Abstract

A theory of broadband matching for an optical heterodyne receiver consisting of a semiconductor photodiode followed by an IF broadband amplifier is presented in this paper. It is assumed that a finite linear lumped lossless interstage network is used for the broadband matching of the optical receiver, and the restrictions thereby imposed by the diode on the gain of the optical receiver are obtained both in integral and nonintegral forms. Several types of rational function approximations to an ideal flat gain characteristic of the optical receiver are then considered, and synthesis procedures for the lossless interstage networks to realize these approximations are presented. Explicit expressions are given for the amount of tolerance of broadband performance obtained with these networks.

© 1968 Optical Society of America

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References

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  1. L. K. Anderson, in Symposium on Optical Masers, J. Fox, Ed, (Polytechnic Press, Brooklyn, 1963).
  2. S. Saito, K. Kurokawa, Y. Fujii, T. Kamura, Y. Uno, Proc. Inst. Radio Eng. 50, 2369 (1962).
  3. H. S. Sommers, Proc. IEEE 51, 140 (1963).
    [Crossref]
  4. M. DiDomenico, O. Svelto, Proc. IEEE 52, 136 (1964).
    [Crossref]
  5. L. K. Anderson, Proc. IEEE 51, 846 (1963).
    [Crossref]
  6. R. P. Riesz, Rev. Sci. Instrum. 33, 994 (1962).
    [Crossref]
  7. H. Inaba, A. E. Seigman, Proc. Inst. Radio Eng. 50, 1823 (1962).
  8. L. U. Kibler, Proc. Inst. Radio Eng. 50, 1834 (1962).
  9. W. M. Sharpless, Proc. IEEE 52, 207 (1964).
    [Crossref]
  10. M. DiDomenico, W. M. Sharpless, J. J. McNicol, Appl. Opt. 4, 677 (1965).
    [Crossref]
  11. M. V. Schneider, Bell Syst. Tech. J. 45, 1611 (1966).
  12. P. Penfield, D. E. Sawyer, Proc. IEEE 53, 340 (1965).
    [Crossref]
  13. H. W. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand and Co., Inc., Princeton, 1945).
  14. R. M. Fano, J. Franklin Inst. 249, 139 (1950).
    [Crossref]
  15. D. C. Youla, IEEE Trans. CT-11, 33 (1964).
  16. Y. T. Chan, E. S. Kuh, IEEE Trans. CT-13, 6 (1966).
  17. V. K. Prabhu, Bell Syst. Tech. J. 47, 429 (1968).
  18. H. Fukui, IEEE Trans. CT-13, 137 (1966).
  19. V. K. Prabhu, C. L. Ruthroff, “Noise Factor of an Optical Heterodyne Receiver,” unpublished technical memorandum, Bell Telephone Labs., Inc., Holmdel, N. J.
  20. R. D. Rosner, Proc. IEEE 56, 126 (1968).
    [Crossref]
  21. F. B. Llewellyn, Proc. Inst. Radio Eng. 40, 271 (1952).
  22. D. C. Youla, Proc. Inst. Radio Eng. 48, 121 (1960).
  23. W. H. Ku, Proc. IEEE 54, 1617 (1966).
    [Crossref]
  24. H. J. Carlin, A. B. Giordano, Network Theory (Prentice-Hall, Inc., Englewood Cliffs, 1964).
  25. E. A. Guillemin, Synthesis of Passive Networks (John Wiley & Sons, Inc., New York, 1962).
  26. R. B. Emmons, G. Lucovsky, Proc. IEEE 52, 865 (1964).
    [Crossref]
  27. L. Weinberg, Network Analysis and Synthesis (McGraw-Hill Book Co., Inc., New York, 1962).
  28. M. V. Schneider, private communication, Bell Telephone Labs., Inc., Holmdel, N. J.
  29. N. Balabanian, Network Synthesis (Prentice-Hall, Inc., Englewood Cliffs, 1958).

