Abstract

A second-quantized theory of the radiation field is used to study the origin of the excess noise observed in gain-guided amplifiers. We find that the reduction of the signal-to-noise ratio is a function of the length of the amplifier, and thus the enhancement of the noise is a propagation effect arising from longitudinally inhomogeneous gain of the noise rather than from an excess of local spontaneous emission. We confirm this conclusion by showing that the microscopic rate of spontaneous emission into a given non-power-orthogonal cavity mode is not enhanced by the Petermann factor. In addition, we illustrate the difficulties associated with photon statistics for this and other open systems by showing that no acceptable family of photon-number operators corresponds to a set of non-power-orthogonal cavity modes.

© 1991 Optical Society of America

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References

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  1. See, for example, A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif.1986).
  2. K. Petermann, IEEE J. Quantum Electron. QE-15, 566 (1979).
    [CrossRef]
  3. H. A. Haus and S. Kawakami, IEEE J. Quantum Electron. QE-21, 63 (1985); see also references therein.
    [CrossRef]
  4. C. H. Henry, IEEE J. Lightwave Technol. LT-4, 288 (1986).
    [CrossRef]
  5. A. E. Siegman, Phys. Rev. A 39, 1252 (1989).
  6. A. E. Siegman, Phys. Rev. A 39, 1264 (1989).
    [CrossRef] [PubMed]
  7. W. A. Hamel and J. P. Woerdman, Phys. Rev. A 40, 2785 (1989).
    [CrossRef] [PubMed]
  8. W. A. Hamel and J. P. Woerdman, Phys. Rev. A 64, 1506 (1990).
  9. I. H. Deutsch and J. C. Garrison, Phys. Rev. A 43, 2498 (1991).
    [CrossRef] [PubMed]
  10. J. C. Garrison, H. Nathel, and R. Y. Chiao, J. Opt. Soc. Am. B 5, 1528 (1988).
    [CrossRef]
  11. T. A. B. Kennedy and E. M. Wright, Phys. Rev. A 38, 212 (1988).
    [CrossRef] [PubMed]
  12. I. H. Deutsch, “A basis independent approach to quantum optics,” submitted to Am. J. Phys. This paper gives a discussion of generalized creation and annihilation operators as they apply to quantum optics. For complete coverage see R. P. Feynman, Statistical Mechanics: A Set of Lectures (Addison-Wesley, Reading, Mass.1972), pp. 167–176; see also J. T. Lewis, “The free boson gas,” in Mathematics of Contemporary Physics, R. F. Streater, ed. (Academic, London, 1972), pp. 213–214.
  13. L. Mandel, Phys. Rev. 144, 1071 (1966).
    [CrossRef]
  14. H. Kogelnik, Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  15. H. Haken, Laser Theory (Springer-Verlag, Berlin, 1984).

1991 (1)

I. H. Deutsch and J. C. Garrison, Phys. Rev. A 43, 2498 (1991).
[CrossRef] [PubMed]

1990 (1)

W. A. Hamel and J. P. Woerdman, Phys. Rev. A 64, 1506 (1990).

1989 (3)

A. E. Siegman, Phys. Rev. A 39, 1252 (1989).

A. E. Siegman, Phys. Rev. A 39, 1264 (1989).
[CrossRef] [PubMed]

W. A. Hamel and J. P. Woerdman, Phys. Rev. A 40, 2785 (1989).
[CrossRef] [PubMed]

1988 (2)

1986 (1)

C. H. Henry, IEEE J. Lightwave Technol. LT-4, 288 (1986).
[CrossRef]

1985 (1)

H. A. Haus and S. Kawakami, IEEE J. Quantum Electron. QE-21, 63 (1985); see also references therein.
[CrossRef]

1979 (1)

K. Petermann, IEEE J. Quantum Electron. QE-15, 566 (1979).
[CrossRef]

1966 (1)

L. Mandel, Phys. Rev. 144, 1071 (1966).
[CrossRef]

1965 (1)

Chiao, R. Y.

Deutsch, I. H.

