Abstract

Two formal methods for finding laser modes and threshold conditions in laser resonators containing as many as N mirrors are presented. The first method is based on an analysis determining the reflectivity and the transmittivity of an N-mirror system with gain. This is an extension of the classical 2 × 2 matrix method. The second method is based on self-consistency equations for the system and directly yields the circulating fields of the individual resonators. A set of rules has been proved to allow these fields to be calculated directly by means of inspection. The laser oscillation condition for an N-mirror system is found. Examples are given for systems with as many as five mirrors.

© 1996 Optical Society of America

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References

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  1. D. A. Kleinman and P. P. Kisliuk, “Discrimination against unwanted orders in the Fabry–Perot resonator,” Bell. Sys. Tech. J. 41, 453–462 (1964).
    [CrossRef]
  2. P. W. Smith, “Stabilized, single frequency output from a long laser cavity,” IEEE J. Quantum. Electron. QE-1, 343–348 (1965).
    [CrossRef]
  3. C. W. Wieman and L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1–20 (1991).
    [CrossRef]
  4. R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 μm distributed feedback lasers,” J. Lightwave Technol. LT-4, 1655–1661 (1986).
    [CrossRef]
  5. M. Munroe, S. E. Hodges, J. Cooper, and M. G. Raymer, “Total intensity modulation and mode hopping in a coupled-cavity laser as a result of external-cavity length variations,” Opt. Lett. 19, 105–107 (1994).
    [CrossRef]
  6. E. P. Ippen, H. A. Haus, and L. Y. Liu, “Additive pulse mode locking,” J. Opt. Soc. Am. B 6, 1736–1745 (1989).
    [CrossRef]
  7. C. Pedersen, P. L. Hansen, T. Skettrup, and P. Buchhave, “Diode-pumped single-frequency Nd:YVO4 laser with a set of coupled resonators,” Opt. Lett. 20, 1389–1392 (1995).
    [CrossRef] [PubMed]
  8. M. Born and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1975), Chap. 7.6.

1995 (1)

1994 (1)

1991 (1)

C. W. Wieman and L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1–20 (1991).
[CrossRef]

1989 (1)

1986 (1)

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 μm distributed feedback lasers,” J. Lightwave Technol. LT-4, 1655–1661 (1986).
[CrossRef]

1965 (1)

P. W. Smith, “Stabilized, single frequency output from a long laser cavity,” IEEE J. Quantum. Electron. QE-1, 343–348 (1965).
[CrossRef]

1964 (1)

D. A. Kleinman and P. P. Kisliuk, “Discrimination against unwanted orders in the Fabry–Perot resonator,” Bell. Sys. Tech. J. 41, 453–462 (1964).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1975), Chap. 7.6.

Buchhave, P.

Chraplyvy, A. R.

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 μm distributed feedback lasers,” J. Lightwave Technol. LT-4, 1655–1661 (1986).
[CrossRef]

Cooper, J.

Hansen, P. L.

Haus, H. A.

Hodges, S. E.

Hollberg, L.

C. W. Wieman and L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1–20 (1991).
[CrossRef]

Ippen, E. P.

Kisliuk, P. P.

D. A. Kleinman and P. P. Kisliuk, “Discrimination against unwanted orders in the Fabry–Perot resonator,” Bell. Sys. Tech. J. 41, 453–462 (1964).
[CrossRef]

Kleinman, D. A.

D. A. Kleinman and P. P. Kisliuk, “Discrimination against unwanted orders in the Fabry–Perot resonator,” Bell. Sys. Tech. J. 41, 453–462 (1964).
[CrossRef]

Liu, L. Y.

Munroe, M.

Pedersen, C.

Raymer, M. G.

Skettrup, T.

Smith, P. W.

P. W. Smith, “Stabilized, single frequency output from a long laser cavity,” IEEE J. Quantum. Electron. QE-1, 343–348 (1965).
[CrossRef]

Tkach, R. W.

