Abstract

A theory of broadband signal and noise performance of a direct detection optical receiver is presented in this paper. Explicit expressions are given for the gain and noise factor of the optical receiver, consisting of a photodiode followed by a high gain, low noise baseband amplifier. It is assumed that a linear lumped lossless interstage network is placed between the diode and the amplifier to obtain broadband performance from the optical receiver. The constraints imposed by the photodiode on the wideband characteristics of the gain GR and noise factor FR of the optical receiver are obtained in integral and nonintegral forms.

© 1968 Optical Society of America

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References

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  1. E. B. Champagne, Appl. Opt. 5, 1843 (1966).
    [CrossRef] [PubMed]
  2. D. L. Fried, J. B. Seidman, Appl. Opt. 6, 245 (1967).
    [CrossRef] [PubMed]
  3. V. K. Prabhu, Appl. Opt. 7, 657 (1968).
    [CrossRef] [PubMed]
  4. V. K. Prabhu, Bell Syst. Tech. J. 47, 429 (1968).
  5. M. V. Schneider, Bell Syst. Tech. J. 45, 1611 (1966).
  6. L. K. Anderson, in Proceedings of the Symposium on Optical Masers (Polytechnic Press, Brooklyn, 1963).
  7. H. Inaba, A. E. Siegman, Proc. Inst. Radio Eng. 50, 1823 (1962).
  8. L. U. Kibler, Proc. Inst. Radio Eng. 50, 1834 (1962).
  9. B. M. Oliver, Proc. IEEE 53, 436 (1965).
    [CrossRef]
  10. R. B. Emmons, G. Lucovsky, Proc. IEEE 52, 865 (1964).
    [CrossRef]
  11. R. M. Fano, J. Franklin Inst. 249, 57, 139 (1950).
    [CrossRef]
  12. D. C. Youla, IEEE Trans. CT-11, 33 (1964).
  13. H. W. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand Co., Inc., Princeton, 1945).
  14. V. K. Prabhu, “Linear Lossless Transformations of Arbitrary Impedances,” unpublished technical memorandum, Bell Telephone Labs., Inc., Holmdel, N. J.
  15. L. Weinberg, Network Analysis and Synthesis (McGraw-Hill Book Company, Inc., New York, 1962).
  16. E. G. Nielsen, Proc. Inst. Radio Eng. 45, 957 (1957).

1968 (2)

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

V. K. Prabhu, Appl. Opt. 7, 657 (1968).
[CrossRef] [PubMed]

1967 (1)

1966 (2)

E. B. Champagne, Appl. Opt. 5, 1843 (1966).
[CrossRef] [PubMed]

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

1965 (1)

B. M. Oliver, Proc. IEEE 53, 436 (1965).
[CrossRef]

1964 (2)

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

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

1962 (2)

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

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

1957 (1)

E. G. Nielsen, Proc. Inst. Radio Eng. 45, 957 (1957).

1950 (1)

R. M. Fano, J. Franklin Inst. 249, 57, 139 (1950).
[CrossRef]

Anderson, L. K.

L. K. Anderson, in Proceedings of the Symposium on Optical Masers (Polytechnic Press, Brooklyn, 1963).

Bode, H. W.

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

Champagne, E. B.

Emmons, R. B.

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

Fano, R. M.

R. M. Fano, J. Franklin Inst. 249, 57, 139 (1950).
[CrossRef]

Fried, D. L.

Inaba, H.

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

Kibler, L. U.

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

Lucovsky, G.

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

Nielsen, E. G.

E. G. Nielsen, Proc. Inst. Radio Eng. 45, 957 (1957).

Oliver, B. M.

B. M. Oliver, Proc. IEEE 53, 436 (1965).
[CrossRef]

Prabhu, V. K.

V. K. Prabhu, Appl. Opt. 7, 657 (1968).
[CrossRef] [PubMed]

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

V. K. Prabhu, “Linear Lossless Transformations of Arbitrary Impedances,” unpublished technical memorandum, Bell Telephone Labs., Inc., Holmdel, N. J.

Schneider, M. V.

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

Seidman, J. B.

Siegman, A. E.

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

Weinberg, L.

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

Youla, D. C.