I. H. Deutsch and J. C. Garrison, Phys. Rev. A 43, 2498 (1991).
[CrossRef] [PubMed]

I. H. Deutsch, “A basis independent approach to quantum optics,” submitted to Am. J. Phys. This paper gives a discussion of generalized creation and annihilation operators as they apply to quantum optics. For complete coverage see R. P. Feynman, Statistical Mechanics: A Set of Lectures (Addison-Wesley, Reading, Mass.1972), pp. 167–176; see also J. T. Lewis, “The free boson gas,” in Mathematics of Contemporary Physics, R. F. Streater, ed. (Academic, London, 1972), pp. 213–214.

Garrison, J. C.

Haken, H.

H. Haken, Laser Theory (Springer-Verlag, Berlin, 1984).

Hamel, W. A.

W. A. Hamel and J. P. Woerdman, Phys. Rev. A 64, 1506 (1990).

W. A. Hamel and J. P. Woerdman, Phys. Rev. A 40, 2785 (1989).
[CrossRef] [PubMed]

Haus, H. A.

H. A. Haus and S. Kawakami, IEEE J. Quantum Electron. QE-21, 63 (1985); see also references therein.
[CrossRef]

Henry, C. H.

C. H. Henry, IEEE J. Lightwave Technol. LT-4, 288 (1986).
[CrossRef]

Kawakami, S.

H. A. Haus and S. Kawakami, IEEE J. Quantum Electron. QE-21, 63 (1985); see also references therein.
[CrossRef]

Kennedy, T. A. B.

T. A. B. Kennedy and E. M. Wright, Phys. Rev. A 38, 212 (1988).
[CrossRef] [PubMed]

Kogelnik, H.

Mandel, L.

L. Mandel, Phys. Rev. 144, 1071 (1966).
[CrossRef]

Nathel, H.

Petermann, K.

K. Petermann, IEEE J. Quantum Electron. QE-15, 566 (1979).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Phys. Rev. A 39, 1252 (1989).

A. E. Siegman, Phys. Rev. A 39, 1264 (1989).
[CrossRef] [PubMed]

See, for example, A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif.1986).

Woerdman, J. P.

W. A. Hamel and J. P. Woerdman, Phys. Rev. A 64, 1506 (1990).

W. A. Hamel and J. P. Woerdman, Phys. Rev. A 40, 2785 (1989).
[CrossRef] [PubMed]

Wright, E. M.

T. A. B. Kennedy and E. M. Wright, Phys. Rev. A 38, 212 (1988).
[CrossRef] [PubMed]

Appl. Opt. (1)

IEEE J. Lightwave Technol. (1)

C. H. Henry, IEEE J. Lightwave Technol. LT-4, 288 (1986).
[CrossRef]

IEEE J. Quantum Electron. (2)

K. Petermann, IEEE J. Quantum Electron. QE-15, 566 (1979).
[CrossRef]

H. A. Haus and S. Kawakami, IEEE J. Quantum Electron. QE-21, 63 (1985); see also references therein.
[CrossRef]

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

Phys. Rev. (1)

L. Mandel, Phys. Rev. 144, 1071 (1966).
[CrossRef]

Phys. Rev. A (6)

T. A. B. Kennedy and E. M. Wright, Phys. Rev. A 38, 212 (1988).
[CrossRef] [PubMed]

A. E. Siegman, Phys. Rev. A 39, 1252 (1989).

A. E. Siegman, Phys. Rev. A 39, 1264 (1989).
[CrossRef] [PubMed]

W. A. Hamel and J. P. Woerdman, Phys. Rev. A 40, 2785 (1989).
[CrossRef] [PubMed]

W. A. Hamel and J. P. Woerdman, Phys. Rev. A 64, 1506 (1990).

I. H. Deutsch and J. C. Garrison, Phys. Rev. A 43, 2498 (1991).
[CrossRef] [PubMed]

Other (3)

I. H. Deutsch, “A basis independent approach to quantum optics,” submitted to Am. J. Phys. This paper gives a discussion of generalized creation and annihilation operators as they apply to quantum optics. For complete coverage see R. P. Feynman, Statistical Mechanics: A Set of Lectures (Addison-Wesley, Reading, Mass.1972), pp. 167–176; see also J. T. Lewis, “The free boson gas,” in Mathematics of Contemporary Physics, R. F. Streater, ed. (Academic, London, 1972), pp. 213–214.

See, for example, A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif.1986).

