Abstract

Propagation characteristics of optically coupled dielectric waveguides including active and passive waveguides are studied in terms of scalar and vector field finite element methods. First, characteristics of couplings for asymmetric passive waveguides, i.e., modal dispersions and field distributions, are obtained using a vector field finite element method. Second, a scalar field finite element method is applied to active coupled slab waveguides, and modal propagation characteristics, i.e., dispersion relationships, modal gains, phase characteristics, and electromagnetic field distributions, are investigated in detail. Characteristics depending on parameters such as gain, symmetrical conditions of waveguides, and numbers of waveguides are obtained. As a result, weak coupling phenomena due to unequal gains are observed and it is found that a modal gain depends on optical power confinement factors. In this paper only TM modes are treated, but our method can be used to obtain TE modes.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. J. Setterlind, L. Thylen, “Directional Coupler Switches with Optical Gain,” IEEE J. Quantum Electron. QE-22, 595–602 (1986).
    [CrossRef]
  2. K. Tsutsui, K. Sakuda, “Analysis of a Phase Mismatched Three Waveguide Coupler with an Optical Amplifier,” IEEE/OSA J. Lightwave Technol. LT-5, 1763–1767 (1987).
    [CrossRef]
  3. K. Hayakawa, K. Sakuda, “Comparison of Numerical Studies on Active Rectangular Waveguide Couplers by Vector, Scalar Finite Element, and Effective Refractive Index Methods,” 1989 Tech. Digest, 3, 46, Numer. Simul. & Anal.
  4. E. A. J. Marcatili, “Dielectric Rectangular Waveguides and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
  5. J. Salzman, R. Lang, A. Yariv, “Frequency Selectivity in Laterally coupled Semiconductor Laser Arrays,” Opt. Lett. 10, 387–389 (1985).
    [CrossRef] [PubMed]
  6. R. K. Watts, “Evanescent Field Coupling of Thin-Film Laser and Passive Waveguide,” J. Appl. Phys. 44, 5635–5636 (1973).
    [CrossRef]
  7. A. Hardy, W. Streifer, “Coupled Mode Theory of Parallel Waveguides,” IEEE/OSA J. Lightwave Technol. LT-3, 1135–1146 (1985).
    [CrossRef]
  8. A. Hardy, W. Streifer, “Coupled Modes of Multiwaveguide Systems and Phased Arrays,” IEEE/OSA J. Lightwave Technol. LT-4, 90–99 (1986).
    [CrossRef]
  9. W. Streifer, M. Osinski, A. Hardy, “Reformulation of the Coupled-Mode Theory of Multiwaveguide Systems,” IEEE/OSA J. Lightwave Technol. LT-5, 1–4 (1987).
    [CrossRef]
  10. S. R. Chinn, R. J. Spiers, “Modal Gain in Coupled-Stripe Lasers,” IEEE J. Quantum Electron. QE-20, 358–363 (1984).
    [CrossRef]
  11. A. Hardy, W. Streifer, “Analysis of Phased-Array Diode Lasers,” Opt. Lett. 10, 335–337 (1985).
    [CrossRef] [PubMed]
  12. K. Hayata, M. Koshiba, M. Suzuki, “Lateral Mode Analysis of Buried Heterostructure Diode Lasers by the Finite-Element Method,” IEEE J. Quantum Electron. QE-22, 781–788 (1986).
    [CrossRef]
  13. A. D. Berk, “Vibrational Principles for Electromagnetic Resonators and Waveguides,” IRE Trans. Antennas Propag. AP-4, 104–111 (1956).
    [CrossRef]
  14. A. Konrad, “Vector Variational Formulation of Electromagnetic Fields in Anisotropic Media,” IEEE Trans. Microwave Theory Tech. MTT-24, 553–559 (1976).
    [CrossRef]
  15. B. M. A. Rahman, J. B. Davies, “Penalty Function Improvement of Waveguide Solution by Finite Elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
    [CrossRef]

1987 (2)