1968 (2)

V. K. Prabhu, Bell Syst. Tech. J. 47, 429 (1968).

R. D. Rosner, Proc. IEEE 56, 126 (1968).
[Crossref]

1966 (4)

H. Fukui, IEEE Trans. CT-13, 137 (1966).

M. V. Schneider, Bell Syst. Tech. J. 45, 1611 (1966).

Y. T. Chan, E. S. Kuh, IEEE Trans. CT-13, 6 (1966).

W. H. Ku, Proc. IEEE 54, 1617 (1966).
[Crossref]

1965 (2)

1964 (4)

W. M. Sharpless, Proc. IEEE 52, 207 (1964).
[Crossref]

M. DiDomenico, O. Svelto, Proc. IEEE 52, 136 (1964).
[Crossref]

D. C. Youla, IEEE Trans. CT-11, 33 (1964).

R. B. Emmons, G. Lucovsky, Proc. IEEE 52, 865 (1964).
[Crossref]

1963 (2)

L. K. Anderson, Proc. IEEE 51, 846 (1963).
[Crossref]

H. S. Sommers, Proc. IEEE 51, 140 (1963).
[Crossref]

1962 (4)

S. Saito, K. Kurokawa, Y. Fujii, T. Kamura, Y. Uno, Proc. Inst. Radio Eng. 50, 2369 (1962).

R. P. Riesz, Rev. Sci. Instrum. 33, 994 (1962).
[Crossref]

H. Inaba, A. E. Seigman, Proc. Inst. Radio Eng. 50, 1823 (1962).

L. U. Kibler, Proc. Inst. Radio Eng. 50, 1834 (1962).

1960 (1)

D. C. Youla, Proc. Inst. Radio Eng. 48, 121 (1960).

1952 (1)

F. B. Llewellyn, Proc. Inst. Radio Eng. 40, 271 (1952).

1950 (1)

R. M. Fano, J. Franklin Inst. 249, 139 (1950).
[Crossref]

Anderson, L. K.

L. K. Anderson, Proc. IEEE 51, 846 (1963).
[Crossref]

L. K. Anderson, in Symposium on Optical Masers, J. Fox, Ed, (Polytechnic Press, Brooklyn, 1963).

Balabanian, N.

N. Balabanian, Network Synthesis (Prentice-Hall, Inc., Englewood Cliffs, 1958).

Bode, H. W.

H. W. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand and Co., Inc., Princeton, 1945).

Carlin, H. J.

H. J. Carlin, A. B. Giordano, Network Theory (Prentice-Hall, Inc., Englewood Cliffs, 1964).

Chan, Y. T.

Y. T. Chan, E. S. Kuh, IEEE Trans. CT-13, 6 (1966).

DiDomenico, M.

Emmons, R. B.

R. B. Emmons, G. Lucovsky, Proc. IEEE 52, 865 (1964).
[Crossref]

Fano, R. M.

R. M. Fano, J. Franklin Inst. 249, 139 (1950).
[Crossref]

Fujii, Y.

S. Saito, K. Kurokawa, Y. Fujii, T. Kamura, Y. Uno, Proc. Inst. Radio Eng. 50, 2369 (1962).

Fukui, H.

H. Fukui, IEEE Trans. CT-13, 137 (1966).

Giordano, A. B.

H. J. Carlin, A. B. Giordano, Network Theory (Prentice-Hall, Inc., Englewood Cliffs, 1964).

Guillemin, E. A.

E. A. Guillemin, Synthesis of Passive Networks (John Wiley & Sons, Inc., New York, 1962).

Inaba, H.

H. Inaba, A. E. Seigman, Proc. Inst. Radio Eng. 50, 1823 (1962).

Kamura, T.

S. Saito, K. Kurokawa, Y. Fujii, T. Kamura, Y. Uno, Proc. Inst. Radio Eng. 50, 2369 (1962).

Kibler, L. U.

L. U. Kibler, Proc. Inst. Radio Eng. 50, 1834 (1962).

Ku, W. H.

