Abstract

Z-domain and digital filter techniques are used to derive and interpret the transfer functions—transmission and reflection dependencies—for several étalonlike structures. Two-, three-, and four-mirror Fabry–Perot étalons and a seven-layer interference filter are analyzed. Although most of the examples have equally spaced mirrors, an example is also presented that illustrates the extension of the technique to structures with dissimilar lengths. Concepts such as group delay and state-variable descriptions may now be naturally applied to the analysis of these structures. The design of these structures may now benefit from algorithms developed for digital signal processing.

© 1994 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 323–367.
  2. E. Delano, R. J. Pegis, “Methods of synthesis for dielectric multilayer filters,” in Progress in Optics VII, E. Wolf, ed. (North-Holland, Amsterdam, 1969).
    [CrossRef]
  3. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 269–363.
  4. H. A. Macleod, “Thin film optical coatings,” in Applied Optics and Optical Engineering X, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987).
  5. Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1988).
    [CrossRef]
  6. J. M. Vigoureux, “Polynomial formulation of reflection and transmission by stratified planar structure,” J. Opt. Soc. Am. A 8, 1697–1701 (1991).
    [CrossRef]
  7. J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
    [CrossRef]
  8. S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
    [CrossRef]
  9. D. L. MacFarlane, K. J. Strozewski, J. A. Tatum, “Mode-locked laser pulse train repetition frequency multiplication: the optical rattler,” Appl. Opt. 30, 1042–1047 (1991).
    [CrossRef] [PubMed]
  10. D. L. MacFarlane, V. Narayan, “A relatively simple way to produce THz bursts of optical pulses,” Rev. Sci. Instrum. 63, 4092–4095 (1992).
    [CrossRef]
  11. J. O. Stoner, “PEPSIOS purely interferometric high resolution scanning spectrometer. III. Calculation of interferometer characteristics by a method of optical transients,”J. Opt. Soc. Am. 56, 370–376 (1966).
    [CrossRef]
  12. G. Hernandez, “Transient response of optical instruments,” Appl. Opt. 24, 928–929 (1985).
    [CrossRef] [PubMed]
  13. J. G. Proakis, D. G. Manolakis, Digital Signal Processing (Macmillan, New York, 1992).
  14. B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
    [CrossRef]
  15. W. R. Skinner, P. B. Hays, V. J. Abreau, “Optimization of a triple etalon interferometer,” Appl. Opt. 26, 2817–2827 (1987).
    [CrossRef] [PubMed]
  16. M. Jinno, T. Matsumoto, “Optical tank circuits used for all-optical timing recovery,” IEEE J. Quantum Electron. 28, 895–900 (1992).
    [CrossRef]

1992 (3)

J. M. Vigoureux, “Use of Einstein’s addition law in studies of reflection by stratified planar structures,” J. Opt. Soc. Am. A 9, 1313–1319 (1992).
[CrossRef]

D. L. MacFarlane, V. Narayan, “A relatively simple way to produce THz bursts of optical pulses,” Rev. Sci. Instrum. 63, 4092–4095 (1992).
[CrossRef]

M. Jinno, T. Matsumoto, “Optical tank circuits used for all-optical timing recovery,” IEEE J. Quantum Electron. 28, 895–900 (1992).
[CrossRef]

1991 (3)

1988 (1)

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1988).
[CrossRef]

1987 (1)

1985 (1)

1984 (1)

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

1966 (1)

Abreau, V. J.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 269–363.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 269–363.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 323–367.

Coldren, L. A.

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

Corzine, S. W.

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

Delano, E.

E. Delano, R. J. Pegis, “Methods of synthesis for dielectric multilayer filters,” in Progress in Optics VII, E. Wolf, ed. (North-Holland, Amsterdam, 1969).
[CrossRef]

Goodman, J. W.

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Hays, P. B.

Hernandez, G.

Jinno, M.

M. Jinno, T. Matsumoto, “Optical tank circuits used for all-optical timing recovery,” IEEE J. Quantum Electron. 28, 895–900 (1992).
[CrossRef]

MacFarlane, D. L.

D. L. MacFarlane, V. Narayan, “A relatively simple way to produce THz bursts of optical pulses,” Rev. Sci. Instrum. 63, 4092–4095 (1992).
[CrossRef]

D. L. MacFarlane, K. J. Strozewski, J. A. Tatum, “Mode-locked laser pulse train repetition frequency multiplication: the optical rattler,” Appl. Opt. 30, 1042–1047 (1991).
[CrossRef] [PubMed]

Macleod, H. A.

