.

© 1989 Optical Society of America

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  1. W. Lukosz and K. Tiefenthaler, IEE Conf. Publ. 227, 152–155 (1983).
  2. K. Tiefenthaler and W. Lukosz, Opt. Lett. 9, 137–139 (1984).
    [CrossRef] [PubMed]
  3. K. Tiefenthaler and W. Lukosz, Proc. Soc. Photo-Opt. Instrum. Eng. 514, 215–218 (1984).
  4. K. Tiefenthaler and W. Lukosz, Thin Solid Films 126, 205–211 (1985).
    [CrossRef]
  5. K. Tiefenthaler and W. Lukosz, IEE Conf. Publ. 227, 108–111 (1983).
  6. W. Lukosz and K. Tiefenthaler, Opt. Lett. 8, 537–539 (1983).
    [CrossRef] [PubMed]
  7. W. Lukosz and K. Tiefenthaler, Sensors Actuators 15, 273–284 (1988); P. Nellen, K. Tiefenthaler, and W. Lukosz, Sensors Actuators 15, 285–295 (1988).
    [CrossRef]
  8. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, Berlin, 1982).
    [CrossRef]
  9. H. Kogelnik, in Integrated Optics, Vol. 7 of Topics in Applied Physics (Springer-Verlag, Berlin, 1979), pp. 13–81.
    [CrossRef]
  10. M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 10–86.
  11. H. Kogelnik and H. P. Weber, J. Opt. Soc. Am. 64, 174–185 (1974).
    [CrossRef]
  12. W. Stutius and W. Streifer, Appl. Opt. 16, 3218–3222 (1977).
    [CrossRef] [PubMed]
  13. M. Minakata, S. Saito, M. Shibata, and S. Miyazawa, J. Appl. Phys. 49, 4677–4682 (1978).
    [CrossRef]
  14. G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, and R. M. De la Rue, IEEE J. Quantum Electron. QE-14, 192–200 (1978).
  15. G. Stewart and P. J. R. Laybourn, IEEE J. Quantum Electron. QE-14, 930–934 (1978).
    [CrossRef]
  16. R. G. Walker, C. D. W. Wilkinson, and J. A. H. Wilkinson, Appl. Opt. 22, 1923–1928 (1983).
    [CrossRef]
  17. H. Kogelnik and V. Ramaswamy, Appl. Opt. 13, 1857–1862 (1974).
    [CrossRef] [PubMed]
  18. P. A. Cuypers, J. W. Corsel, M. P. Janssen, J. M. M. Kop, W. Th. Hermens, and H. C. Henker, J. Biol. Chem. 258, 2426–2431 (1983).
    [PubMed]

1988 (1)

W. Lukosz and K. Tiefenthaler, Sensors Actuators 15, 273–284 (1988); P. Nellen, K. Tiefenthaler, and W. Lukosz, Sensors Actuators 15, 285–295 (1988).
[CrossRef]

1985 (1)

K. Tiefenthaler and W. Lukosz, Thin Solid Films 126, 205–211 (1985).
[CrossRef]

1984 (2)

K. Tiefenthaler and W. Lukosz, Opt. Lett. 9, 137–139 (1984).
[CrossRef] [PubMed]

K. Tiefenthaler and W. Lukosz, Proc. Soc. Photo-Opt. Instrum. Eng. 514, 215–218 (1984).

1983 (5)

K. Tiefenthaler and W. Lukosz, IEE Conf. Publ. 227, 108–111 (1983).

W. Lukosz and K. Tiefenthaler, Opt. Lett. 8, 537–539 (1983).
[CrossRef] [PubMed]

W. Lukosz and K. Tiefenthaler, IEE Conf. Publ. 227, 152–155 (1983).

R. G. Walker, C. D. W. Wilkinson, and J. A. H. Wilkinson, Appl. Opt. 22, 1923–1928 (1983).
[CrossRef]

P. A. Cuypers, J. W. Corsel, M. P. Janssen, J. M. M. Kop, W. Th. Hermens, and H. C. Henker, J. Biol. Chem. 258, 2426–2431 (1983).
[PubMed]

1978 (3)

M. Minakata, S. Saito, M. Shibata, and S. Miyazawa, J. Appl. Phys. 49, 4677–4682 (1978).
[CrossRef]

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, and R. M. De la Rue, IEEE J. Quantum Electron. QE-14, 192–200 (1978).

