Abstract

The relative merits of three designs of compound optical-fiber-based resonators for application to line narrowing in fiber lasers and demultiplexing in optical-communications systems are discussed. The three resonator designs are a set of concatenated rings, a three-reflector resonator with loop mirrors, and a fiber Fox–Smith interferometer.

© 1988 Optical Society of America

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References

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  1. J. A. Stone, D. Marcuse, “Ultrahigh finesse fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. LT-14, 382–385 (1986).
    [CrossRef]
  2. L. F. Stokes, M. Chodorow, H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. 7, 288–290 (1982).
    [CrossRef] [PubMed]
  3. M. Brierley, P. Urquhart, “Transversely coupled fiber Fabry–Perot resonator: performance characteristics,” Appl. Opt. 26, 4841–4845 (1987).
    [CrossRef] [PubMed]
  4. I. D. Miller, D. B. Mortimore, P. Urquhart, B. J. Ainslie, S. P. Craig, C. A. Millar, D. B. Payne, “A Nd3+-doped cw fiber laser using all-fiber reflectors,” Appl. Opt. 26, 2197–2201 (1987).
    [CrossRef] [PubMed]
  5. J. Stone, C. A. Burrus, “Neodymium doped silica lasers in end-pumped geometry,” Appl. Phys. Lett. 23, 388–389 (1973).
    [CrossRef]
  6. S. R. Mallinson, “Wavelength-selective filters for single-mode fiber WDM systems using Fabry–Perot interferometers,” Appl. Opt. 26, 430–436 (1987).
    [CrossRef] [PubMed]
  7. See, for example, L. F. Stokes, M. Chodorow, H. J. Shaw, “All-fiber stimulated Brillouin ring laser with submilliwatt pump threshold,” Opt. Lett. 7, 509–511 (1982).
    [CrossRef] [PubMed]
  8. B. S. Kawasaki, K. O. Hill, R. G. Lamont, “Biconical-taper single-mode fiber coupler,” Opt. Lett. 6, 327–328 (1981).
    [CrossRef] [PubMed]
  9. R. Bergh, G. Kotler, H. J. Shaw, “Single mode fiber optic directional coupler,” Electron. Lett. 16, 260–261 (1982).
    [CrossRef]
  10. R. B. Dyott, J. Bello, “Polarisation holding directional coupler made from elliptically cored fiber having a D-section,” Electron. Lett. 19, 601 (1983).
    [CrossRef]
  11. H. van de Stadt, J. M. Muller, “Multimirror Fabry–Perot interferometers,” J. Opt. Soc. Am. A 2, 1363–1370 (1985).
    [CrossRef]
  12. P. W. Smith, “Stabilized single frequency output from a long laser cavity,” IEEE J. Quantum Electron. QE-1, 343–348 (1965).
    [CrossRef]
  13. G. Merkle, J. Heppner, “CO2waveguide laser with Fox–Smith mode selector,” IEEE J. Quantum Electron. QE-19, 1663–1667 (1983).
    [CrossRef]
  14. R. Wyatt, K. H. Cameron, M. R. Matthews, “Tunable narrow line external cavity lasers for coherent optical systems,” Br. Telecom. Technol. J. 3, 5–12 (1985).
  15. P. Urquhart, P. Barnsley, C. Millar, M. Brierley, “Single longitudinal mode output from a Fox–Smith erbium fiber laser,” in Digest of Topical Meeting on Tunable Solid State Lasers(Optical Society of America, Washington, D.C., 1987), postdeadline paper PD-7.
  16. P. Urquhart, “Transversely coupled fiber Fabry–Perot resonator: theory,” Appl. Opt. 26, 456–463 (1987).
    [CrossRef] [PubMed]
  17. I. M. Jauncey, L. Reekie, D. N. Payne, C. J. Rowe, “Single longitudinal mode operation of a fiber laser,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper ThT9.

1987

1986

J. A. Stone, D. Marcuse, “Ultrahigh finesse fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. LT-14, 382–385 (1986).
[CrossRef]

1985

H. van de Stadt, J. M. Muller, “Multimirror Fabry–Perot interferometers,” J. Opt. Soc. Am. A 2, 1363–1370 (1985).
[CrossRef]

R. Wyatt, K. H. Cameron, M. R. Matthews, “Tunable narrow line external cavity lasers for coherent optical systems,” Br. Telecom. Technol. J. 3, 5–12 (1985).