K. Tsutsui, K. Sakuda, “Analysis of a Phase Mismatched Three Waveguide Coupler with an Optical Amplifier,” IEEE/OSA J. Lightwave Technol. LT-5, 1763–1767 (1987).
[CrossRef]

W. Streifer, M. Osinski, A. Hardy, “Reformulation of the Coupled-Mode Theory of Multiwaveguide Systems,” IEEE/OSA J. Lightwave Technol. LT-5, 1–4 (1987).
[CrossRef]

1986 (3)

C. J. Setterlind, L. Thylen, “Directional Coupler Switches with Optical Gain,” IEEE J. Quantum Electron. QE-22, 595–602 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Coupled Modes of Multiwaveguide Systems and Phased Arrays,” IEEE/OSA J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

K. Hayata, M. Koshiba, M. Suzuki, “Lateral Mode Analysis of Buried Heterostructure Diode Lasers by the Finite-Element Method,” IEEE J. Quantum Electron. QE-22, 781–788 (1986).
[CrossRef]

1985 (3)

1984 (2)

S. R. Chinn, R. J. Spiers, “Modal Gain in Coupled-Stripe Lasers,” IEEE J. Quantum Electron. QE-20, 358–363 (1984).
[CrossRef]

B. M. A. Rahman, J. B. Davies, “Penalty Function Improvement of Waveguide Solution by Finite Elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[CrossRef]

1976 (1)

A. Konrad, “Vector Variational Formulation of Electromagnetic Fields in Anisotropic Media,” IEEE Trans. Microwave Theory Tech. MTT-24, 553–559 (1976).
[CrossRef]

1973 (1)

R. K. Watts, “Evanescent Field Coupling of Thin-Film Laser and Passive Waveguide,” J. Appl. Phys. 44, 5635–5636 (1973).
[CrossRef]

1969 (1)

E. A. J. Marcatili, “Dielectric Rectangular Waveguides and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

1956 (1)

A. D. Berk, “Vibrational Principles for Electromagnetic Resonators and Waveguides,” IRE Trans. Antennas Propag. AP-4, 104–111 (1956).
[CrossRef]

Berk, A. D.

A. D. Berk, “Vibrational Principles for Electromagnetic Resonators and Waveguides,” IRE Trans. Antennas Propag. AP-4, 104–111 (1956).
[CrossRef]

Chinn, S. R.

S. R. Chinn, R. J. Spiers, “Modal Gain in Coupled-Stripe Lasers,” IEEE J. Quantum Electron. QE-20, 358–363 (1984).
[CrossRef]

Davies, J. B.

B. M. A. Rahman, J. B. Davies, “Penalty Function Improvement of Waveguide Solution by Finite Elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[CrossRef]

Hardy, A.

W. Streifer, M. Osinski, A. Hardy, “Reformulation of the Coupled-Mode Theory of Multiwaveguide Systems,” IEEE/OSA J. Lightwave Technol. LT-5, 1–4 (1987).
[CrossRef]

A. Hardy, W. Streifer, “Coupled Modes of Multiwaveguide Systems and Phased Arrays,” IEEE/OSA J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Coupled Mode Theory of Parallel Waveguides,” IEEE/OSA J. Lightwave Technol. LT-3, 1135–1146 (1985).
[CrossRef]

A. Hardy, W. Streifer, “Analysis of Phased-Array Diode Lasers,” Opt. Lett. 10, 335–337 (1985).
[CrossRef] [PubMed]

Hayakawa, K.

K. Hayakawa, K. Sakuda, “Comparison of Numerical Studies on Active Rectangular Waveguide Couplers by Vector, Scalar Finite Element, and Effective Refractive Index Methods,” 1989 Tech. Digest, 3, 46, Numer. Simul. & Anal.

Hayata, K.

K. Hayata, M. Koshiba, M. Suzuki, “Lateral Mode Analysis of Buried Heterostructure Diode Lasers by the Finite-Element Method,” IEEE J. Quantum Electron. QE-22, 781–788 (1986).
[CrossRef]

Konrad, A.