G. Stewart and P. J. R. Laybourn, IEEE J. Quantum Electron. QE-14, 930–934 (1978).
[CrossRef]

1977 (1)

1974 (2)

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 10–86.

Corsel, J. W.

P. A. Cuypers, J. W. Corsel, M. P. Janssen, J. M. M. Kop, W. Th. Hermens, and H. C. Henker, J. Biol. Chem. 258, 2426–2431 (1983).
[PubMed]

Cuypers, P. A.

P. A. Cuypers, J. W. Corsel, M. P. Janssen, J. M. M. Kop, W. Th. Hermens, and H. C. Henker, J. Biol. Chem. 258, 2426–2431 (1983).
[PubMed]

De la Rue, R. M.

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, and R. M. De la Rue, IEEE J. Quantum Electron. QE-14, 192–200 (1978).

Henker, H. C.

P. A. Cuypers, J. W. Corsel, M. P. Janssen, J. M. M. Kop, W. Th. Hermens, and H. C. Henker, J. Biol. Chem. 258, 2426–2431 (1983).
[PubMed]

Hermens, W. Th.

P. A. Cuypers, J. W. Corsel, M. P. Janssen, J. M. M. Kop, W. Th. Hermens, and H. C. Henker, J. Biol. Chem. 258, 2426–2431 (1983).
[PubMed]

Hunsperger, G.

G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, Berlin, 1982).
[CrossRef]

Janssen, M. P.

P. A. Cuypers, J. W. Corsel, M. P. Janssen, J. M. M. Kop, W. Th. Hermens, and H. C. Henker, J. Biol. Chem. 258, 2426–2431 (1983).
[PubMed]

Kogelnik, H.

H. Kogelnik and V. Ramaswamy, Appl. Opt. 13, 1857–1862 (1974).
[CrossRef] [PubMed]

H. Kogelnik and H. P. Weber, J. Opt. Soc. Am. 64, 174–185 (1974).
[CrossRef]

H. Kogelnik, in Integrated Optics, Vol. 7 of Topics in Applied Physics (Springer-Verlag, Berlin, 1979), pp. 13–81.
[CrossRef]

Kop, J. M. M.

P. A. Cuypers, J. W. Corsel, M. P. Janssen, J. M. M. Kop, W. Th. Hermens, and H. C. Henker, J. Biol. Chem. 258, 2426–2431 (1983).
[PubMed]

Laybourn, P. J. R.

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, and R. M. De la Rue, IEEE J. Quantum Electron. QE-14, 192–200 (1978).

G. Stewart and P. J. R. Laybourn, IEEE J. Quantum Electron. QE-14, 930–934 (1978).
[CrossRef]

Lukosz, W.

W. Lukosz and K. Tiefenthaler, Sensors Actuators 15, 273–284 (1988); P. Nellen, K. Tiefenthaler, and W. Lukosz, Sensors Actuators 15, 285–295 (1988).
[CrossRef]

K. Tiefenthaler and W. Lukosz, Thin Solid Films 126, 205–211 (1985).
[CrossRef]

K. Tiefenthaler and W. Lukosz, Proc. Soc. Photo-Opt. Instrum. Eng. 514, 215–218 (1984).

K. Tiefenthaler and W. Lukosz, Opt. Lett. 9, 137–139 (1984).
[CrossRef] [PubMed]

W. Lukosz and K. Tiefenthaler, IEE Conf. Publ. 227, 152–155 (1983).

K. Tiefenthaler and W. Lukosz, IEE Conf. Publ. 227, 108–111 (1983).

W. Lukosz and K. Tiefenthaler, Opt. Lett. 8, 537–539 (1983).
[CrossRef] [PubMed]

Millar, C. A.

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, and R. M. De la Rue, IEEE J. Quantum Electron. QE-14, 192–200 (1978).

Minakata, M.

M. Minakata, S. Saito, M. Shibata, and S. Miyazawa, J. Appl. Phys. 49, 4677–4682 (1978).
[CrossRef]

Miyazawa, S.