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 μm distributed feedback lasers,” J. Lightwave Technol. LT-4, 1655–1661 (1986).
[CrossRef]

Wieman, C. W.

C. W. Wieman and L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1–20 (1991).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1975), Chap. 7.6.

Bell. Sys. Tech. J. (1)

D. A. Kleinman and P. P. Kisliuk, “Discrimination against unwanted orders in the Fabry–Perot resonator,” Bell. Sys. Tech. J. 41, 453–462 (1964).
[CrossRef]

IEEE J. Quantum. Electron. (1)

P. W. Smith, “Stabilized, single frequency output from a long laser cavity,” IEEE J. Quantum. Electron. QE-1, 343–348 (1965).
[CrossRef]

J. Lightwave Technol. (1)

R. W. Tkach and A. R. Chraplyvy, “Regimes of feedback effects in 1.5 μm distributed feedback lasers,” J. Lightwave Technol. LT-4, 1655–1661 (1986).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Lett. (2)

Rev. Sci. Instrum. (1)

C. W. Wieman and L. Hollberg, “Using diode lasers for atomic physics,” Rev. Sci. Instrum. 62, 1–20 (1991).
[CrossRef]

Other (1)

M. Born and E. Wolf, Principles of Optics, (Pergamon, Oxford, 1975), Chap. 7.6.

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Figures (7)

Fig. 1
Fig. 1

N-mirror resonator consisting of an array of dielectric mirrors. A stationary state is assumed; the right-propagating fields (marked i) and left-propagating fields (marked r) are shown. The fields at the right-hand side of a mirror are marked with a prime.

Fig. 2
Fig. 2

N-mirror resonator. Φi,p is defined as the phase shift experienced by one complete round trip of the cavity formed by mirror ri and rp (including mirror phase shifts).

Fig. 3
Fig. 3

Example with N = 4, illustrating how the fields from different parts of the system contribute to the circulating fields of the individual mirror pairs.

Fig. 4
Fig. 4

Longitudinal modes of an N-mirror resonator with an equal mode spacing of 15 GHz for the individual subresonators. The end mirrors have 90% reflectance, while the internal mirrors have 50% reflectance. (a) 2 mirrors, (b) 3 mirrors, (c) 4 mirrors, and (d) 5 mirrors.

Fig. 5
Fig. 5

Three-mirror example as described in Section 7.

Fig. 6
Fig. 6

The kth mirror of an array of mirrors in an N-mirror resonator. A stationary state is assumed, and the right-propagating fields (marked i) and left-propagating fields (marked r) are shown. The fields at the right-hand side of the mirror are marked with a prime.

Fig. 7
Fig. 7

The mirror is considered isolated from the rest of the system. (a) The incoming field is Eik. Hence E r k = 0 . (b) Fields with time reversal. The fields must retrace their paths, and the reflectivity and the transmittivity are complex conjugated.

Equations (101)