W. H. Ku, Proc. IEEE 54, 1617 (1966).
[Crossref]

Kuh, E. S.

Y. T. Chan, E. S. Kuh, IEEE Trans. CT-13, 6 (1966).

Kurokawa, K.

S. Saito, K. Kurokawa, Y. Fujii, T. Kamura, Y. Uno, Proc. Inst. Radio Eng. 50, 2369 (1962).

Llewellyn, F. B.

F. B. Llewellyn, Proc. Inst. Radio Eng. 40, 271 (1952).

Lucovsky, G.

R. B. Emmons, G. Lucovsky, Proc. IEEE 52, 865 (1964).
[Crossref]

McNicol, J. J.

Penfield, P.

P. Penfield, D. E. Sawyer, Proc. IEEE 53, 340 (1965).
[Crossref]

Prabhu, V. K.

V. K. Prabhu, Bell Syst. Tech. J. 47, 429 (1968).

V. K. Prabhu, C. L. Ruthroff, “Noise Factor of an Optical Heterodyne Receiver,” unpublished technical memorandum, Bell Telephone Labs., Inc., Holmdel, N. J.

Riesz, R. P.

R. P. Riesz, Rev. Sci. Instrum. 33, 994 (1962).
[Crossref]

Rosner, R. D.

R. D. Rosner, Proc. IEEE 56, 126 (1968).
[Crossref]

Ruthroff, C. L.

V. K. Prabhu, C. L. Ruthroff, “Noise Factor of an Optical Heterodyne Receiver,” unpublished technical memorandum, Bell Telephone Labs., Inc., Holmdel, N. J.

Saito, S.

S. Saito, K. Kurokawa, Y. Fujii, T. Kamura, Y. Uno, Proc. Inst. Radio Eng. 50, 2369 (1962).

Sawyer, D. E.

P. Penfield, D. E. Sawyer, Proc. IEEE 53, 340 (1965).
[Crossref]

Schneider, M. V.

M. V. Schneider, Bell Syst. Tech. J. 45, 1611 (1966).

M. V. Schneider, private communication, Bell Telephone Labs., Inc., Holmdel, N. J.

Seigman, A. E.

H. Inaba, A. E. Seigman, Proc. Inst. Radio Eng. 50, 1823 (1962).

Sharpless, W. M.

Sommers, H. S.

H. S. Sommers, Proc. IEEE 51, 140 (1963).
[Crossref]

Svelto, O.

M. DiDomenico, O. Svelto, Proc. IEEE 52, 136 (1964).
[Crossref]

Uno, Y.

S. Saito, K. Kurokawa, Y. Fujii, T. Kamura, Y. Uno, Proc. Inst. Radio Eng. 50, 2369 (1962).

Weinberg, L.

L. Weinberg, Network Analysis and Synthesis (McGraw-Hill Book Co., Inc., New York, 1962).

Youla, D. C.

D. C. Youla, IEEE Trans. CT-11, 33 (1964).

D. C. Youla, Proc. Inst. Radio Eng. 48, 121 (1960).

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

M. V. Schneider, Bell Syst. Tech. J. 45, 1611 (1966).

V. K. Prabhu, Bell Syst. Tech. J. 47, 429 (1968).

IEEE Trans. (3)

H. Fukui, IEEE Trans. CT-13, 137 (1966).

D. C. Youla, IEEE Trans. CT-11, 33 (1964).

Y. T. Chan, E. S. Kuh, IEEE Trans. CT-13, 6 (1966).

J. Franklin Inst. (1)

R. M. Fano, J. Franklin Inst. 249, 139 (1950).
[Crossref]

Proc. IEEE (8)