M. Minakata, S. Saito, M. Shibata, and S. Miyazawa, J. Appl. Phys. 49, 4677–4682 (1978).
[CrossRef]

Ramaswamy, V.

Saito, S.

M. Minakata, S. Saito, M. Shibata, and S. Miyazawa, J. Appl. Phys. 49, 4677–4682 (1978).
[CrossRef]

Shibata, M.

M. Minakata, S. Saito, M. Shibata, and S. Miyazawa, J. Appl. Phys. 49, 4677–4682 (1978).
[CrossRef]

Stewart, G.

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, and R. M. De la Rue, IEEE J. Quantum Electron. QE-14, 192–200 (1978).

G. Stewart and P. J. R. Laybourn, IEEE J. Quantum Electron. QE-14, 930–934 (1978).
[CrossRef]

Streifer, W.

Stutius, W.

Tiefenthaler, K.

W. Lukosz and K. Tiefenthaler, Sensors Actuators 15, 273–284 (1988); P. Nellen, K. Tiefenthaler, and W. Lukosz, Sensors Actuators 15, 285–295 (1988).
[CrossRef]

K. Tiefenthaler and W. Lukosz, Thin Solid Films 126, 205–211 (1985).
[CrossRef]

K. Tiefenthaler and W. Lukosz, Proc. Soc. Photo-Opt. Instrum. Eng. 514, 215–218 (1984).

K. Tiefenthaler and W. Lukosz, Opt. Lett. 9, 137–139 (1984).
[CrossRef] [PubMed]

W. Lukosz and K. Tiefenthaler, IEE Conf. Publ. 227, 152–155 (1983).

K. Tiefenthaler and W. Lukosz, IEE Conf. Publ. 227, 108–111 (1983).

W. Lukosz and K. Tiefenthaler, Opt. Lett. 8, 537–539 (1983).
[CrossRef] [PubMed]

Walker, R. G.

Weber, H. P.

Wilkinson, C. D. W.

R. G. Walker, C. D. W. Wilkinson, and J. A. H. Wilkinson, Appl. Opt. 22, 1923–1928 (1983).
[CrossRef]

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, and R. M. De la Rue, IEEE J. Quantum Electron. QE-14, 192–200 (1978).

Wilkinson, J. A. H.

Appl. Opt. (3)

IEE Conf. Publ. (2)

W. Lukosz and K. Tiefenthaler, IEE Conf. Publ. 227, 152–155 (1983).

K. Tiefenthaler and W. Lukosz, IEE Conf. Publ. 227, 108–111 (1983).

IEEE J. Quantum Electron. (2)

G. Stewart, C. A. Millar, P. J. R. Laybourn, C. D. W. Wilkinson, and R. M. De la Rue, IEEE J. Quantum Electron. QE-14, 192–200 (1978).

G. Stewart and P. J. R. Laybourn, IEEE J. Quantum Electron. QE-14, 930–934 (1978).
[CrossRef]

J. Appl. Phys. (1)

M. Minakata, S. Saito, M. Shibata, and S. Miyazawa, J. Appl. Phys. 49, 4677–4682 (1978).
[CrossRef]

J. Biol. Chem. (1)

P. A. Cuypers, J. W. Corsel, M. P. Janssen, J. M. M. Kop, W. Th. Hermens, and H. C. Henker, J. Biol. Chem. 258, 2426–2431 (1983).
[PubMed]

J. Opt. Soc. Am. (1)

Opt. Lett. (2)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

K. Tiefenthaler and W. Lukosz, Proc. Soc. Photo-Opt. Instrum. Eng. 514, 215–218 (1984).

Sensors Actuators (1)

W. Lukosz and K. Tiefenthaler, Sensors Actuators 15, 273–284 (1988); P. Nellen, K. Tiefenthaler, and W. Lukosz, Sensors Actuators 15, 285–295 (1988).
[CrossRef]

Thin Solid Films (1)

K. Tiefenthaler and W. Lukosz, Thin Solid Films 126, 205–211 (1985).
[CrossRef]

Other (3)

G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, Berlin, 1982).
[CrossRef]

H. Kogelnik, in Integrated Optics, Vol. 7 of Topics in Applied Physics (Springer-Verlag, Berlin, 1979), pp. 13–81.
[CrossRef]

M. J. Adams, An Introduction to Optical Waveguides (Wiley, New York, 1981), pp. 10–86.

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Figures (9)

Fig. 1
Fig. 1

Schematic of input grating coupler sensor. P power of incident laser beam; α, angle of incidence; Lx, length of grating; S, substrate; F, waveguiding film; C, cover (gaseous or liquid sample), P′, power of incoupled guided mode monitored by either a detector D or by measuring the straylight. (a) Sensor responding to adsorption of molecules from a gaseous medium. (b) Sensor responding to adsorption of molecules from a liquid medium and/or changes in its refractive index nC. The grating is located either at the film F–cover C interface or at the film F–substrate S interface.