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[ E i k E r k ] = [ exp ( ½ G k ) exp ( ½ i δ k ) 0 0 exp ( ½ G k ) exp ( ½ i δ k ) ] × [ E i k + 1 E r k + 1 ] ,
G k = g k τ k , δ k = 2 q k d k = 4 π λ n k d k ;
[ E i k E r k ] = M ¯ ¯ k [ E i k + 1 E r k + 1 ] ,
M ¯ ¯ k = [ α k β k 1 r ˜ k * α k * β k r ˜ k α k β k 1 α k * β k ] ,
α k = 1 t ˜ k exp ( i δ k 2 ) , β k = exp ( G k 2 ) .
[ E i ( 1 ) E r ( 1 ) ] = [ A B C D ] [ 1 t ˜ N r ˜ N * t ˜ N * r ˜ N t ˜ N 1 t ˜ N * [ ] [ E i N E r N ] ,
[ A B C D ] = M ¯ ¯ 1 M ¯ ¯ 2 M ¯ ¯ 3 M N 1 ¯ ¯
r = E r ( 1 ) E i ( 1 ) = Q 1 , N P 1 , N , where Q 1 , N = C + D r ˜ N , P 1 , N = A + B r ˜ N ,
t = E i N E i ( 1 ) = t ˜ N P 1 , N .
P 1 , N = A + B r ˜ N = 0 .
P 1 , N + 1 = α N β N [ P 1 , N r N r N + 1 exp ( G N + i Φ N ) × P 1 , N , r N 1 r N ] ,
Q 1 , N + 1 = α N β N [ Q 1 , N r N r N + 1 exp ( G N + i Φ N ) × Q 1 , N , r N 1 r N ] ,
Θ i , p + Θ p , j = Θ i , j , Φ i , p + Φ p , j = Φ i , j + π .
E C ( N 1 ) = t N 1 exp ( ½ G N 2 ) exp ( i Θ N 2 , N 1 ) E C ( N 2 ) + r N 1 r N exp ( G N 1 ) exp ( i Φ N 1 , N ) E C ( N 1 ) , E C ( i ) = t i exp ( ½ G i 1 ) exp ( i Θ i 1 , i ) E C ( i 1 ) + r i 1 r i exp ( G i 1 ) exp ( i Φ i 1 , i ) E C ( i ) + + r i t i + 1 t p r p + 1 × exp ( G i , p + 1 ½ G i , p ) exp [ i ( Φ i , p + 1 Θ i , p ) ] × E C ( p ) + + r i t i + 1 t N 1 r N × exp ( G i , N ½ G i , N 1 ) × exp [ i ( Φ i , N Θ i , N 1 ) ] E C ( N 1 ) , E C ( 1 ) = t 1 exp ( i ϕ 1 t ) E i + r 1 r 2 exp ( G 1 ) × exp ( i Φ 1 , 2 ) E C ( 1 ) + + r 1 t 2 t N 1 r N × exp ( G 1 , N ½ G 1 , N 1 ) × exp [ i ( Φ 1 , N Θ 1 , N 1 ) ] E C ( N 1 ) ,
M 1 , N E C = [ t 1 exp ( i ϕ 1 t ) E i 0 0 ] ,
M 1 , N = [ m 1 , 1 m 1 , N m 2 , 1 0 0 0 m i , i 1 m i , i m i , p 0 m N , N ] .
E C ( i ) = N i Det ( M 1 , N ) E i ( 1 ) ,
N i = t 1 t i exp ( ½ G 1 , i ) exp [ i ( ϕ 1 t + Θ 1 , i ) ] Det ( M i + 1 , N )
t = t 1 t N exp [ ½ G 1 , N + i ( ϕ 1 t + Θ 1 , N ) ] Det ( M 1 , N ) .
P 1 , N = A + B r ˜ N = Det ( M 1 , N ) t 1 t N 1 exp [ ½ G 1 , N + i ( ϕ 1 t + Θ 1 , N 1 + δ N ) ] ,
