Abstract

An optical communication channel is analyzed, in which a light beam is amplitude modulated at the source by a filter of continuously variable transmittance, and the detector counts the received photons. Such a communication channel has intrinsic noise limitations because there is not a one-to-one correspondence between the modulated beam power and the number of counts registered. The information rates achievable with single-mode and multimode lasers are evaluated as functions of the mean number N of detected photons per symbol, for several different input statistics. For large N the information rate increases logarithmically with N. It is shown that, when the symbol length is short, there is a minimum number of independent modes for which the multimode laser gives a greater information rate than the single-mode laser, if the laser power is equally divided among all the modes, and the power per mode is regarded as constant. However, for even moderate numbers of detected photons per symbol, this minimum number of modes is so great that the single-mode laser is to be preferred. When the light beam is derived from a thermal source, the information rate in the channel is, in effect, governed by the same equations as those for the single-mode laser, so long as the detector area is limited to a coherence area.

© 1971 Optical Society of America

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References

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  1. D. Gabor, Phil. Mag. 41, 1161 (1950).
  2. T. E. Stern, Trans. IRE IT-6, 435 (1960).
  3. J. P. Gordon, in Advances in Quantum Electronics, edited by J. R. Singer (Columbia University Press, New York, 1961), p. 509.
  4. G. J. Lasher, in Ref. 3, p. 520.
  5. J. P. Gordon, Proc. IRE 50, 1898 (1962).
    [Crossref]
  6. J. P. Gordon, in Quantum Electronics and Coherent Light, edited by C. H. Townes and P. A. Miles (Academic, New York, 1962), p. 156.
  7. L. P. Bolgiano and L. F. Jelsma, Proc. IEEE 52, 218 (1964).
    [Crossref]
  8. B. E. Goodwin and L. P. Bolgiano, Proc. IEEE 53, 1745 (1965).
    [Crossref]
  9. G. Toraldo di Francia, Opt. Acta 2, 5 (1955).
    [Crossref]
  10. R. C. Jones, J. Opt. Soc. Am. 50, 1166 (1960);J. Opt. Soc. Am. 52, 493 (1962).
    [Crossref]
  11. E. Hisdal, J. Opt. Soc. Am. 55, 1446 (1965);J. Opt. Soc. Am. 57, 35 (1967)J. Opt. Soc. Am. 59, 921 (1969).
    [Crossref]
  12. G. L. Fillmore and G. Lachs, IEEE Trans. IT-15, 465 (1969).
  13. C. E. Shannon, Bell System Tech. J. 27, 376, 623 (1948).
  14. See, for example, J. F. Kenney and E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, New York, 1951), Pt. II, Ch. 1.
  15. W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
    [Crossref]
  16. J. P. Gordon, W. H. Louisell, and L. R. Walker, Phys. Rev. 129, 481 (1963).
    [Crossref]
  17. B. R. Mollow and R. J. Walker, Phys. Rev. 160, 1076, 1097 (1967).
    [Crossref]
  18. L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958).
    [Crossref]
  19. L. Mandel, in Progress in Optics, II, edited by E. Wolf (North-Holland, Amsterdam, 1963), p. 181.
    [Crossref]
  20. L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
    [Crossref]
  21. P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
    [Crossref]
  22. R. J. Glauber, in Quantum Optics and Electronics, edited by C. deWitt, A. Blandin, and C. Cohen-Tannoudji, (Gordon and Breach, New York, 1965), p. 63.
  23. L. Mandel and D. Meltzer, Phys. Rev. 188, 198 (1969).
    [Crossref]
  24. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [Crossref]
  25. E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).
    [Crossref]
  26. R. J. Glauber, Phvs. Rev. 131, 2766 (1963).
    [Crossref]
  27. L. Mandel and E. Wolf, Phys. Rev. 149, 1033 (1966).
    [Crossref]
  28. See for example, E. Jahnke and F. Emde, Tables of Functions, 4th ed. (Dover, New York, 1945).
  29. L. Mandel, Phys. Rev. 138, B753 (1965).
    [Crossref]
  30. H. Hodara, Proc. IEEE 53, 696 (1965).
    [Crossref]
  31. L. Mandel, Proc. Phys. Soc. (London) 74, 233 (1959).
    [Crossref]
  32. G. Bédard, J. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
    [Crossref]
  33. L. Mandel, J. Opt. Soc. Am. 51, 797 (1961).
    [Crossref]