A. Konrad, “Vector Variational Formulation of Electromagnetic Fields in Anisotropic Media,” IEEE Trans. Microwave Theory Tech. MTT-24, 553–559 (1976).
[CrossRef]

Koshiba, M.

K. Hayata, M. Koshiba, M. Suzuki, “Lateral Mode Analysis of Buried Heterostructure Diode Lasers by the Finite-Element Method,” IEEE J. Quantum Electron. QE-22, 781–788 (1986).
[CrossRef]

Lang, R.

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric Rectangular Waveguides and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

Osinski, M.

W. Streifer, M. Osinski, A. Hardy, “Reformulation of the Coupled-Mode Theory of Multiwaveguide Systems,” IEEE/OSA J. Lightwave Technol. LT-5, 1–4 (1987).
[CrossRef]

Rahman, B. M. A.

B. M. A. Rahman, J. B. Davies, “Penalty Function Improvement of Waveguide Solution by Finite Elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[CrossRef]

Sakuda, K.

K. Tsutsui, K. Sakuda, “Analysis of a Phase Mismatched Three Waveguide Coupler with an Optical Amplifier,” IEEE/OSA J. Lightwave Technol. LT-5, 1763–1767 (1987).
[CrossRef]

K. Hayakawa, K. Sakuda, “Comparison of Numerical Studies on Active Rectangular Waveguide Couplers by Vector, Scalar Finite Element, and Effective Refractive Index Methods,” 1989 Tech. Digest, 3, 46, Numer. Simul. & Anal.

Salzman, J.

Setterlind, C. J.

C. J. Setterlind, L. Thylen, “Directional Coupler Switches with Optical Gain,” IEEE J. Quantum Electron. QE-22, 595–602 (1986).
[CrossRef]

Spiers, R. J.

S. R. Chinn, R. J. Spiers, “Modal Gain in Coupled-Stripe Lasers,” IEEE J. Quantum Electron. QE-20, 358–363 (1984).
[CrossRef]

Streifer, W.

W. Streifer, M. Osinski, A. Hardy, “Reformulation of the Coupled-Mode Theory of Multiwaveguide Systems,” IEEE/OSA J. Lightwave Technol. LT-5, 1–4 (1987).
[CrossRef]

A. Hardy, W. Streifer, “Coupled Modes of Multiwaveguide Systems and Phased Arrays,” IEEE/OSA J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

A. Hardy, W. Streifer, “Coupled Mode Theory of Parallel Waveguides,” IEEE/OSA J. Lightwave Technol. LT-3, 1135–1146 (1985).
[CrossRef]

A. Hardy, W. Streifer, “Analysis of Phased-Array Diode Lasers,” Opt. Lett. 10, 335–337 (1985).
[CrossRef] [PubMed]

Suzuki, M.

K. Hayata, M. Koshiba, M. Suzuki, “Lateral Mode Analysis of Buried Heterostructure Diode Lasers by the Finite-Element Method,” IEEE J. Quantum Electron. QE-22, 781–788 (1986).
[CrossRef]

Thylen, L.

C. J. Setterlind, L. Thylen, “Directional Coupler Switches with Optical Gain,” IEEE J. Quantum Electron. QE-22, 595–602 (1986).
[CrossRef]

Tsutsui, K.

K. Tsutsui, K. Sakuda, “Analysis of a Phase Mismatched Three Waveguide Coupler with an Optical Amplifier,” IEEE/OSA J. Lightwave Technol. LT-5, 1763–1767 (1987).
[CrossRef]

Watts, R. K.

R. K. Watts, “Evanescent Field Coupling of Thin-Film Laser and Passive Waveguide,” J. Appl. Phys. 44, 5635–5636 (1973).
[CrossRef]

Yariv, A.

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Dielectric Rectangular Waveguides and Directional Coupler for Integrated Optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).