H. A. Macleod, “Thin film optical coatings,” in Applied Optics and Optical Engineering X, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987).

Mandel, L.

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1988).
[CrossRef]

Manolakis, D. G.

J. G. Proakis, D. G. Manolakis, Digital Signal Processing (Macmillan, New York, 1992).

Matsumoto, T.

M. Jinno, T. Matsumoto, “Optical tank circuits used for all-optical timing recovery,” IEEE J. Quantum Electron. 28, 895–900 (1992).
[CrossRef]

Moslehi, B.

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Narayan, V.

D. L. MacFarlane, V. Narayan, “A relatively simple way to produce THz bursts of optical pulses,” Rev. Sci. Instrum. 63, 4092–4095 (1992).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1988).
[CrossRef]

Pegis, R. J.

E. Delano, R. J. Pegis, “Methods of synthesis for dielectric multilayer filters,” in Progress in Optics VII, E. Wolf, ed. (North-Holland, Amsterdam, 1969).
[CrossRef]

Proakis, J. G.

J. G. Proakis, D. G. Manolakis, Digital Signal Processing (Macmillan, New York, 1992).

Shaw, H. J.

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Skinner, W. R.

Stoner, J. O.

Strozewski, K. J.

Tatum, J. A.

Tur, M.

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Vigoureux, J. M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 323–367.

Yan, R. H.

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

Am. J. Phys. (1)

Z. Y. Ou, L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1988).
[CrossRef]

Appl. Opt. (3)

IEEE J. Quantum Electron. (2)

M. Jinno, T. Matsumoto, “Optical tank circuits used for all-optical timing recovery,” IEEE J. Quantum Electron. 28, 895–900 (1992).
[CrossRef]

S. W. Corzine, R. H. Yan, L. A. Coldren, “A tanh substitution technique for the analysis of abrupt and graded interface multilayer dielectric stacks,” IEEE J. Quantum Electron. 27, 2086–2090 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Proc. IEEE (1)

B. Moslehi, J. W. Goodman, M. Tur, H. J. Shaw, “Fiber optic lattice signal processing,” Proc. IEEE 72, 909–930 (1984).
[CrossRef]

Rev. Sci. Instrum. (1)

D. L. MacFarlane, V. Narayan, “A relatively simple way to produce THz bursts of optical pulses,” Rev. Sci. Instrum. 63, 4092–4095 (1992).
[CrossRef]

Other (5)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 323–367.

E. Delano, R. J. Pegis, “Methods of synthesis for dielectric multilayer filters,” in Progress in Optics VII, E. Wolf, ed. (North-Holland, Amsterdam, 1969).
[CrossRef]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), pp. 269–363.

H. A. Macleod, “Thin film optical coatings,” in Applied Optics and Optical Engineering X, R. R. Shannon, J. C. Wyant, eds. (Academic, San Diego, Calif., 1987).

J. G. Proakis, D. G. Manolakis, Digital Signal Processing (Macmillan, New York, 1992).

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Figures (9)

Fig. 1
Fig. 1

(a) Familiar schematic for a Fabry–Perot étalon composed of two mirrors facing each other, separated by a distance d. The transmitted field and the reflected field are made up of a superposition of an infinite number of waves. This infinite sum converges to yield the Airy formula. (b) z-Space block diagram for the étalon. The z−1 blocks represent unit delays that are due to the length d between mirrors. The mirror coefficients are represented by multiplicative constants. The superposition is represented by the summing elements.

Fig. 2
Fig. 2

(a) Familiar transmission, or Airy, curve for the simple étalon, for |r01| = |r12| = 0.8. (b) Phase in transmission. (c) Group delay. The horizontal scale is in frequency normalized so that the free spectral range is π.

Fig. 3
Fig. 3

Block diagram for a three-mirror étalon.

Fig. 4
Fig. 4

Transmission characteristics of a three-mirror étalon with |r01| = |r23| = 0.8 and |r12| = 0.5773: (a) Magnitude, (b) group delay.

Fig. 5
Fig. 5

Magnitude of the transmission for a three-mirror étalon as the reflectivity of the middle mirror approaches 1 (|r12| = 0.97). Here a single-peak per free spectral range appears, with a sharp rolloff behavior. The values for mirrors 1 and 3 remain the same as in Fig. 4.

Fig. 6
Fig. 6

Criteria for a triple pole at ω = 0 for a three-mirror étalon. The necessary valuess for r12 are plotted as equal-value contours against r01 and r23, as determined by Eq. (39).