W. M. Sharpless, Proc. IEEE 52, 207 (1964).
[Crossref]

P. Penfield, D. E. Sawyer, Proc. IEEE 53, 340 (1965).
[Crossref]

H. S. Sommers, Proc. IEEE 51, 140 (1963).
[Crossref]

M. DiDomenico, O. Svelto, Proc. IEEE 52, 136 (1964).
[Crossref]

L. K. Anderson, Proc. IEEE 51, 846 (1963).
[Crossref]

R. D. Rosner, Proc. IEEE 56, 126 (1968).
[Crossref]

R. B. Emmons, G. Lucovsky, Proc. IEEE 52, 865 (1964).
[Crossref]

W. H. Ku, Proc. IEEE 54, 1617 (1966).
[Crossref]

Proc. Inst. Radio Eng. (5)

F. B. Llewellyn, Proc. Inst. Radio Eng. 40, 271 (1952).

D. C. Youla, Proc. Inst. Radio Eng. 48, 121 (1960).

S. Saito, K. Kurokawa, Y. Fujii, T. Kamura, Y. Uno, Proc. Inst. Radio Eng. 50, 2369 (1962).

H. Inaba, A. E. Seigman, Proc. Inst. Radio Eng. 50, 1823 (1962).

L. U. Kibler, Proc. Inst. Radio Eng. 50, 1834 (1962).

Rev. Sci. Instrum. (1)

R. P. Riesz, Rev. Sci. Instrum. 33, 994 (1962).
[Crossref]

Other (8)

L. K. Anderson, in Symposium on Optical Masers, J. Fox, Ed, (Polytechnic Press, Brooklyn, 1963).

H. W. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand and Co., Inc., Princeton, 1945).

V. K. Prabhu, C. L. Ruthroff, “Noise Factor of an Optical Heterodyne Receiver,” unpublished technical memorandum, Bell Telephone Labs., Inc., Holmdel, N. J.

L. Weinberg, Network Analysis and Synthesis (McGraw-Hill Book Co., Inc., New York, 1962).

M. V. Schneider, private communication, Bell Telephone Labs., Inc., Holmdel, N. J.

N. Balabanian, Network Synthesis (Prentice-Hall, Inc., Englewood Cliffs, 1958).

H. J. Carlin, A. B. Giordano, Network Theory (Prentice-Hall, Inc., Englewood Cliffs, 1964).

E. A. Guillemin, Synthesis of Passive Networks (John Wiley & Sons, Inc., New York, 1962).

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Figures (11)

Fig. 1
Fig. 1

A double detection optical receiver. The input to the photodetector is the sum of the local oscillator beam and the incoming signal beam.

Fig. 2
Fig. 2

Equivalent circuit of photodiode. The physical sources of noise that may be present in the diode are not shown. Also the conductance Gp that appears in parallel with C is usually so small (of the order of 10−7 mho) that it can be neglected for all practical purposes if ω ≠ 0.

Fig. 3
Fig. 3

IF amplifier characterized by its admittance parameters.

Fig. 4
Fig. 4

IF amplifier driven by a current source with an internal admittance Ys.

Fig. 5
Fig. 5

Lossless equalizer used in broadband matching of the optical receiver. Y(p) is the admittance of the photodiode as shown in Fig. 2.

Fig. 6
Fig. 6

Normalized available output power Poa/ϕoR as a function of ω for Butterworth approximations of order n = 1,2. It is assumed that fc = ωc/2π = 31.83 GHz, and 2 = 1.

Fig. 7
Fig. 7

A typical plot of 2min/(4RfGogGamax) as a function of n when Butterworth approximations to a flat gain characteristic are used. Even though n is a discrete variable, the plot is given for all n ≥ 1.

Fig. 8
Fig. 8

Lossless interstage network for a Butterworth approximation of order n = 1. The ideal transformer ratio t is given by t = ( R G o g ) 1 2.

Fig. 9
Fig. 9

Normalized available output power PoaoR as a function of ω for Chebyshev approximations of order n = 1,2. It is assumed that fc = ωc/2π = 31.83 GHz, and 2 = 1.

Fig. 10
Fig. 10

A typical plot of min2/(4RfGogGamax)as a function of n when Chebyshev approximations to a flat gain characteristic are used. Even though n is a discrete variable the plot is given for all n ≥ 1.