H. Haken, Laser Theory (Springer-Verlag, Berlin, 1984).

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Figures (1)

Fig. 1
Fig. 1

Evolution of ln[ N 0 (z)], where N 0 is the number of photons per mode, according to Eqs. (3.32 (dashed line) and (3.33) (solid curve). We have chosen a parabolic gain profile for which K0 = 21/2 and have set N1 = 0.

Equations (65)

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[ a n , a m ] = [ a n , a m ] = 0 ,             [ a n , a m ] = δ n m ,
E ( x ) = w 0 ϕ ( x ) ,
w 0 2 = 2 ω / ɛ ,
[ ϕ ( x , t ) , ϕ ( x , t ) ] = δ ( x - x ) ,             [ ϕ ( x , t ) , ϕ ( x , t ) ] = 0.
a [ f ] d 3 x f ( x ) ϕ ( x )
[ a [ f ] , a [ g ] ] = ( f , g ) ,
N [ f ] = a [ f ] a [ f ] f 2 ,
[ N [ f ] , a [ g ] ] = ( f , g ) f 2 a [ f ]
[ N [ f ] , a [ f ] ] = a [ f ] ,
f , n = 1 ( n ! ) 1 / 2 ( a [ f ] f ) n 0
N [ f ] f , n = n f , n .
[ N [ f ] , N [ g ] ] = ( f , g ) f 2 g 2 a [ f ] a [ g ] - H . c .
e ¯ m ( x T , z ) = { 1 [ Δ L ( z ) ] 1 / 2 e m ( x T ) z Δ L ( z ) 0 otherwise , Δ L ( z ) = ( z - Δ L / 2 , z + Δ L / 2 ) ,
N 0 ( z ) = a [ e ¯ 0 ] a [ e ¯ 0 ] .
[ T 2 + 2 i k z - i k G ( x T ) ] ϕ ( x ) = - 2 k 2 Γ ( x ) .
Γ ( x ) Γ ( x ) = n 0 B k 2 c N 2 N 2 - N 1 G ( x T ) δ ( x - x ) ,
G ( x T ) = σ ( x T ) ( N 2 - N 1 ) .
[ T 2 + 2 i k z - i k G ( x T ) ] K ( x T , z ; x T , z ) = 0.
K ( x T , z ; x T , z ) = δ ( x T - x T ) .
ϕ ( x ) = d 2 s K ( x T , z ; s , 0 ) ϕ ( s , 0 ) + i k 0 z d ζ d 2 s K ( x T , z ; s , ζ ) Γ ( s , ζ ) ,
ϕ ( x ) = d 2 s K ( x T , z ; s , 0 ) ϕ ( s , 0 ) - i k 0 z d ζ d 2 s K ( x T , z ; s , ζ ) Γ ( s , ζ ) ,
K ( x T , z ; s , ζ ) K * ( x T , z ; s , ζ ) .
N 0 ( z ) = a [ e ¯ 0 ] a [ e ¯ 0 ] = a [ u ¯ 0 ] a [ u ¯ 0 ] = Δ L d 2 x T d 2 x T u 0 ( x T ) u 0 * ( x T ) × ϕ ( x T , z ) ϕ ( x T , z ) ,
ϕ ( x T , z ) ϕ ( x T , z ) = k 2 0 z d ζ d 2 s 0 z d ζ d 2 s K ( x T , z ; s , ζ ) K ( x T , z ; s , ζ ) Γ ( s , ζ ) Γ ( s , ζ ) , = n 0 B c N 2 N 2 - N 1 0 z d ζ d 2 s K ( x T , z ; s , ζ ) G 0 ( s ) K ( x T , z ; s , ζ ) .
d 2 s K ( x T , z ; s , ζ ) G 0 ( s ) K ( x T , z ; s , ζ ) = - d d ζ d 2 s K ( x T , z ; s , ζ ) K ( x T , z ; s , ζ ) ,
ϕ ( x T , z ) ϕ ( x T , z ) = n 0 B c N 2 N 2 - N 1 × [ d 2 s K * ( x T , z ; s , 0 ) K ( x T , z ; s , 0 ) - δ ( x T - x T ) ] .