Det ( M 1 , N + 1 ) = Det ( M 1 , N ) r N r N + 1 exp ( G N + i Φ N ) × Det ( M 1 , N , r N 1 / r N ) ,
E C ( N ) = t E i + r r ˜ N + 1 exp ( G N + i δ N ) E C ( N ) .
Det ( M 1 , N + 1 ) = Det ( M 1 , N ) [ 1 r r ˜ N + 1 exp ( G N + i δ N ) ] .
Det ( M 1 , N + 1 ) = Det ( M 1 , N ) [ 1 r ˜ N r ˜ N + 1 exp ( G N + i δ N ) × Det ( M 1 , N , r N 1 / r N ) Det ( M 1 , N ) ] .
r = exp ( i ϕ N ) r N Det ( M 1 , N , r N 1 / r N ) Det ( M 1 , N ) .
r = exp ( i ϕ 1 r ) r 1 Det ( M 1 , N , r 1 1 / r 1 ) Det ( M 1 , N ) .
Q 1 , N = C + D r ˜ N = exp ( i ϕ 1 r ) × r 1 Det ( M 1 , N , r 1 1 / r 1 ) t 1 t N 1 exp [ ½ G 1 , N + i ( ϕ 1 t + Θ 1 , N 1 + δ N ) ] .
Det ( M 1 , N ) = 1 + p = 1 q A p exp ( B p ) , where q = n = 1 N 1 [ N 1 n ] = 2 N 1 1 , with A p = r α r β r ψ r ω , B p = s ( G s + i Φ s ) .
α 1 = r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) r 2 r 3 exp ( G 2 ) exp ( i Φ 2 ) r 3 r 4 exp ( G 3 ) exp ( i Φ 3 ) r 4 r 5 exp ( G 4 ) exp ( i Φ 4 ) ;
α 2 = r 1 r 3 exp ( G 1 + G 2 ) exp [ i ( Φ 1 + Φ 2 ) ] + r 1 r 2 r 3 r 4 exp ( G 1 + G 3 ) exp [ i ( Φ 1 + Φ 3 ) ] + r 1 r 2 r 4 r 5 exp ( G 1 + G 4 ) exp [ i ( Φ 1 + Φ 4 ) ] + r 2 r 4 exp ( G 2 + G 3 ) exp [ i ( Φ 2 + Φ 3 ) ] + r 2 r 3 r 4 r 5 exp ( G 2 + G 4 ) exp [ i ( Φ 2 + Φ 4 ) ] + r 3 r 5 exp ( G 3 + G 4 ) exp [ i ( Φ 3 + Φ 4 ) ] ;
α 3 = r 1 r 4 exp ( G 1 + G 2 + G 3 ) exp [ i ( Φ 1 + Φ 2 + Φ 3 ) ] r 1 r 3 r 4 r 5 exp ( G 1 + G 2 + G 4 ) exp [ i ( Φ 1 + Φ 2 + Φ 4 ) ] r 1 r 2 r 3 r 5 exp ( G 1 + G 3 + G 4 ) exp [ i ( Φ 1 + Φ 3 + Φ 4 ) ] r 2 r 5 exp ( G 2 + G 3 + G 4 ) exp [ i ( Φ 2 + Φ 3 + Φ 4 ) ] ;
α 4 = r 1 r 5 exp ( G 1 + G 2 + G 3 + G 4 ) × exp [ i ( Φ 1 + Φ 2 + Φ 3 + Φ 4 ) ] .
Det ( M 1 , 5 ) = 1 + α 1 + α 2 + α 3 + α 4 .
Det ( M 1 , N ) = 1 + p A p ( G ) exp [ B p ( v ) ] = 0 ,
r ˜ L = r L ( v ) exp [ i ϕ L ( v ) ] , r ˜ R = r R ( v ) exp [ i ϕ R ( v ) ] ,
exp ( G m ) = 1 r L ( v ) r R ( v ) ,
2 π ( v / Δ v m ) + ϕ L ( v ) + ϕ R ( v ) = 2 p π ( p integer ) ,
Φ 1 = 2 p π ( p integer ) , exp ( G 1 ) = 1 / r 1 r 2 .
1 ( a + b ) ( cos Φ ) + c ( cos 2 Φ ) = 0 ,