1983

G. Merkle, J. Heppner, “CO2waveguide laser with Fox–Smith mode selector,” IEEE J. Quantum Electron. QE-19, 1663–1667 (1983).
[CrossRef]

R. B. Dyott, J. Bello, “Polarisation holding directional coupler made from elliptically cored fiber having a D-section,” Electron. Lett. 19, 601 (1983).
[CrossRef]

1982

1981

1973

J. Stone, C. A. Burrus, “Neodymium doped silica lasers in end-pumped geometry,” Appl. Phys. Lett. 23, 388–389 (1973).
[CrossRef]

1965

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

Ainslie, B. J.

Barnsley, P.

P. Urquhart, P. Barnsley, C. Millar, M. Brierley, “Single longitudinal mode output from a Fox–Smith erbium fiber laser,” in Digest of Topical Meeting on Tunable Solid State Lasers(Optical Society of America, Washington, D.C., 1987), postdeadline paper PD-7.

Bello, J.

R. B. Dyott, J. Bello, “Polarisation holding directional coupler made from elliptically cored fiber having a D-section,” Electron. Lett. 19, 601 (1983).
[CrossRef]

Bergh, R.

R. Bergh, G. Kotler, H. J. Shaw, “Single mode fiber optic directional coupler,” Electron. Lett. 16, 260–261 (1982).
[CrossRef]

Brierley, M.

M. Brierley, P. Urquhart, “Transversely coupled fiber Fabry–Perot resonator: performance characteristics,” Appl. Opt. 26, 4841–4845 (1987).
[CrossRef] [PubMed]

P. Urquhart, P. Barnsley, C. Millar, M. Brierley, “Single longitudinal mode output from a Fox–Smith erbium fiber laser,” in Digest of Topical Meeting on Tunable Solid State Lasers(Optical Society of America, Washington, D.C., 1987), postdeadline paper PD-7.

Burrus, C. A.

J. Stone, C. A. Burrus, “Neodymium doped silica lasers in end-pumped geometry,” Appl. Phys. Lett. 23, 388–389 (1973).
[CrossRef]

Cameron, K. H.

R. Wyatt, K. H. Cameron, M. R. Matthews, “Tunable narrow line external cavity lasers for coherent optical systems,” Br. Telecom. Technol. J. 3, 5–12 (1985).

Chodorow, M.

Craig, S. P.

Dyott, R. B.

R. B. Dyott, J. Bello, “Polarisation holding directional coupler made from elliptically cored fiber having a D-section,” Electron. Lett. 19, 601 (1983).
[CrossRef]

Heppner, J.

G. Merkle, J. Heppner, “CO2waveguide laser with Fox–Smith mode selector,” IEEE J. Quantum Electron. QE-19, 1663–1667 (1983).
[CrossRef]

Hill, K. O.

Jauncey, I. M.

I. M. Jauncey, L. Reekie, D. N. Payne, C. J. Rowe, “Single longitudinal mode operation of a fiber laser,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper ThT9.

Kawasaki, B. S.

Kotler, G.

R. Bergh, G. Kotler, H. J. Shaw, “Single mode fiber optic directional coupler,” Electron. Lett. 16, 260–261 (1982).
[CrossRef]

Lamont, R. G.

Mallinson, S. R.

Marcuse, D.

J. A. Stone, D. Marcuse, “Ultrahigh finesse fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. LT-14, 382–385 (1986).
[CrossRef]

Matthews, M. R.

R. Wyatt, K. H. Cameron, M. R. Matthews, “Tunable narrow line external cavity lasers for coherent optical systems,” Br. Telecom. Technol. J. 3, 5–12 (1985).

Merkle, G.

G. Merkle, J. Heppner, “CO2waveguide laser with Fox–Smith mode selector,” IEEE J. Quantum Electron. QE-19, 1663–1667 (1983).
[CrossRef]

Millar, C.

P. Urquhart, P. Barnsley, C. Millar, M. Brierley, “Single longitudinal mode output from a Fox–Smith erbium fiber laser,” in Digest of Topical Meeting on Tunable Solid State Lasers(Optical Society of America, Washington, D.C., 1987), postdeadline paper PD-7.