N 0 ( z ) = n 0 Δ L B c N 2 N 2 - N 1 × [ d 2 x T | d 2 s K ( x T , z ; s , 0 ) u 0 * ( s ) | 2 - 1 ] .
N 0 ( z ) = N 2 N 2 - N 1 [ d 2 x T | d 2 s K ( x T , z ; s , 0 ) u 0 * ( s ) | 2 - 1 ] ,
N 0 N 2 N 1 - N 2 = 1 exp ( ω / k B T ) - 1 .
N 0 ( z ) = N 2 N 2 - N 1 [ exp ( g 0 z ) - 1 ] ,
N 0 ( z ) = N 2 N 2 - N 1 exp ( g 0 z ) .
[ 1 2 k T 2 - i G ( x T ) ] u n ( x T ) = δ n u n ( x T ) ,
[ 1 2 k T 2 + i G ( x T ) ] v n ( x T ) = δ n * v n ( x T ) ,
d 2 x T v n * ( x T ) u m ( x T ) = δ n m ,
d 2 x T u n ( x T ) 2 = 1 .
d 2 x T v n ( x T ) 2 = K n > 1 .
v n ( x T ) = K n 1 / 2 u n * ( x T ) .
K ( x T , z ; s , 0 ) = m exp ( i δ m z ) u m ( x T ) v m * ( s ) .
K ( x T , z ; s , 0 ) K 0 1 / 2 exp ( i δ 0 z ) u 0 ( x T ) u 0 ( s ) ,
N 0 ( z ) = K 0 N 2 N 2 - N 1 exp ( g 0 z ) ,
N 0 ( z ) = N 2 N 2 - N 1 d 2 x T U ( x T , z ) 2 ,
U ( x T , z ) = d 2 s K ( x T , z ; s , 0 ) u 0 * ( s ) .
i = e 0 .
f = g a [ u ¯ 0 ] 0 .
dW f i = 2 π f H int i 2 δ ( k c / n 0 - ν eg ) Δ L d k 2 π ,
H int = - μ w 0 2 [ ϕ ( x ) exp ( i k z ) + ϕ ( x ) exp ( - i k z ) ] ,
f H int i = - μ eg ( w 0 / 2 ) 0 [ a [ u ¯ 0 ] , ϕ ( x ) ] 0 exp ( - i k z ) = - μ eg ( w 0 / 2 ) u ¯ 0 * ( x T , z ) exp ( - i k z ) .
W f i = W sp u n ( x T ) 2 ,
W sp = μ eg 2 ( n 0 ω / c ) ɛ .
F ( x ) = n C n ( z ) u n ( x T ) .
f = ( 1 / A f ) g Δ L ( z ) d z c n ( z ) 0 ,
Δ L ( z ) d z c n ( z ) = a [ v ¯ n ] .
m V m n u m ( x T ) = v n ( x T ) ,             V m n d 2 x T v m * ( x T ) v n ( x T )
W f i = K n W sp u n ( x ) 2 + W sp l , m n V ln V nm K n u l ( x ) u m * ( x ) .
W f i = W sp v n ( x ) 2 K n .
[ T 2 + 2 i k ( z + n 0 c t ) - i k G ( x T ) ] × E ( x , t ) = - μ 0 ω 2 P N ( x , t ) ,
[ T 2 + 2 i k ( z + n 0 c t ) - i k G ( x T ) ] ϕ ( x , t ) = - 2 k 2 Γ ( x , t ) ,
P N ( x , t ) = ( 8 ω ɛ ) 1 / 2 Γ ( x , t ) .
[ ϕ ( x , t ) , ϕ ( x , t ) ] = δ ( x - x ) ,
[ Γ ( x , t ) , Γ ( x , t ) ] = - n 0 k 2 c G ( x T ) δ ( x - x ) δ ( t - t ) .
ϕ ( x ) = B - 1 / ( 2 B ) 1 / ( 2 B ) d t ϕ ( x , t ) .
[ T 2 + 2 i k z - i k G ( x T ) ] ϕ ( x ) = - 2 k 2 Γ ( x ) ,
[ Γ ( x ) , Γ ( x ) ] = - n 0 B k 2 c G ( x T ) δ ( x - x ) .
Γ ( x ) Γ ( x ) = n 0 B k 2 c N 2 N 2 - N 1 G ( x T ) δ ( x - x ) ,
Γ ( x ) Γ ( x ) = n 0 B k 2 c N 1 N 2 - N 1 G ( x T ) δ ( x - x ) ,

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