Fig. 7
Fig. 7

Magnitude of the transmittance of a four-mirror étalon for |r01| = |r34| = 0.8 and different values for the middle mirrors. In (a) the two middle mirrors have a large reflectivity, |r12| = |r23| = 0.95, and the transmittance is a very high Q (finesse) filter with a single peak at 0 and a single peak at π. As we decrease the reflectivities of the middle mirrors, multiple maxima in the transmission function occur—at higher frequencies for lower r’s. In (b), |r12| = |r23| = 0.8. In (c), |r12| = |r23| = 0.5.

Fig. 8
Fig. 8

Calculated transmission characteristics for an HLH-2L-HLH interference filter made with zinc sulfide (nH = 2.3) and magnesium fluoride (nL = 1.35): (a) magnitude, (b) group delay, (c) pole-zero plot.

Fig. 9
Fig. 9

Magnitude of the transmittance of a coupled cavity configuration with |r01| = |r23| = 0.8 and |r12| = 0.5773, and for different cavity-length ratios. The mirror coefficients are the same as those for Fig. 4, and hence these plots may be compared directly with the transmission magnitude plot in Fig. 4(a). (a) n = 1 and m = 3, (b) n = 1 and m = 8, (c) n = 3 and m = 8.

Equations (74)

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E r ( z ) = t 10 r 12 z - 2 A ( z ) + r 01 E i ( z ) ,
A ( z ) = t 01 E i ( z ) + r 10 r 12 z - 2 A ( z ) ,
E t ( z ) = t 12 z - 1 A ( z ) .
E t ( z ) = E i ( z ) [ t 01 t 12 z - 1 1 - r 10 r 12 z - 2 ] ,
E r ( z ) = E i ( z ) [ r 01 + t 01 t 10 r 12 z - 2 1 - r 10 r 12 z - 2 ] .
H ( z ) = k = 0 M b k z - k 1 + k = 1 N a k z - k ,
E r ( z ) E i ( z ) = r 01 + r 12 z - 2 1 - r 10 r 12 z - 2 .
y [ n ] - r 10 r 12 y [ n - 2 ] = t 01 t 12 x [ n - 1 ] ,
y [ n ] - r 10 r 12 y [ n - 2 ] = r 01 x [ n ] + r 12 x [ n - 2 ] .
H ( z ) = r 01 [ 1 - ( r 12 / r 10 ) 1 / 2 z - 1 ] [ 1 + ( r 12 / r 10 ) 1 / 2 z - 1 ] [ 1 - ( r 10 r 12 ) 1 / 2 z - 1 ] [ 1 + [ r 10 r 12 ] 1 / 2 z - 1 ] .
H ( z ) = 1 2 [ r 01 + ( r 12 / r 10 ) 1 / 2 z - 1 1 - ( r 10 r 12 ) 1 / 2 z - 1 + r 01 - ( r 12 / r 10 ) 1 / 2 z - 1 1 + ( r 10 r 12 ) 1 / 2 z - 1 ] .
E r ( z ) = r 01 E i ( z ) + t 10 z - 1 R 1 ( z ) ,
T 1 ( z ) = t 01 E i ( z ) + r 10 z - 1 R 1 ( z ) ,
R 1 ( z ) = r 12 z - 1 T 1 ( z ) + t 21 z - 1 R 2 ( z ) ,
T 2 ( z ) = t 12 z - 1 T 1 ( z ) + r 21 z - 1 R 2 ( z ) ,
R 2 ( z ) = r 23 z - 1 T 2 ( z ) ,
E t ( z ) = t 23 z - 1 T 2 ( z ) .
E t ( z ) E i ( z ) = t 01 t 12 t 23 z - 2 ( 1 - r 21 r 23 z - 2 ) [ 1 - r 10 r 12 z - 2 - r 10 t 12 t 21 r 23 z - 4 1 - r 21 r 23 z - 2 ] ,