Millar, C. A.

Miller, I. D.

Mortimore, D. B.

Muller, J. M.

Payne, D. B.

Payne, D. N.

I. M. Jauncey, L. Reekie, D. N. Payne, C. J. Rowe, “Single longitudinal mode operation of a fiber laser,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper ThT9.

Reekie, L.

I. M. Jauncey, L. Reekie, D. N. Payne, C. J. Rowe, “Single longitudinal mode operation of a fiber laser,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper ThT9.

Rowe, C. J.

I. M. Jauncey, L. Reekie, D. N. Payne, C. J. Rowe, “Single longitudinal mode operation of a fiber laser,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper ThT9.

Shaw, H. J.

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]

Stokes, L. F.

Stone, J.

J. Stone, C. A. Burrus, “Neodymium doped silica lasers in end-pumped geometry,” Appl. Phys. Lett. 23, 388–389 (1973).
[CrossRef]

Stone, J. A.

J. A. Stone, D. Marcuse, “Ultrahigh finesse fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. LT-14, 382–385 (1986).
[CrossRef]

Urquhart, P.

van de Stadt, H.

Wyatt, R.

R. Wyatt, K. H. Cameron, M. R. Matthews, “Tunable narrow line external cavity lasers for coherent optical systems,” Br. Telecom. Technol. J. 3, 5–12 (1985).

Appl. Opt.

Appl. Phys. Lett.

J. Stone, C. A. Burrus, “Neodymium doped silica lasers in end-pumped geometry,” Appl. Phys. Lett. 23, 388–389 (1973).
[CrossRef]

Br. Telecom. Technol. J.

R. Wyatt, K. H. Cameron, M. R. Matthews, “Tunable narrow line external cavity lasers for coherent optical systems,” Br. Telecom. Technol. J. 3, 5–12 (1985).

Electron. Lett.

R. Bergh, G. Kotler, H. J. Shaw, “Single mode fiber optic directional coupler,” Electron. Lett. 16, 260–261 (1982).
[CrossRef]

R. B. Dyott, J. Bello, “Polarisation holding directional coupler made from elliptically cored fiber having a D-section,” Electron. Lett. 19, 601 (1983).
[CrossRef]

IEEE J. Lightwave Technol.

J. A. Stone, D. Marcuse, “Ultrahigh finesse fiber Fabry–Perot interferometers,” IEEE J. Lightwave Technol. LT-14, 382–385 (1986).
[CrossRef]

IEEE J. Quantum Electron.

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

G. Merkle, J. Heppner, “CO2waveguide laser with Fox–Smith mode selector,” IEEE J. Quantum Electron. QE-19, 1663–1667 (1983).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Other

P. Urquhart, P. Barnsley, C. Millar, M. Brierley, “Single longitudinal mode output from a Fox–Smith erbium fiber laser,” in Digest of Topical Meeting on Tunable Solid State Lasers(Optical Society of America, Washington, D.C., 1987), postdeadline paper PD-7.

I. M. Jauncey, L. Reekie, D. N. Payne, C. J. Rowe, “Single longitudinal mode operation of a fiber laser,” in Digest of Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1987), paper ThT9.

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Figures (12)

Fig. 1
Fig. 1

Double-coupler fiber-ring resonator.

Fig. 2
Fig. 2

(a) Double-coupler fiber-ring resonators in series. Compound channel-dropping responses are observed from the output arm. (b) Double-coupler fiber-ring resonators in parallel. Compound channel-passing responses are observed at the output arm.

Fig. 3
Fig. 3

Spectral-response function for (a) one ring and (b) two identical rings in parallel. All the couplers have a coupling ratio K = 0.3. All the components used are assumed to be lossless.

Fig. 4
Fig. 4

Mechanisms for the suppression of spectral orders: (a) the Vernier principle; (b) one long cavity and one short cavity. The arrows indicate the coincidences of resonance peaks.

Fig. 5
Fig. 5

Spectral response function for two parallel rings of unequal lengths. The ratio of lengths is 21(l11 + l12) = 20(l21 + l22). All components are assumed to be lossless. The coupling ratios of all four couplers are 0.15.