Fig. 11
Fig. 11

Lossless interstage network for a Chebyshev approximation of order n = 2. The ideal transformer ratio t is given by t = ( R G o g ) 1 2.

Equations (104)

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ω 0 = [ ( Ω 0 - W ) ( Ω 0 + W ) ] 1 2 .
I s = 2 η q / h ν ( P 0 P s ) 1 2 ,
S a d = I s 2 / 8 ω 2 C 2 R .
[ I 1 I 2 ] = [ y 11 y 12 y 21 y 22 ] [ V 1 V 2 ] .
G a = S 0 a / S i a
S i a = I s 2 / 2 ( Y s + Y s * ) .
G a = y 21 2 ( Y s + Y s * ) ( y 22 + y 22 * ) y 11 + Y s 2 - Re [ y 12 y 21 ( y 11 * + Y s * ) ] .
G a max = y 21 / y 12 1 / [ λ + ( λ 2 - 1 ) 1 2 ] ,
G 0 g = [ y 12 y 21 / ( y 22 + y 22 * ) ] ( λ 2 - 1 ) 1 2
B 0 g = - Im y 11 + Im ( y 12 y 21 ) / ( y 11 + y 11 * ) ,
λ = 2 Re ( y 11 ) Re ( y 22 ) - Re ( y 12 y 21 ) y 12 y 21 .
λ 1.
G 0 g 0.
( 1 / G a ) = ( 1 / G a max ) + ( R f / G s ) Y s - Y 0 g 2 ,
R f = Re y 22 / y 21 2 .
( G s - G h ) 2 + ( B s - B h ) 2 = G t ,
P 0 max = 1 2 G a max ( η q / h ν ) 2 P 0 P s / ω 2 C 2 R ,
P 0 max = Φ 0 R ( ω c / ω ) 2 ,
ω c = 1 / R C ,
Φ 0 = 1 2 G a max ( η q / h ν ) 2 P 0 P s .
S = [ s 11 s 12 s 21 s 22 ]
ρ 1 ( p ) = [ Y 1 ( p ) - Y ( - p ) ] / [ Y 1 ( p ) + Y ( p ) ] ,
ρ 2 ( p ) = [ Y 2 ( p ) - 1 ] / [ Y 2 ( p ) + 1 ] ,
ρ 1 ( j ω ) = ρ 2 ( j ω ) .
1 ρ 1 ( j ω ) 2 = 1 ρ 2 ( j ω ) 2 = 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) .
β ( p ) = r = 1 m ( p - α r ) / ( p + α r * ) .
β ( p ) { 1 - ρ 1 ( p ) } = β ( p ) [ Y ( p ) + Y ( - p ) ] / [ Y 1 ( p ) + Y ( p ) ] .
ζ ( p ) = β ( p ) [ 1 - ρ 1 ( p ) ] .
s ( p ) = β ( p ) ρ 1 ( p ) = k = 0 S k ( p - p 0 ) k ,
ln s ( p ) = k = 0 s k ( p - p 0 ) k ,
β ( p ) = k = 0 B k ( p - p 0 ) k ,
ln β ( p ) = k = 0 b k ( p - p 0 ) k .
F ( p ) = β ( p ) [ Y ( p ) + Y ( - p ) ] = k = 0 F k ( p - p 0 ) k ,
1 π p p 2 + ω 2 + k = 0 f k ( p - p 0 ) k ,
g ( p ) = F ( p ) 2 β ( p ) = k = 0 θ k ( p - p 0 ) k .
η ( p ) = l = 1 ν p - μ l p + μ l * ,
ln η ( p ) = k = 0 ν η k ( p - p 0 ) k .
1 s 0 ( p ) s 0 ( - p ) = 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ,
β ( p ) ρ 1 ( p ) = s ( p ) = η ( p ) s 0 ( p ) ,
1 s ( j ω ) 2 = 1 ρ 1 ( j ω ) 2 = 1 s 0 ( j ω ) 2 = 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) .
S r = B r , 0 r k - 1 ;