1969 (2)

G. L. Fillmore and G. Lachs, IEEE Trans. IT-15, 465 (1969).

L. Mandel and D. Meltzer, Phys. Rev. 188, 198 (1969).
[Crossref]

1967 (2)

G. Bédard, J. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[Crossref]

B. R. Mollow and R. J. Walker, Phys. Rev. 160, 1076, 1097 (1967).
[Crossref]

1966 (1)

L. Mandel and E. Wolf, Phys. Rev. 149, 1033 (1966).
[Crossref]

1965 (5)

L. Mandel, Phys. Rev. 138, B753 (1965).
[Crossref]

H. Hodara, Proc. IEEE 53, 696 (1965).
[Crossref]

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

B. E. Goodwin and L. P. Bolgiano, Proc. IEEE 53, 1745 (1965).
[Crossref]

E. Hisdal, J. Opt. Soc. Am. 55, 1446 (1965);J. Opt. Soc. Am. 57, 35 (1967)J. Opt. Soc. Am. 59, 921 (1969).
[Crossref]

1964 (3)

L. P. Bolgiano and L. F. Jelsma, Proc. IEEE 52, 218 (1964).
[Crossref]

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[Crossref]

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[Crossref]

1963 (3)

J. P. Gordon, W. H. Louisell, and L. R. Walker, Phys. Rev. 129, 481 (1963).
[Crossref]

E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).
[Crossref]

R. J. Glauber, Phvs. Rev. 131, 2766 (1963).
[Crossref]

1962 (1)

J. P. Gordon, Proc. IRE 50, 1898 (1962).
[Crossref]

1961 (2)

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[Crossref]

L. Mandel, J. Opt. Soc. Am. 51, 797 (1961).
[Crossref]

1960 (2)

1959 (1)

L. Mandel, Proc. Phys. Soc. (London) 74, 233 (1959).
[Crossref]

1958 (1)

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958).
[Crossref]

1955 (1)

G. Toraldo di Francia, Opt. Acta 2, 5 (1955).
[Crossref]

1950 (1)

D. Gabor, Phil. Mag. 41, 1161 (1950).

1948 (1)

C. E. Shannon, Bell System Tech. J. 27, 376, 623 (1948).

Bédard, G.

G. Bédard, J. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[Crossref]

Bolgiano, L. P.

B. E. Goodwin and L. P. Bolgiano, Proc. IEEE 53, 1745 (1965).
[Crossref]

L. P. Bolgiano and L. F. Jelsma, Proc. IEEE 52, 218 (1964).
[Crossref]

Chang, J.

G. Bédard, J. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[Crossref]

Emde, F.

See for example, E. Jahnke and F. Emde, Tables of Functions, 4th ed. (Dover, New York, 1945).

Fillmore, G. L.

G. L. Fillmore and G. Lachs, IEEE Trans. IT-15, 465 (1969).

Gabor, D.

D. Gabor, Phil. Mag. 41, 1161 (1950).

Glauber, R. J.

R. J. Glauber, Phvs. Rev. 131, 2766 (1963).
[Crossref]

R. J. Glauber, in Quantum Optics and Electronics, edited by C. deWitt, A. Blandin, and C. Cohen-Tannoudji, (Gordon and Breach, New York, 1965), p. 63.

Goodwin, B. E.

B. E. Goodwin and L. P. Bolgiano, Proc. IEEE 53, 1745 (1965).
[Crossref]

Gordon, J. P.

J. P. Gordon, W. H. Louisell, and L. R. Walker, Phys. Rev. 129, 481 (1963).
[Crossref]

J. P. Gordon, Proc. IRE 50, 1898 (1962).
[Crossref]

J. P. Gordon, in Quantum Electronics and Coherent Light, edited by C. H. Townes and P. A. Miles (Academic, New York, 1962), p. 156.

J. P. Gordon, in Advances in Quantum Electronics, edited by J. R. Singer (Columbia University Press, New York, 1961), p. 509.

Hisdal, E.

Hodara, H.

H. Hodara, Proc. IEEE 53, 696 (1965).
[Crossref]

Jahnke, E.