Fig. 6
Fig. 6

Zero and first spectral orders of response functions for two parallel rings with a length ratio 21(l11 + l12) = 20(l21 + l22). All components are assumed to be lossless. The coupling ratios are (a) 0.15, (b) 0.12, (c) 0.09, (d) 0.06, and (e) 0.03. Curve (a) corresponds to Fig. 5.

Fig. 7
Fig. 7

All-fiber loop mirror. The coupler may be of any type; fused, polished, or D fiber.

Fig. 8
Fig. 8

Multimirror all-fiber Fabry–Perot resonator with loop reflectors. The device can be made from a continuous length of fiber.

Fig. 9
Fig. 9

Demonstration of mode suppression in an all-fiber three-mirror resonator. The values of the Xi, as defined by Eqs. (31)(33), are X1 = 0.68, X2 = 0.535, and X3 = 0.68. Cavity lengths are 21ℒ1 = 19ℒ2.

Fig. 10
Fig. 10

Fiber Fox–Smith resonator illustrated with the fiber ends butted against dielectric mirrors. An alternative arrangement would be to use one or more loop reflectors.

Fig. 11
Fig. 11

Demonstration of mode suppression in a fiber Fox–Smith resonator by the Vernier principle. K = 0.45, r 1 2 = r 2 2 = r 3 2 = 0.99. Cavity lengths are 21ℒ1 = 20ℒ2.

Fig. 12
Fig. 12

Fiber Fox–Smith resonator: variation of the heights of the three most prominent peaks, as shown in Fig. 11, when the cavity lengths are not exactly in the ratio of 21ℒ1 = 20ℒ2. (a) Left-hand peak, (b) central peak, (c) right-hand peak.

Tables (2)

Tables Icon

Table 1 Numerator Terms for All the Possible Arrangements of Input and Output Ports for the Fiber Fox–Smith Resonator

Tables Icon

Table 2 Classification of Solutions for the Fiber Fox–Smith Resonatora

Equations (64)