( a + b ) ( sin Φ ) + c ( sin 2 Φ ) = 0 ,
a = r 1 r 2 exp ( G ) , b = r 2 r 3 , c = r 1 r 3 exp ( G ) .
( A ) sin Φ = 0 Φ = p π ( p integer ) , ( B ) cos Φ = r 2 2 1 + r 3 2 r 3 ,
( A ) exp ( G ) = 1 r 2 r 3 r 1 ( r 2 r 3 ) for r 2 > r 3 p even , ( B ) exp ( G ) = 1 / r 1 r 3 .
1 A ( cos Φ ) + B ( cos 2 Φ ) C ( cos 3 Φ ) = 0 , A ( sin Φ ) + B ( sin 2 Φ ) C ( sin 3 Φ ) = 0 ,
A = r 1 r 2 exp ( G ) + r 2 r 3 + r 3 r 4 , B = r 1 r 3 exp ( G ) + r 1 r 2 r 3 r 4 exp ( G ) + r 2 r 4 , C = r 1 r 4 exp ( G ) .
( A ) sin Φ = 0 Φ = p π ( p integer ) , ( B ) cos Φ = r 1 r 3 ( 1 + r 2 r 4 ) exp ( G ) + r 2 r 4 1 2 r 1 r 4 exp ( G ) ,
( A ) exp ( G ) = 1 r 2 r 3 r 3 r 4 + r 2 r 4 r 1 ( r 2 r 3 + r 4 r 2 r 3 r 4 ) for r 3 < r 2 + r 4 1 + r 2 r 4 p even , ( B ) exp ( G ) = r 1 2 r 3 2 ( 1 r 4 2 ) 2 + 4 ( 1 r 2 r 4 ) r 1 2 r 4 ( r 4 r 2 ) ] 1 / 2 r 1 r 3 ( 1 r 4 2 ) 2 r 1 2 r 4 ( r 4 r 2 ) .
exp ( G ) = 1 r 2 r 3 r 3 r 4 r 4 r 5 + r 2 r 4 r 2 r 5 + r 3 r 5 + r 2 r 3 r 4 r 5 r 1 ( r 2 r 3 + r 4 r 5 r 2 r 3 r 4 + r 2 r 3 r 5 r 2 r 4 r 5 + r 3 r 4 r 5 ) .
σ ( Δ v / π ) ( 1 α i 2 ) 1 / 2 ,
σ ( Δ v / π ) α i .
E k ( x k ) = E i k exp ( i q k 1 x k ) + E r k exp ( i q k 1 x k ) ,
q k = ( 2 π / λ ) n k .
E k ( x k + ) = E i k exp ( i q k x k ) + E r k exp ( i q k x k ) .
E r k / E i k = r ˜ k = r k exp ( i ϕ k r ) , E i k / E i k = t ˜ k = t k exp ( i ϕ k t ) .
E i k E r k = r ˜ k = r k exp ( i ϕ k r ) , E r k E r k = t ˜ k = t k exp ( i ϕ k t ) .
R k = r k 2 = r k 2 , T k = n k n k 1 t k 2 = n k 1 n k t k 2 , R k + T k = 1 .
E i k = t ˜ k E i k + r ˜ k E r k , E r k = t ˜ k E r k + r ˜ k E i k ,
[ E i k E r k ] = 1 t ˜ k [ 1 r ˜ k r ˜ k ( t ˜ k t ˜ k r ˜ k r ˜ k ) ] [ E i k E r k ] .
D k = t ˜ k / t ˜ k ,
E i k = t ˜ k E i k , E r k = r ˜ k E i k , E r k = 0 .
E r k = t ˜ k * E r k + r ˜ k * E i k , E i k = t ˜ k * E i k + r ˜ k * E r k ,
t ˜ k r ˜ k * + t ˜ k * r ˜ k = 0 , t ˜ k * t ˜ k + r ˜ k * r ˜ k = 1 .
r ˜ k = r ˜ k * ( t ˜ k / t ˜ k * ) , t ˜ k = 1 r ˜ k * r ˜ k t ˜ k * = T k t ˜ k * t ˜ k t ˜ k = n k n k 1 t ˜ k ,