Fig. 2
Fig. 2

Measured normalized incoupling efficiency η ^ as a function of the variable N ¯N − (nair sin α + lλ/Λ),which describes the detuning from optimal incoupling. The resonance width is δ N ¯ ≈ 6 × 10−4. Embossed grating with 1/Λ = 1200 lines/mm on a SiO2–TiO2 waveguide with aqueous cover medium C.

Fig. 3
Fig. 3

Detailed waveguide cross section of the coupler region in case of adsorption of molecules to the waveguide surface. The adsorbate is modeled as a homogeneous adlayer F′ with refractive index nF and thickness dF: F, planar waveguiding film with refractive index nF and thickness dF; S, substrate with refractive index nS; C, cover with refractive index nC.

Fig. 4
Fig. 4

Sensor sensitivities ∂N/∂nC for refractive-index changes versus waveguide thickness dF in the case of an aqueous solution nC = 1.333 < nS. Wavelength λ = 632.8 nm. (a) SiO2–TiO2 waveguide (nF = 1.750) on glass substrate (nS = 1.471), (b) Si3N4 waveguide (nF = 2.01) on a thick SiO2 buffer layer (nS = 1.458) on a Si substrate.

Fig. 5
Fig. 5

Sensor sensitivities ∂N/∂nC as in Fig. 4 but for the case of a medium C with nC = 1.500 > nS.

Fig. 6
Fig. 6

Sensor sensitivities ∂N/∂ (dF) for adsorption or chemisorption of H2O molecules from the gas phase (nC = 1) versus waveguide thickness dF. The molecules are assumed to form a homogeneous adlayer of thickness dF and refractive index nF = 1.333 on the waveguide surface. Wavelength λ = 632.8 nm. (a) SiO2–TiO2 waveguide on glass substrate, (b) Si3N4 waveguide on SiO2.

Fig. 7
Fig. 7

Sensor sensitivities ∂N/∂ (dF) for adsorption or chemisorption of (bio)molecules (nF = 1.500) from an aqueous solution (nC = 1.333) on the waveguide surface versus waveguide thickness dF. Wavelength λ = 632.8 nm. (a) SiO2–TiO2 waveguide on glass substrate, (b) Si3N4 waveguide on a SiO2 buffer layer on a Si substrate.

Fig. 8
Fig. 8

Sensor sensitivities ∂N/∂nF for refractive-index changes of the waveguiding film versus waveguide thickness dF: nS = 1.47, nF = 1.75, nC = 1, λ = 632.8 nm.

Fig. 9
Fig. 9

Elements of the inverse sensitivity matrix Sx,y−1 versus waveguide thickness dF. It permits differentiation between refractive-index changes Δx = ΔnC of the liquid cover and adsorbate thickness changes Δy = ΔdF. SiO2–TiO2 waveguide; nS = 1.47, nF = 1.75, nF = 1.50, nC = 1.33, λ = 632.8 nm. dcutoff(TM0), cutoff thickness for the TM0 mode.