IEEE J. Quantum Electron. (3)

C. J. Setterlind, L. Thylen, “Directional Coupler Switches with Optical Gain,” IEEE J. Quantum Electron. QE-22, 595–602 (1986).
[CrossRef]

S. R. Chinn, R. J. Spiers, “Modal Gain in Coupled-Stripe Lasers,” IEEE J. Quantum Electron. QE-20, 358–363 (1984).
[CrossRef]

K. Hayata, M. Koshiba, M. Suzuki, “Lateral Mode Analysis of Buried Heterostructure Diode Lasers by the Finite-Element Method,” IEEE J. Quantum Electron. QE-22, 781–788 (1986).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

A. Konrad, “Vector Variational Formulation of Electromagnetic Fields in Anisotropic Media,” IEEE Trans. Microwave Theory Tech. MTT-24, 553–559 (1976).
[CrossRef]

B. M. A. Rahman, J. B. Davies, “Penalty Function Improvement of Waveguide Solution by Finite Elements,” IEEE Trans. Microwave Theory Tech. MTT-32, 922–928 (1984).
[CrossRef]

IEEE/OSA J. Lightwave Technol. (4)

A. Hardy, W. Streifer, “Coupled Mode Theory of Parallel Waveguides,” IEEE/OSA J. Lightwave Technol. LT-3, 1135–1146 (1985).
[CrossRef]

A. Hardy, W. Streifer, “Coupled Modes of Multiwaveguide Systems and Phased Arrays,” IEEE/OSA J. Lightwave Technol. LT-4, 90–99 (1986).
[CrossRef]

W. Streifer, M. Osinski, A. Hardy, “Reformulation of the Coupled-Mode Theory of Multiwaveguide Systems,” IEEE/OSA J. Lightwave Technol. LT-5, 1–4 (1987).
[CrossRef]

K. Tsutsui, K. Sakuda, “Analysis of a Phase Mismatched Three Waveguide Coupler with an Optical Amplifier,” IEEE/OSA J. Lightwave Technol. LT-5, 1763–1767 (1987).
[CrossRef]

IRE Trans. Antennas Propag. (1)

A. D. Berk, “Vibrational Principles for Electromagnetic Resonators and Waveguides,” IRE Trans. Antennas Propag. AP-4, 104–111 (1956).
[CrossRef]

J. Appl. Phys. (1)

R. K. Watts, “Evanescent Field Coupling of Thin-Film Laser and Passive Waveguide,” J. Appl. Phys. 44, 5635–5636 (1973).
[CrossRef]

Opt. Lett. (2)

Other (1)

K. Hayakawa, K. Sakuda, “Comparison of Numerical Studies on Active Rectangular Waveguide Couplers by Vector, Scalar Finite Element, and Effective Refractive Index Methods,” 1989 Tech. Digest, 3, 46, Numer. Simul. & Anal.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (42)

Fig. 1
Fig. 1

Asymmetric two slab coupling waveguide model; n1 and n2 are refractive indices in waveguides and cladding, respectively.

Fig. 2
Fig. 2

Dispersion characteristics for asymmetric two slab coupling waveguides. The dotted lines indicate individual slab waveguide dispersion characteristics.

Fig. 3
Fig. 3

Magnetic field profiles for the lowest three TM supermodes of asymmetric two slab waveguides are shown in (a) and (b) at a and b in Fig. 2, respectively.

Fig. 4
Fig. 4

Enlarged dispersion curves in the vicinity of the arrow in Fig. 2. The coupling characteristics of the TM2nd and TM3rd modes are clearly observed.

Fig. 5
Fig. 5

Asymmetric three slab waveguide model; n1 and n2 are refractive indices in waveguides and cladding, respectively.

Fig. 6
Fig. 6

Dispersion characteristics for asymmetric three slab coupling waveguides. The dotted lines indicate individual slab waveguide dispersion characteristics.

Fig. 7
Fig. 7

Magnetic field profiles for the lowest four TM supermodes of asymmetric three slabs waveguides are shown in (a) and (b) at a and b in Fig. 6, respectively.