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E 1 t = E 6 a exp [ ( α + j β ) l i 2 ] ,
E 2 t = E in ,
E 7 t = E 4 a exp [ ( α + j β ) l i 1 ] ,
E 3 a = ( 1 K i 1 ) 1 / 2 ( 1 γ i 1 ) 1 / 2 E 2 t + j K i 1 1 / 2 ( 1 γ i 1 ) 1 / 2 E 1 t ,
E 4 a = ( 1 K i 1 ) 1 / 2 ( 1 γ i 1 ) 1 / 2 E 1 t + j K i 1 1 / 2 ( 1 γ i 1 ) 1 / 2 E 2 t ,
E 5 a = ( 1 K i 2 ) 1 / 2 ( 1 γ i 2 ) 1 / 2 E 8 t + j K i 2 1 / 2 ( 1 γ i 2 ) 1 / 2 E 7 t ,
E 6 a = ( 1 K i 2 ) 1 / 2 ( 1 γ i 2 ) 1 / 2 E 7 t + j K i 2 1 / 2 ( 1 γ i 2 ) 1 / 2 E 8 t ,
I 3 out / I in = ( 1 { i 1 i 2 + i 1 i 2 [ ( 1 K i 2 ) 1 / 2 ( 1 γ i 2 ) 1 / 2 ( 1 K i 1 ) 1 / 2 ( 1 γ i 1 ) 1 / 2 ] } ) 2 + 4 { i 1 i 2 + i 1 i 2 [ ( 1 K i 2 ) 1 / 2 ( 1 γ i 2 ) 1 / 2 ( 1 K i 1 ) 1 / 2 ( 1 γ i 1 ) 1 / 2 ] } sin 2 [ β ( l i 1 + l i 2 ) ] ( 1 i 1 i 2 ) 2 + 4 i 1 i 2 sin 2 [ β ( l i 1 + l i 2 ) ] ,
i j = ( 1 K i j ) 1 / 2 ( 1 γ i j ) 1 / 2 exp ( 2 α l i j ) , j = 1 , 2 ,
i j = K i j 1 / 2 ( 1 γ i j ) 1 / 2 exp ( 2 α l i j ) , j = 1 , 2 ,
I 5 out / I in = ( i 1 i ) 2 / exp [ 2 α ( l i 1 + 2 l i 2 ) ] ( 1 i 1 i 2 ) 2 + 4 i 1 i 2 sin 2 [ β ( l i 1 + l i 2 ) ]
( I out I in ) i rings = i = 1 N ( I out / I in ) ring i i = 1 N 1 exp ( 2 α L i ) .
C N = i = 1 N ( 1 + i 1 i 2 1 i 1 i 2 ) 2 = i = 1 N [ 1 + ( 2 F i π ) 2 ] .
F i = π ( i 1 i 2 ) 1 / 2 ( 1 i 1 i 2 ) .
F ( N rings ) = F 1 / ( 2 1 / N 1 ) 1 / 2 .
F ( 2 rings ) = 2 1 / 2 F 1 F 2 { [ ( F 1 2 + F 1 2 ) 2 + 4 F 1 2 F 2 2 ] 1 / 2 ( F 1 2 + F 1 2 ) } 1 / 2
t i = ( 1 2 K i ) ( 1 γ i ) exp [ ( α + j β ) l i ] ,
r i = 2 j K i 1 / 2 ( 1 K i ) 1 / 2 ( 1 γ i ) exp [ ( α + j β ) l i ] .
T i = ( 1 2 K i ) 2 ( 1 γ i ) 2 exp ( 2 α l i ) ,
R i = 4 K i ( 1 K i ) ( 1 γ i ) 2 exp ( 2 α l i ) ,
A i = 1 ( T i + R i ) .
E 1 t = E in t 1 exp [ ( α + j β ) l 2 ] + E 1 a r 1 exp [ 2 ( α + j β ) l 4 ] ,
E 1 a = r 2 E 1 t + t 2 E 2 t ,
E 2 t = E 2 a r 3 exp [ 2 ( α + j β ) l 4 ] ,
E 2 a = E 2 t r 2 + E 2 t t 2 ,
E out = E 2 a t 3 exp [ ( α + j β ) l 4 ] .
I out / I in = T 1 T 2 T 3 exp [ 2 α ( l 2 + l 4 ) ] / D ,
D = ( 1 + X 1 X 2 + X 2 X 3 + X 1 X 3 ) 2 4 X 1 X 2 ( 1 + X 3 2 ) × sin 2 ( β 1 ) 4 X 2 X 3 ( 1 + X 1 2 ) sin 2 ( β 2 ) 4 X 1 X 2 2 X 3 sin 2 [ β ( 1 2 ) ] 4 X 1 X 3 × sin 2 [ β ( 1 2 ) ] .
1 = ½ ( l 1 + 2 l 2 = l 3 ) ,
2 = ½ ( l 3 + 2 l 4 = l 5 ) .
X 1 = 2 K 1 1 / 2 ( 1 K 1 ) 1 / 2 ( 1 γ 1 ) ( 1 γ 2 ) exp ( 2 α 1 ) ,