See for example, E. Jahnke and F. Emde, Tables of Functions, 4th ed. (Dover, New York, 1945).

Jelsma, L. F.

L. P. Bolgiano and L. F. Jelsma, Proc. IEEE 52, 218 (1964).
[Crossref]

Jones, R. C.

Keeping, E. S.

See, for example, J. F. Kenney and E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, New York, 1951), Pt. II, Ch. 1.

Kelley, P. L.

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[Crossref]

Kenney, J. F.

See, for example, J. F. Kenney and E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, New York, 1951), Pt. II, Ch. 1.

Kleiner, W. H.

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[Crossref]

Lachs, G.

G. L. Fillmore and G. Lachs, IEEE Trans. IT-15, 465 (1969).

Lasher, G. J.

G. J. Lasher, in Ref. 3, p. 520.

Louisell, W. H.

J. P. Gordon, W. H. Louisell, and L. R. Walker, Phys. Rev. 129, 481 (1963).
[Crossref]

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[Crossref]

Mandel, L.

L. Mandel and D. Meltzer, Phys. Rev. 188, 198 (1969).
[Crossref]

G. Bédard, J. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[Crossref]

L. Mandel and E. Wolf, Phys. Rev. 149, 1033 (1966).
[Crossref]

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

L. Mandel, Phys. Rev. 138, B753 (1965).
[Crossref]

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[Crossref]

L. Mandel, J. Opt. Soc. Am. 51, 797 (1961).
[Crossref]

L. Mandel, Proc. Phys. Soc. (London) 74, 233 (1959).
[Crossref]

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958).
[Crossref]

L. Mandel, in Progress in Optics, II, edited by E. Wolf (North-Holland, Amsterdam, 1963), p. 181.
[Crossref]

Meltzer, D.

L. Mandel and D. Meltzer, Phys. Rev. 188, 198 (1969).
[Crossref]

Mollow, B. R.

B. R. Mollow and R. J. Walker, Phys. Rev. 160, 1076, 1097 (1967).
[Crossref]

Shannon, C. E.

C. E. Shannon, Bell System Tech. J. 27, 376, 623 (1948).

Siegman, A. E.

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[Crossref]

Stern, T. E.

T. E. Stern, Trans. IRE IT-6, 435 (1960).

Sudarshan, E. C. G.

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[Crossref]

E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).
[Crossref]

Toraldo di Francia, G.

G. Toraldo di Francia, Opt. Acta 2, 5 (1955).
[Crossref]

Walker, L. R.

J. P. Gordon, W. H. Louisell, and L. R. Walker, Phys. Rev. 129, 481 (1963).
[Crossref]

Walker, R. J.

B. R. Mollow and R. J. Walker, Phys. Rev. 160, 1076, 1097 (1967).
[Crossref]

Wolf, E.

L. Mandel and E. Wolf, Phys. Rev. 149, 1033 (1966).
[Crossref]

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[Crossref]

Yariv, A.

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[Crossref]

Bell System Tech. J. (1)

C. E. Shannon, Bell System Tech. J. 27, 376, 623 (1948).

IEEE Trans. (1)

G. L. Fillmore and G. Lachs, IEEE Trans. IT-15, 465 (1969).

J. Opt. Soc. Am. (3)

Opt. Acta (1)

G. Toraldo di Francia, Opt. Acta 2, 5 (1955).
[Crossref]

Phil. Mag. (1)

D. Gabor, Phil. Mag. 41, 1161 (1950).

Phvs. Rev. (1)

R. J. Glauber, Phvs. Rev. 131, 2766 (1963).
[Crossref]

Phys. Rev. (8)

L. Mandel and E. Wolf, Phys. Rev. 149, 1033 (1966).
[Crossref]

L. Mandel, Phys. Rev. 138, B753 (1965).
[Crossref]

G. Bédard, J. Chang, and L. Mandel, Phys. Rev. 160, 1496 (1967).
[Crossref]

P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 (1964).
[Crossref]

L. Mandel and D. Meltzer, Phys. Rev. 188, 198 (1969).
[Crossref]

W. H. Louisell, A. Yariv, and A. E. Siegman, Phys. Rev. 124, 1646 (1961).
[Crossref]

J. P. Gordon, W. H. Louisell, and L. R. Walker, Phys. Rev. 129, 481 (1963).
[Crossref]

B. R. Mollow and R. J. Walker, Phys. Rev. 160, 1076, 1097 (1967).
[Crossref]

Phys. Rev. Letters (1)