Fig. 8
Fig. 8

Enlarged dispersion curves in the vicinity of the arrow in Fig. 6. The coupling characteristics of the three supermodes, namely, TM2nd, TM3rd, and TM4th, are clearly observed.

Fig. 9
Fig. 9

Active two slab coupling waveguide model (a) geometrically symmetrical and (b) asymmetrical structures. Parameters of the imaginary parts of the refractive indices, namely, n 1 , n 2, and n 3 are listed in Table I; n1 and n2 are waveguide refractive indices and n3 is a cladding refractive index.

Fig. 10
Fig. 10

Modal gains of the lowest two supermodes for models A and B in Table I.

Fig. 11
Fig. 11

Magnetic field profiles of the lowest two TM supermodes (a) TM1st and (b) TM2nd for model A at k0 = 1.5 rad/μm. The dotted line in (a) indicates the imaginary part of the fields.

Fig. 12
Fig. 12

Magnetic field profiles of the lowest two TM supermodes (a) TM1st and (b) TM2nd for model A at k0 = 4.5 rad/μm. The dotted line indicates the imaginary part of the fields.

Fig. 13
Fig. 13

Dispersion characteristics of the lowest two supermodes for model C.

Fig. 14
Fig. 14

Modal gain curves of the lowest two supermodes for model C.

Fig. 15
Fig. 15

Phase characteristics of the lowest two supermodes for model C.

Fig. 16
Fig. 16

Magnetic field profiles of the lowest two TM supermodes (a) and (c) TM1st and (b) and (d) TM2nd for model C (a) and (b) at k0 = 1.5 rad/μm, and (c) and (d) at k0 = 4.5 rad/μm, respectively. The dotted line indicates the imaginary part of the fields.

Fig. 17
Fig. 17

Dispersion characteristics of the lowest two supermodes for model D.

Fig. 18
Fig. 18

Modal gain curves of the lowest two supermodes for model D.

Fig. 19
Fig. 19

Phase characteristics of the lowest two supermodes for model D.

Fig. 20
Fig. 20

Magnetic field profiles of the lowest two TM supermodes (a) and (c) TM1st and (b) and (d) TM2nd for model D (a) and (b) at k0 = 1.5 rad/μm, and (c) and (d) at k0 = 4.5 rad/μm, respectively. The dotted line indicates the imaginary part of the fields.

Fig. 21
Fig. 21

Modal gain curves of the lowest three TM supermodes for model F.

Fig. 22
Fig. 22

Enlarged modal gain curves in the vicinity of k0 = 3.35 rad/μm for model F.

Fig. 23
Fig. 23

Enlarged dispersion characteristics in the vicinity of k0 = 3.35 rad/μm for model F.

Fig. 24
Fig. 24

Magnetic field profiles of the lowest three TM supermodes (a) and (d) TM1st, (b) and (e) TM2nd, and (c) and (f) TM3rd for model F (a), (b), and (c) at k0 = 1.5 rad/μm, and (d), (e), and (f) at k0 = 4.5 rad/μm, respectively. The dotted line indicates the imaginary part of the fields.

Fig. 25
Fig. 25

Modal gain curves of the lowest three TM supermodes for model G.

Fig. 26
Fig. 26

Modal gain curves of the lowest three TM supermodes for model H.

Fig. 27
Fig. 27

Active three slab coupling waveguide model, geometrically symmetric type. Here n1, n2 and n3 are refractive indices of the waveguides and n4 is the refractive index of the cladding. The imaginary parts of the refractive indices, n 1 , n 2, and n 3 are listed in Table II.

Fig. 28
Fig. 28

Active three slab coupling waveguide model, geometrically asymmetric type. Here n1, n2, and n3 are refractive indices of the waveguides and n4 is the refractive index of the cladding. The imaginary parts of the refractive indices, n 1 , n 2, and n 3, are listed in Table II.

Fig. 29
Fig. 29

Modal gain curves of the lowest three TM supermodes for model I.