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

Appl. Opt. (3)

Bell Syst. Tech. J. (2)

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

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

IEEE Trans. (1)

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

J. Franklin Inst. (1)

R. M. Fano, J. Franklin Inst. 249, 57, 139 (1950).
[CrossRef]

Proc. IEEE (2)

B. M. Oliver, Proc. IEEE 53, 436 (1965).
[CrossRef]

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

Proc. Inst. Radio Eng. (3)

E. G. Nielsen, Proc. Inst. Radio Eng. 45, 957 (1957).

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

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

Other (4)

L. K. Anderson, in Proceedings of the Symposium on Optical Masers (Polytechnic Press, Brooklyn, 1963).

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

V. K. Prabhu, “Linear Lossless Transformations of Arbitrary Impedances,” unpublished technical memorandum, Bell Telephone Labs., Inc., Holmdel, N. J.

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

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Figures (10)

Fig. 1
Fig. 1

A direct detection optical receiver.

Fig. 2
Fig. 2

Equivalent circuit of photodiode. Is is the signal component of current and I0 is the dc component. ish is the shot noise source present in the diode, and inp and ens are thermal noise sources associated with Gp and Gs, respectively.

Fig. 3
Fig. 3

Lossless interstage network used in broadbanding the signal and noise performance of the optical receiver.

Fig. 4
Fig. 4

Normalized gain function GR as a function of ω for Butterworth approximations of order n = 1,2. It is assumed that W = 108 rad/sec, and s2. = 1, Ks = 1.

Fig. 5
Fig. 5

Normalized noise factor FR as a function of ω for Butterworth approximations of order n = 1,2. It is assumed that W = 108 rad/sec, and u2 = 1, Ku = 1. FR′ = FR − 2/η and FRO′ = FRO − 2/η.

Fig. 6
Fig. 6

Normalized plots of s2 and u2 as functions of n. Even though n is a discrete variable, the plots are given for all n ≥ 1. It is assumed that ωp = 108 rad/sec, and W = 104 rad/sec.

Fig. 7
Fig. 7

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

Fig. 8
Fig. 8

Normalized gain function GR as a function of ω for Chebyshev approximations of order n = 1,2. It is assumed that W = 108 rad/sec, and s2 = 1, Ks = 1.

Fig. 9
Fig. 9

Normalized noise factor FR as a function of ω for Chebyshev approximations of order n = 1,2. It is assumed that W = 108 rad/sec, and u2 = 1, Ku = 1. FR′ = FR − 2/η, and FRO′ = FRO − 2/η.

Fig. 10
Fig. 10

Normalized plots of s2 and u2 as functions of n. Even though n is a discrete variable, the plots are given for all n ≥ 1. It is assumed that ωp = 106 rad/sec, and W = 108 rad/sec.

Equations (66)