E. C. G. Sudarshan, Phys. Rev. Letters 10, 277 (1963).
[Crossref]

Proc. IEEE (3)

H. Hodara, Proc. IEEE 53, 696 (1965).
[Crossref]

L. P. Bolgiano and L. F. Jelsma, Proc. IEEE 52, 218 (1964).
[Crossref]

B. E. Goodwin and L. P. Bolgiano, Proc. IEEE 53, 1745 (1965).
[Crossref]

Proc. IRE (1)

J. P. Gordon, Proc. IRE 50, 1898 (1962).
[Crossref]

Proc. Phys. Soc. (London) (3)

L. Mandel, Proc. Phys. Soc. (London) 72, 1037 (1958).
[Crossref]

L. Mandel, Proc. Phys. Soc. (London) 74, 233 (1959).
[Crossref]

L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. (London) 84, 435 (1964).
[Crossref]

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[Crossref]

Trans. IRE (1)

T. E. Stern, Trans. IRE IT-6, 435 (1960).

Other (7)

J. P. Gordon, in Advances in Quantum Electronics, edited by J. R. Singer (Columbia University Press, New York, 1961), p. 509.

G. J. Lasher, in Ref. 3, p. 520.

J. P. Gordon, in Quantum Electronics and Coherent Light, edited by C. H. Townes and P. A. Miles (Academic, New York, 1962), p. 156.

L. Mandel, in Progress in Optics, II, edited by E. Wolf (North-Holland, Amsterdam, 1963), p. 181.
[Crossref]

R. J. Glauber, in Quantum Optics and Electronics, edited by C. deWitt, A. Blandin, and C. Cohen-Tannoudji, (Gordon and Breach, New York, 1965), p. 63.

See, for example, J. F. Kenney and E. S. Keeping, Mathematics of Statistics, 2nd ed. (Van Nostrand, New York, 1951), Pt. II, Ch. 1.

See for example, E. Jahnke and F. Emde, Tables of Functions, 4th ed. (Dover, New York, 1945).

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Figures (5)

F. 1
F. 1

Outline of the optical communication channel.

F. 2
F. 2

Variation of the information rate per symbol R with input statistics, as characterized by the parameter a, and with the mean number of photons detected per symbol N (with x = 1), for a single-mode laser. Because the logarithms were taken to the base e, the R values need to be multiplied by log2e to be in units of bits/symbol.

F. 3
F. 3

Variation of the maximum information R rate with the mean number of photons detected per symbol (with x = 1), for a single-mode and a multimode laser. Because the logarithms were taken to the base e, the R values need to be multiplied by log2e to be in units of bits/symbol.

F. 4
F. 4

Variation of the information rate per symbol R with input statistics, as characterized by the parameter a, and with the mean number of photons detected per symbol N (with x = 1), for a multimode laser. Because the logarithms were taken to the base e, the R values need to be multiplied by log2e to be in units of bits/symbol.

F. 5
F. 5

The minimum mode number m for which a multimode laser leads to a greater information rate than a single-mode laser, plotted against the mean number N of photons detected per symbol per mode (with x = 1).

Equations (38)

Equations on this page are rendered with MathJax. Learn more.