Equations (64)

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( k x k y ) = ( k x k y ) + ( κ 0 ) ,
± N = n sin α l + l ( λ / Λ )
η P / P
η ( N ¯ ) = η m η ^ ( N ¯ )
N ¯ N - n sin α - l ( λ / Λ )
L x δ k x 2 π ,
δ N ¯ = δ N = δ k x / k λ / L x .
Δ N = n [ sin ( α 1 + Δ α 1 ) - sin α 1 ] n cos α 1 Δ α 1 ,
Δ P / P = [ η ^ ( N ¯ ) ] - 1 ( d η ^ / d N ¯ ) Δ N .
Φ 3 2 k z , F d F + Φ F , S + Φ F , C = 2 π m ,
r F , J = ( k z , F / n F 2 ρ - k z , J / n J 2 ρ ) / ( k z , F / n F 2 ρ + k z , J / n J 2 ρ ) ,
Φ F , J = 2 arctan [ i ( 1 - r F , J ) / ( 1 + r F , J ) ] = - 2 arctan [ ( n F / n J ) 2 ρ k z , J / k z , F ] ,
Φ 4 2 k z , F d F + Φ F , S + Φ F , F , C = 2 π m .
r F , F , C = r F , F + r F , C exp ( 2 i k z , F d F ) 1 + r F , F r F , C exp ( 2 i k z , F d F ) ,
r F , F , C = exp ( i Φ F , F , C ) ,
Φ F , F , C = 2 arctan [ i ( 1 - r F , F , C ) / ( 1 + r F , F C ) ] = 2 arctan [ i 1 - r F , F 1 + r F , F 1 - r F , C exp ( 2 i k z , F d F ) 1 + r F , C exp ( 2 i k z , F d F ) ] .
Φ F , F , C = 2 arctan { ( n F n F ) 2 ρ k z , F k z , F tan [ k z , F d F + ( Φ F , C + i Φ ¯ F , C ) / 2 ] } .
Φ F , F , C = - 2 arctan { ( n F n F ) 2 ρ k z , F k z , F [ tanh ( k z , F d F + Φ ¯ F , C / 2 ) ] σ } ,
[ tanh ( Φ ¯ F , C / 2 ) ] σ = ( n F / n C ) 2 ρ k z , C / k z , F = ( n F / n C ) 2 ρ [ ( N 2 - n C 2 ) / ( N 2 - n F 2 ) ] 1 / 2 ,
Φ F , F , C = Φ F , F ,
r F , F , C r F , F + r F , C ( 1 + 2 i k z , F d F ) 1 + r F , F r F , C ( 1 + 2 i k z , F d F ) .
r F , F , C r F , F + r F , C 1 + r F , F r F , C × [ 1 + i 2 r F , C ( 1 - r F , F 2 ) ( 1 + r F , F r F , C ) ( r F , F + r F , C ) k z , F d F ) ] .
r F , F , C r F , C exp [ i 2 r F , C ( 1 - r F , F 2 ) ( 1 + r F , F r F , C ) ( r F , F + r F , C ) k z , F d F ] .
Φ F , F , C Φ F , C + 2 r F , C ( 1 - r F , F 2 ) ( 1 + r F , F r F , C ) ( r F , F + r F , C ) k z , F d F .
Φ F , F , C Φ F , C + 2 k z , F ( n F / n F ) 2 ρ k z , F 2 / n F 4 ρ - k z , C 2 / n C 2 ρ k z , F 2 / n F 4 ρ - k z , C 2 / n C 4 ρ d F .
Φ F , F , C Φ F , C + 2 k z , F n F 2 - n C 2 n F 2 - n C 2 × { ( N / n C ) 2 + ( N / n F ) 2 - 1 ( N / n C ) 2 + ( N / n F ) 2 - 1 } ρ d F ,
Φ 4 2 k z , F ( d F + Δ d F ) + Φ F , S + Φ F , C 2 π m ,
Δ d F = n F 2 - n C 2 n F 2 - n C 2 { ( N / n C ) 2 + ( N / n F ) 2 - 1 ( N / n C ) 2 + ( N / n F ) 2 - 1 } ρ d F .
P = P C + P S + P F
P J / P = [ ( n F 2 - N 2 ) / ( n F 2 - n J 2 ) ] ( Δ z F , J / d eff ) ,
d eff = d F + J = C , S Δ z F , J