Fig. 30
Fig. 30

Magnetic field profiles of the lowest three TM supermodes (a) and (d) TM1st (b) and (e) TM2nd, and (c) and (f) TM3rd for model I (a), (b), and (c) at k0 = 1.5 rad/μm, and (d), (e), and (f) at k0 = 4.5 rad/μm, respectively. The dotted line in (a)–(c) and (f) indicates the imaginary parts of the fields.

Fig. 31
Fig. 31

Modal gain curves of the lowest three TM supermodes for model J.

Fig. 32
Fig. 32

Magnetic field profiles of the lowest three TM supermodes (a) and (d) TM1st, (b) and (e) TM2nd, and (c) and (f) TM3rd for model J (a), (b), and (c) at k0 = 1.5 rad/μm, and (d), (e), and (f) at k0 = 4.5 rad/μm, respectively. The dotted line indicates the imaginary parts of the fields.

Fig. 33
Fig. 33

Modal gain curves of the lowest four TM supermodes for model K.

Fig. 34
Fig. 34

Magnetic field profiles of the lowest four TM supermodes (a) and (e) TM1st, (b) and (f) TM2nd, (c) and (g) TM3rd, and (d) and (h) TM4th for model K (a), (b), (c), and (d) at k0 = 1.5 rad/μm, and (e), (f), (g) and (h) at k0 = 4.5 rad/μm, respectively. The dotted line indicates the imaginary parts of the fields.

Fig. 35
Fig. 35

Enlarged dispersion characteristics in the vicinity of k0 = 3.38 rad/μm for three supermodes, namely, TM2nd, TM3rd, and TM4th.

Fig. 36
Fig. 36

Enlarged modal gain curves in the vicinity of k0 = 3.38 rad/μm for the same supermodes as in Fig. 35.

Fig. 37
Fig. 37

Modal gain curves of the lowest three TM supermodes for model L.

Fig. 38
Fig. 38

Magnetic field profiles of the lowest four TM supermodes (a) and (e) TM1st, (b) and (f) TM2nd, (c) and (g) TM3rd, and (d) and (h) TM4th for model L (a), (b), (c), and (d) at k0 = 1.5 rad/μm, and (e), (f), (g), and (h) at k0 = 4.5 rad/μm, respectively. The dotted line indicates the imaginary parts of the fields.

Fig. 39
Fig. 39

Comparison between modal gains obtained by the confinement factors (dotted lines) and modal gains obtained by the finite element method (solid lines) for model F.

Fig. 40
Fig. 40

Enlarged dispersion characteristics in the vicinity of the coupling starting point for model C compared with passive dispersion characteristics (dotted lines).

Fig. 41
Fig. 41

Enlarged dispersion characteristics in the vicinity of the coupling starting point for model D compared with passive dispersion characteristics (dotted lines).

Fig. 42
Fig. 42

Coordinate systems for the finite element analysis.

Tables (2)

Tables Icon

Table I Parameters for Imaginary Parts of Refractive Indices of Active Two Waveguide Couplers

Tables Icon

Table II Parameters for Imaginary Parts of Refractive Indices of Active Three Waveguide Couplers

Equations (23)