r k = r k , ϕ k r = 2 ϕ k t ϕ k r + π , t k = n k n k 1 t k , ϕ k t = ϕ k t .
D k = n k / n k 1 = 1 for n k = n k 1 ,
M ¯ ¯ k = [ 1 t ˜ k r ˜ k * t ˜ k * r ˜ k t ˜ k 1 t ˜ k * ] .
[ A B C D ] = ( M ¯ ¯ 1 M p 1 ¯ ¯ ) ( M ¯ ¯ p M N 1 ¯ ¯ ) = [ A p B p C p D p ] × [ A N B N C N D N ] .
P 1 , N = A + B r ˜ N = A p A N + B p C N + A p B N r ˜ N + B p D N r ˜ N = ( A p + B p C N + D N r ˜ N A N + B N r ˜ N ) ( A N + B N r ˜ N ) = P 1 , p ( r ˜ p r ˜ p N ) P p , N ,
r ˜ p N = C N + D N r ˜ N A N + B N r ˜ N .
Q 1 , N = C + D r ˜ N = Q 1 , p ( r ˜ p r ˜ p N ) P p , N .
r ˜ 1 N = Q 1 , N P 1 , N = r ˜ 1 p ( r ˜ p r ˜ p N ) ;
A N = α N β N 1 , B N = r ˜ N * α N * β N , C N = r ˜ N α N β N 1 , D N = α N * β N
r N , N 1 = exp ( i ϕ N r ) r N r N + 1 exp ( G N + i Φ N ) 1 r N r N + 1 exp ( G N + i Φ N )
P 1 , N + 1 = α N β N [ P 1 , N r N r N + 1 exp ( G N + i Φ N ) × P 1 , N ( r N 1 r N ) ] ,
Q 1 , N + 1 = α N β N [ Q 1 , N r N r N + 1 exp ( G N + i Φ N ) × Q 1 , N ( r N 1 r N ) ] .
M ¯ ¯ = α 1 β 1 1 [ 1 γ 1 r ˜ 1 * r ˜ 1 γ 1 ] ,
γ 1 = α 1 * α 1 β 1 2 = r ˜ 1 r ˜ 1 * exp ( G 1 + i θ 1 ) .
r = r ˜ 1 + γ 1 r ˜ 2 1 + γ 1 r ˜ 1 * r ˜ 2 ,
t = t ˜ 1 t ˜ 2 exp [ ½ ( G 1 + i θ 1 ) ] 1 + γ 1 r ˜ 1 * r ˜ 2
r = exp ( i ϕ 1 r ) r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) 1 r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) ,
t = ( n 0 n 2 ) 1 / 2 × [ ( 1 r 1 2 ) ( 1 r 2 2 ) ] 1 / 2 exp ( ½ G 1 ) exp [ i ( ϕ 1 t + ϕ 2 t ) ] exp ( i q 1 d 1 ) 1 r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) ,
Φ 1 = Φ 12 = ( 4 π / λ ) n 1 d 1 + ϕ 1 r + ϕ 2 r .
r = Q 1 , 3 P 1 , 3 = Q 1 , 2 r 2 r 3 exp ( G 2 ) exp ( i Φ 2 ) Q 1 , 2 ( r 2 1 / r 2 ) P 1 , 2 r 2 r 3 exp ( G 2 ) exp ( i Φ 2 ) P 1 , 2 ( r 2 1 / r 2 ) ,
Q 1 , 2 = ( α 1 / β 1 ) exp ( i ϕ 1 r ) [ r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) ] , P 1 , 2 = ( α 1 / β 1 ) [ 1 r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) ] .
r = exp ( i ϕ 1 r ) r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) + r 3 exp ( G 1 + G 2 ) exp [ i ( Φ 1 + Φ 2 ) ] r 1 r 2 r 3 exp ( G 2 ) exp ( i Φ 2 ) 1 r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) r 2 r 3 exp ( G 2 ) exp ( i Φ 2 ) + r 1 r 3 exp ( G 1 + G 2 ) exp [ i ( Φ 1 + Φ 2 ) ] .