Δ z F , J ( TE ) = ( λ / 2 π ) ( N 2 - n J 2 ) - 1 / 2
Δ z F , J ( TM ) = ( λ / 2 π ) [ N 2 - n J 2 ] - 1 / 2 [ ( N / n F ) 2 + ( N / n J ) 2 - 1 ] - 1
Δ z F , J = ( 1 / 2 ) ( Φ F , J / k z , F ) ,
P F / P = ( 1 / d eff ) { d F + J = C , S [ ( N 2 - n J 2 ) / ( n F 2 - n J 2 ) ] Δ z F , J } .
Δ N = ( N / n C ) Δ n C .
Φ [ x , N ( x ) ] = 2 π m ,
Φ / x + ( Φ / N ) ( N / x ) = 0 ,
N / x = - ( Φ / x ) / ( Φ / N ) .
Φ 3 / N = k ( Φ 1 / k z , F ) ( d k z , F / d k x ) = 2 k d eff ( - k x / k z , F ) ,
Φ 3 / n C = Φ F , C / n C = k z , F Δ z F , C [ 2 n C / ( n F 2 - n C 2 ) ] [ 2 ( N / n C ) 2 - 1 ] ρ ,
N / n C = ( n C / N ) [ ( n F 2 - N 2 ) / ( n F 2 - n C 2 ) ] × ( Δ z F , C / d eff ) [ 2 ( N / n C ) 2 - 1 ] ρ .
N / n C = ( n C / N ) ( P C / P ) [ 2 ( N / n C ) 2 - 1 ] ρ .
Φ 4 / d F | d F = 0 = Φ F , F , C / d F | d F = 0 ,
Φ 4 / N | d F = 0 = Φ 3 / N ,
N / d F = n F 2 - N 2 N d eff n F 2 - n C 2 n F 2 - n C 2 { ( N / n C ) 2 + ( N / n F ) 2 - 1 ( N / n C ) 2 + ( N / n F ) 2 - 1 } ρ ,
N / d F = ( n F 2 - N 2 ) / ( N d eff ) ,
n F = q n ( solid ) + ( 1 - q ) n ( pores ) ,
Δ n F = ( 1 - q ) Δ n C .
( d N / d n C ) eff = N / n C + ( 1 - q ) N / n F .
Δ n F = ( Δ d F / d F ) [ n ( sorbent ) - n C ] .
N / d F = d F - 1 [ n ( sorbent ) - n C ] ( N / n F ) .
Φ 3 / n F = ( 2 k 2 n F / k z , F ) { d F + [ 2 ( N / n F ) 2 - 1 ] ρ × J = C , S [ ( N 2 - n J 2 ) / ( n F 2 - n J 2 ) ] Δ z F , J } ,
N / n F = ( n F / N ) ( 1 / d eff ) { d F + [ 2 ( N / n F ) 2 - 1 ] ρ × J = C , S [ ( N 2 - n J 2 ) / ( n F 2 - n J 2 ) ] Δ z F , J } .
N / n F = ( n F / N ) { ( P F / P ) - J = C , S [ [ 2 ( N / n J ) 2 - 1 ] ρ - 1 ] ( n J / n F ) 2 ( P J / P ) } .
Δ N = ( N / n C ) Δ n C + ( N / d F ) Δ d F .
Δ N ( TE 0 ) = N x ( TE 0 ) Δ x + N y ( TE 0 ) Δ y , Δ N ( TM 0 ) = N x ( TM 0 ) Δ x + N y ( TM 0 ) Δ y ,
S x , y [ N x ( TE 0 ) N y ( TE 0 ) N x ( TM 0 ) N y ( TM 0 ) ]
Δ N = S x , y Δ x .
Δ x = S x , y - 1 Δ N ,
S x , y - 1 = ( 1 / det S x , y ) [ N y ( TM 0 ) - N y ( TE 0 ) - N x ( TM 0 ) N x ( TE 0 ) ]
det S x , y = N x ( TE 0 ) N y ( TM 0 ) - N y ( TE 0 ) N x ( TM 0 )
Δ x = ( 1 / det S x , y ) [ N y ( TM 0 ) Δ N ( TE 0 ) - N y ( TE 0 ) Δ N ( TM 0 ) ] , Δ y = ( 1 / det S x , y ) [ - N x ( TM 0 ) Δ N ( TE 0 ) + N x ( TE 0 ) Δ N ( TM 0 ) ] .
1 - q = Δ n F / Δ n C = N x ( TE 0 ) Δ N ( TM 0 ) - N x ( TM 0 ) Δ N ( TE 0 ) N y ( TM 0 ) Δ N ( TE 0 ) - N y ( TE 0 ) Δ N ( TM 0 ) | x = n C , y = n F .

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