E r ( z ) E i ( z ) = r 01 + t 01 t 10 r 12 z - 2 + t 01 t 12 t 21 t 10 r 23 z - 4 1 - r 21 r 23 z - 2 [ 1 - r 10 r 12 z - 2 - r 10 t 12 t 21 r 23 z - 4 1 - r 21 r 23 z - 2 ] .
T k ( z ) = t k - 1 , k z - 1 T k - 1 ( z ) + r k , k - 1 z - 1 R k ( z ) ,
R k ( z ) = r k , k + 1 z - 1 T k ( z ) + t k + 1 , k z - 1 R k + 1 ( z ) ,
T 0 ( z ) = E i ( z ) ,
R N + 1 ( z ) = 0 ,
- r 01 E i ( z ) = - E r ( z ) + t 10 z - 1 R 1 ( z ) ,
- t 01 E i ( z ) = - T 1 ( z ) + r 10 z - 1 R 1 ( z ) ,
0 = r 12 z - 1 T 1 ( z ) - R 1 ( z ) + t 21 z - 1 R 2 ( z ) ,
0 = t 12 z - 1 T 1 ( z ) - T 2 ( z ) + r 21 z - 1 R 2 ( z ) ,
0 = r 23 z - 1 T 2 ( z ) - R 2 ( z ) ,
0 = t 23 z - 1 T 2 ( z ) - E t ( z ) .
Y = M 0 X ,
Y = { - r 01 E i , - t 01 E i , 0 , 0 , 0 , 0 } ,
X = { E r , T 1 , R 1 , T 2 , R 2 , E t } ,
M 0 = | - 1 0 t 10 z - 1 0 0 0 0 - 1 r 10 z - 1 0 0 0 0 r 12 z - 1 - 1 0 t 21 z - 1 0 0 t 12 z - 1 0 - 1 r 21 z - 1 0 0 0 0 r 23 z - 1 - 1 0 0 0 0 t 23 z - 1 0 - 1 | .
E t ( z ) E i ( z ) = t 01 t 12 t 23 z - 1 1 - ( r 10 r 12 + r 21 r 23 ) z - 2 - r 10 r 23 z - 4 .
E r ( z ) E i ( z ) = [ r 01 + ( r 12 - r 01 r 21 r 23 ) z - 2 + r 23 z - 4 ] 1 - ( r 10 r 12 + r 21 r 23 ) z - 2 - r 10 r 23 r - 4 .
T ( ω ) = t 01 t 12 t 23 [ 1 + ( r 10 r 12 + r 21 r 23 ) 2 + r 10 2 r 23 2 - 2 ( r 10 r 12 + r 21 r 23 ) ( 1 - r 10 r 23 ) cos ( 2 ω ) - 2 r 10 r 23 cos ( 4 ω ) ] 1 / 2 .
0 = - 4 sin ( 2 ω ) [ - ( r 10 r 12 + r 21 r 23 ) × ( 1 - r 10 r 23 ) - 4 r 10 r 23 cos ( 2 ω ) ] .
ω m = 1 2 arccos [ - ( r 10 r 12 + r 21 r 23 ) ( 1 - r 10 r 23 ) 4 r 10 r 23 ] .
( r 10 r 12 + r 21 r 23 ) ( 1 - r 10 r 23 ) = - 4 r 10 r 23
E t ( z ) E i ( z ) = t 01 t 12 t 23 t 34 z - 3 ( 1 - r 32 r 34 z - 2 ) [ 1 - r 21 r 23 z - 2 - r 21 t 32 t 23 r 34 z - 4 ( 1 - r 32 r 34 z - 2 ) ] { 1 - r 01 r 12 z - 2 - r 01 t 12 t 21 r 23 z - 4 - r 01 t 12 t 21 t 23 t 32 r 34 z - 6 ( 1 - r 32 r 34 z - 2 ) [ 1 - r 21 r 23 z - 2 - r 21 t 32 t 23 r 34 z - 4 ( 1 - r 32 r 34 z - 2 ) ] } ,
E r ( z ) E i ( z ) = r 01 + t 01 t 10 r 12 z - 2 + t 01 t 10 t 12 t 21 r 23 z - 4 + t 01 t 10 t 12 t 21 t 23 t 32 r 34 z - 6 ( 1 - r 32 r 34 z - 2 ) [ 1 - r 21 r 23 z - 2 - r 21 t 23 t 32 r 34 z - 4 ( 1 - r 32 r 34 z - 2 ) ] { 1 - r 01 r 12 z - 2 - r 01 t 12 t 21 r 23 z - 4 + r 01 t 12 t 21 t 23 t 32 r 34 z - 6 ( 1 - r 32 r 34 z - 2 ) [ 1 - r 21 r 23 z - 2 - r 21 t 23 t 32 r 34 z - 4 ( 1 - r 32 r 34 z - 2 ) ] } .