b 0 = π j + η 0 - 0 f 0 ln [ 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ] d ω , = 0 , if d d ω [ 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ] 0 , = 1 , if d d ω [ 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ] = 0 ,
b r = η r - 0 f r ln [ 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ] d ω ,             1 r k - 1.
S r = B r ,             0 r k - 1 ,
( S k - B k ) / ( F k + 1 ) 0 ;
b r = η r - 0 f r ln [ 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ] d ω ,             0 r k - 1.
( b k - s k ) / ( θ k + 1 ) 0.
b k - η k + 0 f k ln [ 1 + 4 R f G 0 g 1 / G a - ( 1 / G a max ) ] d ω θ k + 1 0.
S r = B r ,             0 r k - 2 ,
( S k - 1 - B k - 1 ) / F k 0 ;
b r = η r - 0 f r ln [ 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ] d ω ,             0 r k - 1 ,
( b k - 1 - s k - 1 ) / θ k 0 ,
b k - 1 - η k - 1 + 0 f r ln [ 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ] d ω θ k 0.
S r = B r ,             0 r k - 1 ,
( F k - 1 ) / ( S k - B k ) a - 1 ,
b r = η r - 0 f r ln [ 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ] d ω ,             0 r k - 1 ,
( 2 θ k - 1 ) / ( b k - s k ) a - 1 ,
2 θ k - 1 b k - η k + 0 f k ln [ 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ] d ω a - 1 .
Y ( p ) = ( 1 / R G 0 g ) ( p / p + ω c ) .
β ( p ) = ( p - ω c ) / ( p + ω c )
= - 1 + ( 2 p / ω c ) - ( 2 p 2 / ω c 2 ) + .
F ( p ) = 1 G 0 g 2 p 2 C ω c ( 1 + p / ω c ) 2
= 1 G 0 g [ 2 C p 2 ω c - 4 C p 3 ω c 2 + ] .
S 0 = - 1 ,
S 1 2 / ω c ,
s ( p ) = ± η ( p ) s 0 ( p ) ,
s 0 ( p ) s 0 ( - p ) = [ 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ] - 1 .
1 π 0 1 ω 2 ln [ 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ] d ω 2 ω c - l = 1 v μ l μ l * ( 1 μ l + 1 μ l * ) ,
1 π 0 1 ω 2 ln [ 1 + 4 R f G 0 g ( 1 / G a ) - ( 1 / G a max ) ] d ω 2 ω c .
1 π 0 1 ω 2 ln [ 1 + ( h ν η q ) 2 8 R f G 0 g R P 0 P s ( ω / ω c ) 2 ( 1 / P 0 a ) - ( 1 / P 0 max ) ] d ω 2 ω c .
P 0 a = P 0 max 1 1 + 2 [ ( ω 2 - ω 0 2 ) / 2 ω W ] 2 n ,
s 0 ( p ) s 0 ( - p ) = 1 1 + 4 R f G 0 g G a max 2 [ ( ω 2 - ω 0 2 ) / 2 ω W ] 2 n .
s 0 ( p ) = ± ( p 2 + ω 0 2 ) n ( p 2 + ω 0 2 ) n + a n - 1 ( 2 p W ) ( p 2 + ω 0 2 ) n - 1 + + a 0 ( 2 p W ) n