t = ( n 0 n 3 ) 1 / 2 [ ( 1 r 1 2 ) ( 1 r 2 2 ) ( 1 r 3 2 ) ] 1 / 2 exp [ ½ ( G 1 + G 2 ) ] exp [ i ( q 1 d 1 + q 2 d 2 ) ] exp [ i ( ϕ 1 t + ϕ 2 t + ϕ 3 t ) ] 1 r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) r 2 r 3 exp ( G 2 ) exp ( i Φ 2 ) + r 1 r 3 exp ( G 1 + G 2 ) exp [ i ( Φ 1 + Φ 2 ) ]
Φ 1 = Φ 12 = ( 4 π / λ ) n 1 d 1 + Φ 1 r + ϕ 2 r , Φ 2 = Φ 23 = ( 4 π / λ ) n 2 d 2 + Φ 2 r + ϕ 3 r .
Φ 1 + Φ 2 = Φ 12 + Φ 23 = ( 4 π / λ ) ( n 1 d 1 + n 2 d 2 ) + ϕ 1 r + ϕ 3 r + ϕ 2 r + ϕ 2 r = ( 4 π / λ ) ( n 1 d 1 + n 2 d 2 ) + ϕ 1 r + ϕ 3 r + 2 ϕ 2 t + π = Φ 13 + π ,
M 1 , 4 = [ 1 r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) r 1 t 2 r 3 exp ( G 1 , 3 ½ G 1 ) exp [ i ( Φ 1 , 3 Θ 1 , 2 ) ] r 1 t 2 t 3 r 4 exp ( G 1 , 4 ½ G 1 , 3 ) exp [ i ( Φ 1 , 4 Θ 1 , 3 ) ] t 2 exp ( ½ G 1 ) exp ( i Θ 12 ) 1 r 2 r 3 exp ( G 2 ) exp ( i Φ 2 ) r 2 t 3 r 4 exp ( G 2 , 4 ½ G 2 ) exp [ i ( Φ 2 , 4 Θ 2 , 3 ) ] 0 t 3 exp ( ½ G 2 ) exp ( i Θ 23 ) 1 r 3 r 4 exp ( G 3 ) exp ( i Φ 3 ) ] .
Det ( M 1 , 4 )= [ 1 r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) ] Det ( M 2 , 4 ) r 1 t 2 t 2 r 3 exp ( G 1 , 3 ) exp ( i Φ 1 , 3 ) Det ( M 3 , 4 ) r 1 t 2 t 2 t 3 t 3 r 4 exp ( G 1 , 4 ) exp ( i Φ 1 , 4 ) .
Det ( M 1 , 4 ) = 1 r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) r 2 r 3 exp ( G 2 ) exp ( i Φ 2 ) r 3 r 4 exp ( G 3 ) ( i Φ 3 ) r 1 r 3 exp ( G 1 , 3 ) exp ( i Φ 1 , 3 ) r 2 r 4 exp ( G 2 , 4 ) exp ( i Φ 2 , 4 ) r 1 r 4 exp ( G 1 , 4 ) exp ( i Φ 1 , 4 ) + r 1 r 2 r 3 r 4 exp ( G 1 + G 3 ) exp ( i Φ 1 + Φ 3 ) ] .
E C ( 1 ) = t 1 exp ( i ϕ 1 t ) 1 r 2 r 3 exp ( G 2 ) exp ( i Φ 2 ) r 3 r 4 exp ( G 3 ) exp ( i Φ 3 ) r 2 r 4 exp ( G 2 , 4 ) exp ( i Φ 2 , 4 ) Det ( M 1 , 4 ) E i ( 1 ) , E C ( 2 ) = t 1 t 2 exp ( ½ G 1 ) exp ( i q 1 d 1 ) exp [ i ( ϕ 1 t + ϕ 2 t ) ] 1 r 3 r 4 exp ( G 3 ) exp ( i Φ 3 ) Det ( M 1 , 4 ) E i ( 1 ) , E C ( 3 ) = t 1 t 2 t 3 exp [ ½ ( G 1 + G 2 ) ] exp [ i ( q 1 d 1 + q 2 d 2 ) ] exp [ i ( ϕ 1 t + ϕ 2 t + ϕ 3 t ) ] 1 Det ( M 1 , 4 ) E i ( 1 ) .