E t ( z ) E i ( z ) = t 01 t 12 t 23 t 34 z - 3 1 - ( r 10 r 12 + r 21 r 23 + r 32 r 34 ) z - 2 + ( r 10 r 23 + r 21 r 34 - r 10 r 12 r 32 r 34 ) z - 4 - r 10 r 34 z - 6 ,
E r ( z ) E i ( z ) = r 01 + ( r 12 - r 01 r 21 r 23 - r 01 r 32 r 34 ) z - 2 + ( r 23 - r 01 r 21 r 34 - r 12 r 32 r 34 ) z - 4 - r 34 z - 6 1 - ( r 10 r 12 + r 21 r 23 + r 32 r 34 ) z - 2 + ( r 10 r 23 + r 21 r 34 - r 10 r 12 r 32 r 34 ) z - 4 - r 10 r 34 z - 6 .
T k [ n + 1 ] = t k - 1 , k T k - 1 [ n ] + r k , k - 1 R k [ n ] ,
R k [ n + 1 ] = r k , k + 1 T k [ n ] + t k + 1 , k R k + 1 [ n ] .
v [ n + 1 ] = Fv [ n ] + qx [ n ] ,
F = | 0 r 10 0 0 0 0 r 12 0 0 t 21 0 0 t 12 0 0 r 21 0 0 0 0 r 23 0 0 t 32 0 0 t 23 0 0 r 32 0 0 0 0 r 34 0 | ,
q = { t 01 , 0 , 0 , 0 , 0 , 0 } .
E r = r 01 E i + t 10 R 1 ,
E t = t 34 T 3 ,
y [ n ] = g t v [ n ] + dx [ n ] ,
y = { E r , E t } .
g t = | 0 t 10 0 0 0 0 0 0 t 34 0 0 0 | ,
d = { r 01 , 0 } .
a 0 = 1 ,
a 1 = - 2 r 01 r 12 - 4 r 12 2 ,
a 4 = 2 r 01 r 12 + r 01 2 r 12 2 + 6 r 01 r 12 3 + r 12 2 + 4 r 12 4 ,
a 6 = - 4 r 01 r 12 5 - 2 r 01 2 r 12 4 - 2 r 12 4 - 4 r 01 r 12 3 - 2 r 01 2 r 12 2 + 2 r 12 2 - 2 r 01 r 12 ,
a 8 = - r 12 2 ( 1 - r 01 2 ) ( 3 + r 12 2 + r 12 4 ) ,
a 10 = 4 r 01 r 12 5 + 2 r 01 2 r 12 4 + 2 r 12 4 + 4 r 01 r 12 3 - 2 r 01 2 r 12 2 + 2 r 01 2 + 2 r 01 r 12 ,
a 12 = - 2 r 01 r 12 - r 01 2 r 12 2 - 6 r 01 r 12 3 - r 12 2 - 4 r 01 2 r 12 4 ,
a 14 = 2 r 01 r 12 + 4 r 01 2 r 12 2 ,
a 16 = - r 01 2 .
b 8 = t 01 t 12 t 21 t 12 t 21 t 12 t 21 t 10 .
E r ( z ) = r 01 E i ( z ) + t 10 z - n R 1 ( z ) ,
T 1 ( z ) = t 01 E i ( z ) + r 10 z - n R 1 ( z ) ,
R 1 ( z ) = r 12 z - n T 1 ( z ) + t 21 z - m R 2 ( z ) ,
T 2 ( z ) = t 12 z - n T 1 ( z ) + r 21 z - m R 2 ( z ) ,
R 2 ( z ) = r 23 z - m T 2 ( z ) ,
E t ( z ) = t 23 z - m T 2 ( z ) .
E t E i = t 01 t 12 t 23 z - ( m + n ) ( 1 - r 21 r 23 z - 2 m ) [ 1 - r 10 r 12 z - 2 n - r 10 t 12 t 21 r 23 z - 2 ( m + n ) 1 - r 21 r 23 z - 2 m ] ,
E r E i = r 01 + t 01 t 10 r 12 z - 2 n + t 01 t 10 t 12 t 21 r 23 z - 2 ( n + m ) 1 - r 21 r 23 z - 2 m 1 - r 10 r 12 z - 2 n - r 10 t 12 t 21 r 23 z - 2 ( n + m ) 1 - r 21 r 23 z - 2 m ,
E t E i = t 01 t 12 t 23 z - ( n + m ) 1 - r 10 r 12 z - 2 n - r 21 r 23 z - 2 m - r 10 r 23 z - 2 ( m + n ) ,
E r E i = r 01 + r 12 z - 2 n - r 01 r 21 r 23 z - 2 m + r 23 z - 2 ( m + n ) 1 - r 10 r 12 z - 2 n - r 21 r 23 z - 2 m - r 10 r 23 z - 2 ( m + n ) .

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