a n - 1 = ( 4 R f G 0 g G a max / 2 ) 1 / 2 n sin ( π / 2 n ) .
s 0 ( p ) = - 1 + a n - 1 ( 2 W / ω 0 2 ) p - .
η ( p ) = i = 1 ν p - μ i p + μ i *
= ( - 1 ) ν [ 1 - p l = 1 ν { 1 μ l + 1 μ l * } μ l μ l * + ] .
a n - 1 2 W ω 0 2 + l = 1 ν μ l μ l * { 1 μ l + 1 μ l * } 2 ω c .
2 4 R f G 0 g G a max [ ( ω 0 / ω c ) ( ω 0 / W ) sin ( π / 2 n ) ] 2 n .
min 2 = 4 R f G 0 g G a max [ ( ω 0 / ω c ) ( ω 0 / W ) ] 2 ,
P 0 a = Φ 0 R ( ω c ω ) 2 1 1 + 2 [ ( ω 2 - ω 0 2 ) / 2 ω W ] 2 ,
s ( p ) = s 0 ( p ) = - p 2 + ω 0 2 p 2 + 2 p ( ω 0 2 / ω c ) + ω 0 2 .
Y 1 ( p ) = 1 R G 0 g ω 0 2 / ω c p + ω 0 2 / ω c .
L = 1 C ( Ω 0 - W ) ( Ω 0 + W ) ,
n = ( G 0 g R ) 1 2 .
P 0 a = P 0 max 1 1 + 2 T n 2 [ ( ω 2 - ω 0 2 ) / 2 ω W ] ,
T n ( x ) = cos ( n cos - 1 x ) .
s 0 ( p ) = - ( p 2 + ω 0 2 ) n + b n - 2 ( 2 p W ) 2 ( p 2 + ω 0 2 ) n - 2 + ( p 2 + ω 0 2 ) n + a n - 1 ( 2 p W ) ( p 2 + ω 0 2 ) n - 1 + ,
a n - 1 = sinh { ( 1 / n ) sinh - 1 [ 2 ( R f G 0 g G a max ) 1 2 / ] } sin π / 2 n .
s 0 ( p ) = - 1 + p a n - 1 ( 2 W / ω 0 2 ) + .
min 2 = 4 R f G 0 g G a max sinh 2 { n sinh - 1 [ ( ω 0 / ω c ) ( ω 0 / W ) sin ( π / 2 n ) ] } .
min 2 = 16 R f G 0 g G a max π 2 ( ω 0 / ω c ) ( ω 0 / W ) [ ( sin π / 2 n ) / ( π / 2 n ) ] 2 .
[ min 2 ] n = 1 [ min 2 ] n = = π 2 4 2.5 ,
[ min 2 ] n = 2 [ min 2 ] n = π 2 8 1.25.
min 2 = 2 R f G 0 g G a max ( ω c / ω 0 ) 2 ( W / ω 0 ) 2 ,
S o ( p ) = - ( p 2 + ω 0 2 ) 2 + 1 2 ( 2 p W ) 2 ( p 2 + ω 0 2 ) 2 + ( ω 0 / ω c ) ( ω 0 / W ) ( 2 p W ) ( p 2 + ω 0 2 ) + 1 2 ( 2 p W ) 2 .
Y 1 ( p ) = 1 R G 0 g ω 0 2 ω c p 2 + ω 0 2 p 3 + p 2 ( ω 0 2 / ω c ) + p ( ω 0 2 + 2 W 2 ) + ( ω 0 4 / ω c ) .
0 < K 1.
P 0 a = P 0 max K 1 + [ ( ω 2 - ω 0 2 ) / 2 ω W ] 2 n ,
P 0 a = P max K 1 + 2 T n 2 [ ( ω 2 - ω 0 2 ) / ( 2 ω W ) ] ,
[ P 0 a ] max = K P 0 max P 0 max .
( 1 - K + 4 K R f G 0 g G a max ) / 2 n 1 - ( 1 - K ) / 2 n 1 ( ω 0 / ω c ) ( ω 0 / W ) sin ( π / 2 n ) .
sinh [ 1 n sinh - 1 ( 1 - K + 4 K R f G 0 g G a max ) 1 2 ] - sinh [ ( 1 / n ) sinh - 1 ( 1 - K ) 1 2 / ] ( ω 0 / ω c ) ( ω 0 / W ) sin π / 2 n .
1 π 0 1 ω 2 ln [ 1 + ( h ν η q ) 2 8 R f G 0 g R P 0 P s ( ω / ω c ) 2 ( 1 / P 0 a ) - ( 1 / P 0 max ) ] × d ω 2 / ω c .

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