P ( x i ) P ( y j x i ) = p ( y j ) P ( x i y j ) ,
p ( y j ) = i P ( x i ) P ( y j x i ) .
R = i P ( x i ) log P ( x i ) + j p ( y j ) × i P ( x i y j ) log P ( x i y j ) . H ( X ) H ( X Y ) . }
0 H ( X Y ) H ( X ) ,
R = j p ( y j ) log p ( y j ) + i P ( x i ) × j i P ( y j x i ) log P ( y j x i ) H ( Y ) H ( Y ) ( Y X ) . }
p ( y j ) = dxP ( x ) P ( y j x ) .
R = j d x P ( y j x ) P ( x ) log [ d x P ( x ) P ( y j x ) ] + dxP ( x ) j P ( y j x ) log [ P ( y j x ) ] .
R = j d x P ( x ) P ( y j x ) × log [ P ( y j x ) / d x P ( x ) P ( y j x ) ] .
j P ( y j x ) log [ P ( y j x ) / d x P ( y j x ) P ( x ) ] = const ,
y j = j Δ y ,
P ( y j x ) = 1 , if y j 1 2 Δ y x y j + 1 2 Δ y , = 0 , otherwise }
P ( y j + z ) P ( y j ) , if | z | 1 2 Δ y ,
R = j y j 1 2 Δ y y j + 1 2 Δ y dxP ( x ) log [ y j 1 2 Δ y y j + 1 2 Δ y d x P ( x ) ] P ( y j ) Δ y log [ P ( y j ) Δ y ] dxP ( x ) log [ P ( x ) ] log ( Δ y ) .
P ( n x ) = 1 n ! [ α x t t + τ I ( t ) d t ] n × exp [ α x t t + τ I ( t ) d t ] ,
P ( n x ) = 0 1 n ! ( x α E ) n exp ( x α E ) Ψ ( E ) d E ,
N = n = 0 n P ( n 1 ) = 0 α E Ψ ( E ) d E α E .
R = n = 0 0 1 dxP ( x ) P ( n x ) × log [ P ( n x ) / 0 1 d x P ( x ) ( n x ) ] .
N = α E ,
P ( n x ) = ( 1 / n ! ) ( N x ) n exp ( N x ) ,
P ( x ) = [ a / ( 1 e a ) ] exp ( a x ) , for 0 x 1 = 0 , otherwise }
R = a 1 e a n 0 1 d x ( N x ) n n ! exp [ x ( N + a ) ] × log [ x n ( N + a ) n + 1 e N x ( 1 e a ) a Γ ( n + 1 , N + a ) ] .
R = [ 1 + N / a N / ( e a 1 ) ] log ( N + a ) + log [ ( 1 e a ) / a ] + N / [ a ( 1 e a ) ] ( E i ( a ) log γ a + a e a ) a / [ ( N + a ) ( 1 e a ) ] n 1 / [ n ! ( 1 + a / N ) n ] × Γ ( n + 1 , N + a ) log Γ ( n + 1 , N + a ) ,
E i ( y ) = y e x / xdx , ( 0 < y ) ,
R 0.14 + log N ,
V ( t ) = r ( I r ) 1 2 exp [ i ω r t + i ϕ r ( t ) ] ,
I ( t ) = V * ( t ) V ( t ) = r I r + r s ( I r I s ) 1 2 × exp { i ( ω s ω r ) t + i [ ϕ s ( t ) ϕ r ( t ) ] } ,
E = t t + τ I ( t ) d t = τ r I r + r s ( I r I s ) 1 2 × t t + τ d t exp [ i ( ω s ω r ) t + i ϕ s ( t ) i ϕ r ( t ) ] .
E τ r I r r E r = const ;
N = α r E r r N r .
E = τ I ,
B ( I ) = π 1 [ I 2 ( I I ) 2 ] 1 2 , for 0 I 2 I , = 0 , otherwise , }
P ( n x ) = 1 π n ! 0 2 I ( x α τ I ) n [ I 2 ( I I ) 2 ] 1 2 exp ( x α τ I ) d I = ( x N ) n π n ! 0 2 d y y n exp ( xNy ) ( 2 y y 2 ) 1 2 ,
B ( I ) = ( 1 / I ) exp ( I / I ) ,
P ( n x ) = 1 I n ! 0 ( x α τ I ) n exp [ x α τ I I / I ] d I = ( 1 + x N m ) 1 ( 1 + 1 / x N m ) n ,
R = a 1 e a n = 0 0 1 d x exp ( a x ) ( 1 + x N m ) ( 1 + 1 / x N m ) n × log ( 1 e a ( 1 + x N m ) ( 1 + 1 / x N m ) n / 0 1 d x a exp ( a x ) ( 1 + x N m ) ( 1 + 1 / x N m ) n ) .
N m = m N ,
R multimode ( m N ) > R singlemode ( N ) .
P ( n x ) = { [ Γ ( n + τ / T c ) ] / [ n ! Γ ( τ / T c ) ] } × [ ( 1 + x δ ) τ / T c ( 1 + 1 / x δ ) n ] ,