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I s = ( η q / h ν ) P s ,
i s h 2 ¯ = 2 ( η q 2 / h ν ) P s ( Δ ω / 2 π ) ,
i n p 2 ¯ = 4 G p k T d ( Δ ω / 2 π ) ,
e n s 2 ¯ = 4 R s k T d ( Δ ω / 2 π ) ,
P d a = ( η q / h ν ) 2 P s 2 4 [ R s C 2 ω 2 + G p ( 1 + G p / G s ) ]
N d a = k T d Δ ω 2 π + 2 ( η q 2 / h ν ) P s ( Δ ω / 2 π ) 4 [ R s C 2 ω 2 + G p ( 1 + G p / G s ) ] .
ω ω m = ( G p G s ) 1 2 / C .
P d a N d a = ( η q h ν ) 2 2 π Δ ω × P s 2 2 ( η q 2 / h ν ) P s + 4 k T d [ R s C 2 ω 2 + G p ( 1 + G p / G s ) ] .
1 G a = 1 G a o + R f Re Y s Y s - Y o g 2 ,
F = F o + R n Re Y s Y s - Y o f 2 ,
G R = P d a G a P s ,
G R = ( η q / h ν ) 2 P s 4 [ R s C 2 ω 2 + G p ( 1 + G p / G s ) ] × 1 ( 1 / G a o ) + ( R f / Re Y s ) Y s - Y o g 2 .
F R = 2 η { 1 + 2 k T o q I o [ R s C 2 ω 2 + G p ( 1 + G p / G s ) ] × ( T d - T o T o + F o + R n Re Y s Y s - Y o f 2 ) } ,
G R O = ( η q / h ν ) 2 P s 4 [ R s C 2 ω 2 + G p ( 1 + G p / G s ) ] G a o ,
F R O = 2 π { 1 + 2 k T o q I o [ R s C 2 ω 2 + G p ( 1 + G p / G s ) ] × ( T d - T o T o + F o ) } .
Y ( p ) = G s p + G p / C p + ( G s + G p ) / C ,
p o = φ o ( 1 + G p / G s ) 1 2 ,
φ o = ( G s G p ) 1 2 / C .
ln | 1 + [ ( G p / G s ) / ( 1 + G p / G s ) ] 1 2 1 - [ ( G p / G s ) / ( 1 + G p / G s ) ] 1 2 | 1 π φ o ( 1 + G p / G s ) 1 2 × 0 1 φ o 2 ( 1 + G p / G s ) + ω 2 ln [ 1 + 4 R f G o g ( 1 / G a ) - ( 1 / G a o ) ] d ω ,
ln | 1 + [ ( G p / G s ) / ( 1 + G p / G s ) ] 1 2 1 - [ ( G p / G s ) / ( 1 + G p / G s ) ] 1 2 | 1 π φ o ( 1 + G p / G s ) 1 2 × 0 1 φ o 2 ( 1 + G p / G s ) + ω 2 ln [ 1 + 4 R n G o f F - F o ] d ω .
1 π φ o ( 1 + G p / G s ) 1 2 0 1 φ o 2 ( 1 + G p / G s ) + ω 2 × ln [ 1 + ( h ν η q ) 2 16 R f G o g R s P s ( ω / ω c ) 2 + ( G p / G s ) ( 1 + G p / G s ) ( 1 / G R ) - ( 1 / G R O ) ] d ω ln | 1 + [ ( G p / G s ) / ( 1 + G p / G s ) ] 1 2 1 - [ ( G p / G s ) / ( 1 + G p / G s ) ] 1 2 | ,
1 π φ o ( 1 + G p / G s ) 1 2 0 1 φ o 2 ( 1 + G p / G s ) + ω 2 × ln [ 1 + k T o 16 R n G o f η q I o R s ( ω / ω c ) 2 + ( G p / G s ) ( 1 + G p / G s ) F - F o ] d ω ln | 1 + [ ( G p / G s ) / ( 1 + G p / G s ) ] 1 2 1 - [ ( G p / G s ) / ( 1 + G p / G s ) ] 1 2 | ,
ω c = 1 / R s C .
G p / G s 1.
1 π 0 1 φ o 2 + ω 2 ln [ 1 + ( h ν η q ) 2 16 R f G o g R s P s ( ω / ω c ) 2 + ( G p / G s ) ( 1 / G R ) - ( 1 / G R O ) ] d ω 2 ω c ,
1 π 0 1 φ 0 2 + ω 2 ln [ 1 + k T o η q I o 16 R n G o f R s ( ω / ω c ) 2 + ( G p / G s ) F R - F R O ] d ω 2 ω c .
s ( p ) = ± η ( p ) s o ( p ) ,
s o ( p ) s o ( - p ) = 1 1 + 4 R f G o g ( 1 / G a ) - ( 1 / G a o )
s o ( p ) s o ( - p ) = 1 1 + 4 R n G o f / ( F - F o ) .
η ( p ) = l = 1 ν p - μ l p + μ l * ,
s ( p ) = β ( p ) Y 1 ( p ) - Y ( - p ) Y 1 ( p ) + Y ( p ) ,
β ( p ) = r = 1 m p - α r p + α r * .
β ( p ) = p - ( G s + G p ) / C p + ( G s + G p ) / C .