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

Γ = [ core P ( x ) d x ] / [ P ( x ) d x ] .
P ( x ) H x ( x ) · H x * ( x ) = | H x ( x ) | 2 .
Γ 1 = ( G 1 | H x | 2 d x ) / ( + | H x | 2 d x ) , Γ 2 = ( G 2 | H x | 2 d x ) / ( + | H x | 2 d x ) .
G net = i Γ i g i ,
× E = j ω μ 0 H ,
× H = j ω ε 0 κ · E ,
× ( κ 1 · × H ) k 0 2 H = 0 ,
F = Ω [ ( × H * ) · ( κ 1 · × H ) k 0 2 H * · H ] d Ω ,
F = Ω [ ( × H * ) ( κ 1 · × H ) + P ( · H * ) ( · H ) k 0 2 H * · H ] d Ω ,
F = i F i = i Ω i [ × H * ] · [ κ 1 · × H ] + P ( · H * ) ( · H * ) k 0 2 H * · H ] d Ω .
H x i = N i T · h x i exp ( j β z ) , H y i = N i T · h y i exp ( j β z ) , H z i = j N i T · h z i · exp ( j β z ) ,
F i = { H } i T · { [ A ] i + P [ C ] i k 0 2 [ B ] i } · { H } i ,
F = { H } T · { [ A ] + P [ C ] k 0 2 [ B ] } · { H } .
δ F = F / { H } T = { [ A ] + P [ C ] k 0 2 [ B ] } · { H } = 0 .
{ [ A ] + P [ C ] } · { H } = k 0 2 [ B ] · { H } ,
[ A ] = | A x x , A x y , A x z A x y T , A y y , A y z A x z T A y z T , A z z | , [ B ] = | B x x , 0 , 0 , 0 , B y y , 0 0 , 0 B z z | , [ C ] = | C x x , C x y , C x z C x y T , C y y , C y z C x z T , C y z T , C z z | , A x x = i Ω i [ ( 1 / n z i 2 ) ( N i / y ) ( N i T / y ) + ( β 2 / n y i 2 ) ( N i N i T ) ] d Ω , A x y = i Ω i [ ( 1 / n z i 2 ) ( N i / y ) ( N i T / x ) ] d Ω , A x z = i Ω i [ ( β / n y i 2 ) N i ( N i T / x ) ] d Ω , A y y = i Ω i [ ( 1 / n z i 2 ) ( N i / x ) ( N i T / x ) + ( β 2 / n x i 2 ) ( N i N i T ) ] d Ω , A y z = i Ω i [ ( β / n x i 2 ) N i ( N i T / y ) ] d Ω , A z z = i Ω i [ ( 1 / n y i 2 ) ( N i / x ) ( N i T / x ) + ( 1 / n x i 2 ) ( N i / y ) ( N i T / y ) ] d Ω , B x x = B y y = B z z = i Ω i [ N i N i T ] d Ω , C x x = i Ω i [ ( N i / x ) ( N i T / x ) ] d Ω , C x y = i Ω i [ ( N i / x ) ( N i T / y ) ] d Ω , C x z = i Ω i [ β ( N i / x ) N i T ] d Ω , C y y = i Ω i [ ( N i / y ) ( N i T / y ) ] d Ω , C y z = i Ω i [ β ( N i / y ) N i T ] d Ω , C z z = i Ω i [ β 2 N i N i T ] d Ω ,
TE : 2 E x / y 2 + ( k 0 2 n 2 + γ 2 ) E x = 0 ,
TM : 2 H x / y 2 + ( k 0 2 n 2 + γ 2 ) H x = 0 ,
TE : 2 E x / x 2 + 2 E x / y 2 + γ 2 E x + k 0 2 n 2 E x = 0 ,
TM : ( 1 / n ) 2 { H x / x 2 + 2 H x / y 2 + γ 2 H x + k 0 2 n 2 H x } = 0 .
[ A ] · { Ψ } = γ 2 [ B ] · { Ψ } ,
[ A ] = i Ω i [ ( N i / x ) ( N i T / x ) + ( N i / y ) ( N i T / y ) n i 2 k 0 2 N i N i T ] d Ω , [ B ] = i Ω i [ N i N i T ] d Ω , Ψ = E x = | E x | × exp [ j θ ] , θ = tan 1 [ ( Im E x ) / ( Re E x ) ] ,
[ A ] = i Ω i [ ( 1 / n i 2 ) { ( N i / x ) ( N i T / x ) + ( N i / y ) ( N i T / y ) n i 2 k 0 2 N i N i T } ] d Ω , [ B ] = i Ω i [ ( 1 / n i 2 ) ( N i N i T ) ] d Ω , Ψ = H x = | H x | × exp [ j ϕ ] , ϕ = tan 1 [ ( Im H x ) / ( Re H x ) ] .

Metrics