X 2 = 2 K 2 1 / 2 ( 1 K 2 ) 1 / 2 ,
X 3 = 2 K 3 1 / 2 ( 1 K 3 ) 1 / 2 ( 1 γ 2 ) ( 1 γ 3 ) exp ( 2 α 2 ) .
I out / I in = t 1 2 t 4 2 ( 1 K ) ( 1 γ ) exp [ 2 α ( l 1 + l 4 ) ] / D ,
D = ( 1 R 1 R 4 + R 1 R 3 ) 2 + 4 R 1 R 4 sin 2 ( β 1 ) 4 R 1 R 3 sin 2 ( β 2 ) + 4 R 1 R 4 R 1 R 3 sin 2 [ β ( 1 2 ) ] .
R i = r i ( 1 K ) 1 / 2 ( 1 γ ) 1 / 2 exp ( 2 α l i ) = ρ i ( 1 K ) 1 / 2 i = 1 , 3 , 4 ,
R i = r i K 1 / 2 ( 1 γ ) 1 / 2 exp ( 2 α l i ) = ρ i K 1 / 2 , i = 1 , 3 , 4 .
1 = l 1 + l 4 ,
2 = l 1 + l 3 .
D = ( 1 R 1 R 4 + R 1 R 3 ) 2 + 4 ( R 1 R 4 + R 1 R 3 ) sin 2 ( β ) .
F I = π ( R 1 R 4 R 1 R 3 ) 1 / 2 ( 1 R 1 R 4 + R 1 R 3 )
K 0 = ρ 4 / ( ρ 3 + ρ 4 ) .
F II = π ( R 1 R 3 R 1 R 4 ) 1 / 2 ( 1 R 1 R 3 R 1 R 4 ) .
r 1 2 { ( 1 τ 1 R 1 R 4 + τ 1 R 1 R 3 ) 2 + 4 τ 1 R 1 R 4 sin 2 ( β 1 ) 4 τ 1 R 1 R 3 sin 2 ( β 2 ) + 4 τ 1 2 R 1 R 4 R 1 R 3 sin 2 [ β ( 1 2 ) ] }
T 1 2 { ( R 3 + R 4 ) 2 4 R 3 R 4 sin 2 [ β ( 1 2 ) ] } / t 1 2
T 1 T 3
[ ( 1 K ) ρ 3 K ρ 4 ρ 1 ρ 3 ρ 4 / ( 1 γ ) 1 / 2 ] 2 + 4 K ( 1 K ) ρ 3 ρ 4 sin 2 [ β ( 1 2 ) ] + [ 4 ( 1 K ) ρ 1 ρ 3 2 ρ 4 / ( 1 γ ) 1 / 2 ] sin 2 ( β 1 ) [ 4 K ρ 1 ρ 3 ρ 4 2 / ( 1 γ ) 1 / 2 ] sin 2 ( β 2 )
T 3 2 [ ( 1 ρ 1 ρ 4 ) 2 + 4 ρ 1 ρ 4 sin 2 ( β 1 ) ] / t 3 2
( T 4 ) 2 [ ( 1 + ρ 1 ρ 3 ) 2 4 ρ 1 ρ 3 sin 2 ( β 2 ) ] / t 4 2
r 3 2 { ( 1 + τ 3 R 1 R 3 R 1 R 4 ) 2 + 4 R 1 R 4 sin 2 ( β 1 ) 4 τ 3 R 1 R 3 sin 2 ( β 2 ) + 4 R 1 R 4 R 1 R 3 τ 3 sin 2 [ β ( 1 2 ) ] }
R 1 2 T 3 T 4
r 4 2 { ( 1 + R 1 R 3 τ 4 R 1 R 4 ) 2 + 4 R 1 R 4 τ 4 sin 2 ( β 1 ) 4 R 1 R 3 sin 2 ( β 2 ) + 4 R 1 R 4 R 1 R 3 τ 4 sin 2 [ β ( 1 4 ) ] }
E 1 a = ( 1 K ) 1 / 2 ( 1 γ ) 1 / 2 E 4 t + j K 1 / 2 ( 1 γ ) 1 / 2 E 3 t ,
E 2 a = ( 1 K ) 1 / 2 ( 1 γ ) 1 / 2 E 3 t + j K 1 / 2 ( 1 γ ) 1 / 2 E 4 t ,
E 3 a = ( 1 K ) 1 / 2 ( 1 γ ) 1 / 2 E 2 t + j K 1 / 2 ( 1 γ ) 1 / 2 E 1 t ,
E 4 a = ( 1 K ) 1 / 2 ( 1 γ ) 1 / 2 E 1 t + j K 1 / 2 ( 1 γ ) 1 / 2 E 2 t ,
E 1 t = E in t 1 exp [ ( α + j β ) l 1 ] + E 1 a r 1 exp [ 2 ( α + j β ) l 1 ] ,
E 2 t = 0 ,
E 3 t = E 3 a r 3 exp [ 2 ( α + j β ) l 3 ] ,
E 4 t = E 4 a r 4 exp [ 2 ( α + j β ) l 4 ] ,
E 4 out = E 4 a t 4 exp [ ( α + j β ) l 4 ] ,
T i = t i 2 ( 1 K ) 1 / 2 ( 1 γ ) 1 / 2 exp ( α l i ) i = 1 , 2 , 3 ,
T i = t i 2 K 1 / 2 ( 1 γ ) 1 / 2 exp ( α l i ) i = 1 , 3 , 4 ,
τ i = ( 1 r i 2 / t i 2 ) , i = 1 , 3 , 4.

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