S o = [ ( G p / G s ) / ( 1 + G p / G s ) ] ¹ / - 1 [ ( G p / G s ) / ( 1 + G p / G s ) ] ¹ / + 1 ,
s ( p ) = k = 0 S k ( p - p o ) k .
G R = G R O K s 1 + s 2 ( ω / W ) 2 n
F R + 2 η + 1 K u ( F R O - 2 η ) [ 1 + u 2 ( ω W ) 2 n ] ,
0 < K s 1
0 < K u 1.
s o ( p ) = ± ( 1 - K s ) ¹ / ( 1 - K s + 4 K s R f G o g G a o ) ¹ / × Δ n [ p s 1 / n W ( 1 - K s ) 1 / n ] Δ n [ p s 1 / n W ( 1 - K s + 4 K s R f G o g G a o ) / 2 n 1 ]
s o ( p ) = ± ( 1 - K u ) ¹ / ( 1 - K u + 4 K u R n G o f M o ) ¹ / × Δ n [ p u 1 / n W ( 1 - K u ) / 2 n 1 ] Δ n [ p u 1 / n W ( 1 - K u + 4 K u R n G o f M o ) / 2 n 1 ] ,
M o = 1 / F o ,
± η ( p o ) ( 1 - K s ) ¹ / ( 1 - K s + 4 K s R f G o g G a o ) ¹ / × Δ n [ p o s 1 / n W ( 1 - K s ) / 2 n 1 ] Δ n [ p o s 1 / n W ( 1 - K s + 4 K s R f G o g G a o ) / 2 n 1 ] = ( G p / G s 1 + G p / G s ) ¹ / - 1 ( G p / G s 1 + G p / G s ) ¹ / + 1 ,
± η ( p o ) ( 1 - K u ) ¹ / ( 1 - K u + 4 K u R n G o f M o ) ¹ / × Δ n [ p o u 1 / n W ( 1 - K u ) / 2 n 1 ] Δ n [ p o u 1 / n W ( 1 - K u + 4 K u R n G o f M o ) / 2 n 1 ] = ( G p / G s 1 + G p / G s ) ¹ / - 1 ( G p / G s 1 + G p / G s ) ¹ / + 1 .
φ o / W = ( G s G p ) ¹ / / C W 1 ,
1 s 1 / n { [ 1 - K s + 4 K s R f G o g G a o ] 1 / 2 n - [ 1 - K s ] 1 / 2 n } 2 ω p W sin π 2 n
1 u 1 / n { [ 1 - K u + 4 K u R n G o f M o ] 1 / 2 n - [ 1 - K u ] 1 / 2 n } 2 ω p W sin π 2 n ,
ω p = 1 / R p C .
s 2 4 R f G o g G a o [ ( 2 ω p / W ) sin ( π / 2 n ) ] 2 n ,
u 2 4 R n G o f M o [ ( 2 ω p / W ) sin ( π / 2 n ) ] 2 n .
Z 1 ( p ) = 1 Y 1 ( p ) = C G p ( G s G p ) 1 / 2 p + 1 G p .
G R = G R O K s 1 + s 2 T n 2 ( ω / W )
F R = 2 η + 1 K u ( F R O - 2 η ) [ 1 + u 2 T n 2 ( ω W ) ] ,
T n ( x ) = cos ( n cos - 1 x ) ,
0 < K s 1 ,
0 < K u 1.
sinh [ 1 n sinh - 1 ( 1 - K s + 4 K s R f G o g G a o ) 1 / 2 s ] - sinh [ 1 n sinh - 1 ( 1 - K s ) 1 / 2 s ] 2 ω p W sin π 2 n ,
sinh [ 1 n sinh - 1 ( 1 - K u + 4 K u R n G o f M o ) 1 / 2 u ] - sinh [ 1 n sinh - 1 ( 1 - K u ) 1 / 2 u ] 2 ω p W sin π 2 n .
[ s 2 ] min = 4 R f G o g G a o sinh 2 { n sinh - 1 [ ( 2 ω p / W ) sin ( π / 2 n ) ] } ,
[ u 2 ] min = 4 R n G o f M o sinh 2 { n sinh - 1 [ ( 2 ω p / W ) sin ( π / 2 n ) ] } ,
[ s 2 ] min = 16 R f G o g G a o π 2 ( 2 ω p / W ) 2 [ ( sin π / 2 n ) / ( π / 2 n ) ] 2 ,
[ u 2 ] min = 16 R n G o f M o π 2 ( 2 ω p / W ) 2 [ ( sin π / 2 n ) / ( π / 2 n ) ] 2 .
{ [ s 2 ] min } n = 1 { [ s 2 ] min } n = = { [ u 2 ] min } n = 1 { [ u 2 ] min } n = = π 2 4 2.5 ,
{ [ s 2 ] min } n = 2 { [ s 2 ] min } n = = { [ u 2 ] min } { [ u 2 ] min } n = = π 2 8 1.25.
1 π o 1 φ o 2 + ω 2 × ln [ 1 + ( h ν η q ) 2 16 R f G o g R s P s ( ω / ω c ) 2 + ( G p / G s ) 1 / G R - 1 / G R O ] d ω 2 ω c ,
1 π o 1 φ o 2 + ω 2 × ln [ 1 + k T o η q I o 16 R n G o f R s ( ω / ω c ) 2 + ( G p / G s ) F R - F R O ] d ω 2 ω c .

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