r = exp ( i ϕ 1 r ) r 1 Det ( M 1 , 4 , r 1 1 / r 1 ) Det ( M 1 , 4 ) , t = N t Det ( M 1 , 4 ) ,
r 1 Det ( M 1 , 4 , r 1 1 / r 1 ) = r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) + r 3 exp ( G 1 + G 2 ) exp [ i ( Φ 1 + Φ 2 ) ] r 1 r 2 r 3 exp ( G 2 ) exp ( i Φ 2 ) r 1 r 3 r 4 exp ( G 3 ) exp ( i Φ 3 ) + r 2 r 3 r 4 exp ( G 1 + G 3 ) exp [ i ( Φ 1 + Φ 3 ) ] + r 1 r 2 r 4 exp ( G 2 + G 3 ) exp [ i ( Φ 2 + Φ 3 ) ] r 4 exp ( G 1 + G 2 + G 3 ) exp [ i ( Φ 1 + Φ 2 + Φ 3 ) ] , N t = t 1 t 2 t 3 t 4 exp [ ½ ( G 1 + G 2 + G 3 ) ] exp [ i ( q 1 d 1 + q 2 d 2 + q 3 d 3 ) ] exp [ i ( ϕ 1 t + ϕ 2 t + ϕ 3 t + ϕ 4 t ) ] Det ( M 1 , 4 ) = 1 r 1 r 2 exp ( G 1 ) exp ( i Φ 1 ) r 2 r 3 exp ( G 2 ) exp ( i Φ 2 ) + r 1 r 3 exp ( G 1 + G 2 ) exp [ i ( Φ 1 + Φ 2 ) ] r 3 r 4 exp ( G 3 ) exp ( i Φ 3 ) + r 1 r 2 r 3 r 4 exp ( G 1 + G 3 ) exp [ i ( Φ 1 + Φ 3 ) ] + r 2 r 4 exp ( G 2 + G 3 ) exp [ i ( Φ 2 + Φ 3 ) ] r 1 r 4 exp ( G 1 + G 2 + G 3 ) exp [ i ( Φ 1 + Φ 2 + Φ 3 ) ] .
Det ( M 1 , N + 1 ) = 1 + C 1 + C 2 + C 3 .
C 1 = p = 1 p = 2 N 1 1 A p exp ( B p ) .
C 2 = r N r N + 1 exp ( G N + i Φ N ) .
p = 1 p = 2 N 1 1 A p exp ( B p ) = p = 1 p = 2 N 1 1 A p exp ( B p ) | r N A p + p = 1 p = 2 N 1 1 A p exp ( B p ) | r N A p
C 3 = exp ( G N + i Φ N ) × p = 1 p = 2 N 1 1 A p ( r N r N + 1 ) exp ( B p ) | r N A p r N r N + 1 exp ( G N + i Φ N ) p = 1 p = 2 N 1 1 A p exp ( B p ) | r N A p
C 3 = r N r N + 1 exp ( G N + i Φ N ) × [ p = 1 p = 2 N 1 1 A p ( r N 1 / r N ) exp ( B p ) | r N A p + p = 1 p = 2 N 1 1 A p ( r N 1 / r N ) exp ( B p ) | r N A p ] C 3 = r N r N + 1 exp ( G N + i Φ N ) × p = 1 p = 2 N 1 1 A p ( r N 1 / r N ) exp ( B N ) .
Det ( M 1 , N + 1 ) = 1 + p = 1 p = 2 N 1 1 A p exp ( B p ) r N r N + 1 exp ( G N + i Φ N ) × [ 1 + p = 1 p = 2 N 1 1 A p ( r N 1 / r N ) exp ( B p ) ] Det ( M 1 , N + 1 ) = Det ( M 1 , N ) r N r N + 1 exp ( G N + i Φ N ) × Det ( M 1 , N , r N 1 / r N ) .

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