Abstract

Methods for estimating the spectral width of a narrowband optical signal are investigated. Spectral analysis and Fourier spectroscopy are compared. Optimum and close-to-optimum estimators are developed under the constraint of having only one photodetector.

© 1980 Optical Society of America

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References

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  1. G. Fiocco et al., Nat. Phys. Sci. 229, 78 (1971).
  2. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 1 (Wiley, New York, 1968).
  3. A. D. Wahlen, Detection of Signals in Noise (Academic, New York, 1971), Chap. 10.
  4. A. S. Jensen, L. Lading, “The Optimum Code in Single Particle Velocity Detection with a Laser Anemometer” (submitted for publication).
  5. P. Fellgett, Thesis, U. Cambridge (1951)(here quoted from G. A. Vanasse, H. Sakai, Progr. Opt. 6, 261 (1967).

1971 (1)

G. Fiocco et al., Nat. Phys. Sci. 229, 78 (1971).

Fellgett, P.

P. Fellgett, Thesis, U. Cambridge (1951)(here quoted from G. A. Vanasse, H. Sakai, Progr. Opt. 6, 261 (1967).

Fiocco, G.

G. Fiocco et al., Nat. Phys. Sci. 229, 78 (1971).

Jensen, A. S.

A. S. Jensen, L. Lading, “The Optimum Code in Single Particle Velocity Detection with a Laser Anemometer” (submitted for publication).

Lading, L.

A. S. Jensen, L. Lading, “The Optimum Code in Single Particle Velocity Detection with a Laser Anemometer” (submitted for publication).

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 1 (Wiley, New York, 1968).

Wahlen, A. D.

A. D. Wahlen, Detection of Signals in Noise (Academic, New York, 1971), Chap. 10.

Nat. Phys. Sci. (1)

G. Fiocco et al., Nat. Phys. Sci. 229, 78 (1971).

Other (4)

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part 1 (Wiley, New York, 1968).

A. D. Wahlen, Detection of Signals in Noise (Academic, New York, 1971), Chap. 10.

A. S. Jensen, L. Lading, “The Optimum Code in Single Particle Velocity Detection with a Laser Anemometer” (submitted for publication).

P. Fellgett, Thesis, U. Cambridge (1951)(here quoted from G. A. Vanasse, H. Sakai, Progr. Opt. 6, 261 (1967).

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Figures (3)

Fig. 1
Fig. 1

∂S(ω)/∂ω indicates how much an individual spectral component contributes to information about spectral width. An optimum wideband filter should only pass the power from ω0 − √θ to ω0 + √θ. Collected power from outside this range would (partly) cancel the effect of power from the central part of the spectrum, that is, if a negative transmission is unobtainable. Otherwise the difference in the power of the positive and negative rectangles should be used for determining spectral width. Note S(√θ) is a lever point: its value is independent of θ.

Fig. 2
Fig. 2

System utilizing full spectral power. H1(ω) has a transmission as given by the positive rectangle in Fig. 1, H2(ω) as given by the negative rectangles.

Fig. 3
Fig. 3

Two-arm interferometer with prefilter. The prefilter incorporates a wide bandpass filter to reduce background and a narrow notch filter to estimate light at the exciting frequency. The detector will see a signal of which the mean is given by the signal power passed by H(ω), and the ac component is proportional to the correlation function for the electric field at a time lag given by the difference in path length between the two arms. The ac signal oscillates with a frequency given by the difference between the Bragg cell drive frequencies.

Tables (2)

Tables Icon

Table I Comparison of FS and SA

Tables Icon

Table II Performance of Different Filters

Equations (72)

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S ( ω ) = R ( 2 π θ ) 1 / 2 exp { 1 2 ( ω ω 0 ) 2 θ } + B + M δ ( ω ω 0 ) ,
I ( τ ) = R + B Δ ω p + M + [ R exp ( ½ θ τ 2 ) + B δ ( τ ) + M ] exp ( j ω 0 τ ) ,
p ( n α ) or ln p ( n α ) ,
ln p ( n α ) = i ln p ( n i α ) .
var { α ̂ i } ( [ Γ ] 1 ) i i ,
[ Γ ] = { Γ l l Γ l k Γ k l Γ k k } ,
Γ j i = Γ i j = E { 2 ln p ( n α ) α i α j } .
p ( n i ) = exp ( n i ) n i n i n i ! ,
Γ i j = m 1 n m n m α i n m α j .
H ( ω , ω ) = { a ω = ω ± Δ ω / 2 , b otherwise ,
n ( ω ) = [ a S ( ω ) Δ ω + b M + b B Δ ω p + D ] T a
Γ 11 = Γ R = 1 2 π T a a 2 K θ 1 / 2 Γ 22 = Γ M = T a N b 2 K Δ ω Γ 33 = Γ θ = 3 32 π T a a 2 R 2 K θ 5 / 2 Γ 12 = Γ R , M = a b T a K Δ ω Γ 13 = Γ R , θ = 1 8 π a 2 R T a K θ 3 / 2 Γ 23 = Γ M , θ = 0 } ,
var { R ̂ } = 1 Γ R 1 1 [ Γ R , θ 2 / ( Γ R Γ θ ) ] = 1 Γ R 3 2 ,
var { θ ̂ } = 1 Γ δ 3 2 ,
n ( τ ) = Re { I ( τ ) + D } T / m .
Γ R = T + 1 4 a f 2 ( π / θ ) 1 / 2 ( T / τ 0 ) I 0 Γ M = ( 1 + 1 2 a f 2 ) T I 0 Γ θ = 3 32 π a f 2 R 2 ( T / τ 0 ) I 0 θ 5 / 2 Γ R , M = T + 1 2 a f 2 [ ( 2 π ) / θ ] 1 / 2 ( T / τ 0 ) I 0 Γ R , θ = π 8 a f 2 R ( T / τ 0 ) I 0 θ 3 / 2 Γ M , θ = ( 2 π ) 1 / 2 8 a f 2 R ( T / τ 0 ) I 0 θ 3 / 2 } ,
ρ θ , R = Γ θ , R 2 Γ θ Γ R = π 6 a f 2 θ τ 0 ,
ρ θ , R = Γ θ , M 2 Γ θ Γ M = π 3 a f 2 1 + 1 2 a f 1 θ τ 0 .
0 τ 0 τ 4 τ 0 exp ( θ τ 2 ) d τ ,
0 ω m ω 4 ω m exp ( ω 2 θ ) d ω .
Γ θ = 1 2 exp ( 2 ) a f 2 R 2 I 0 T a θ 2 ,
0.4 θ τ 0 .
n ( θ ) = T | u ( t , θ ) * h ( t ) | 2 ,
Γ θ = { [ / ( θ ) ] n ( θ ) } 2 n ( θ ) .
n ( θ ) = T S ( ω ) | H ( ω ) | 2 d ω ,
0 | H ( ω ) | 2 1 .
| H ( ω ) | 2 = { 1 if | ω ω 0 | < Δ ω 0 , 0 otherwise ,
S ( ω , θ ) θ | ω 0 + Δ ω 0 = 0 or ( Δ ω 0 ) 2 = θ
Δ ω 1 = ( D / B ) + [ ( D / B ) 2 + 4 ( Δ ω 0 ) 2 ] 1 / 2 ( Δ ω 1 > Δ ω 0 ) .
n ( θ ) = T | R ( τ , θ ) * G ( τ ) | τ = 0 ,
Γ θ = T { [ / ( θ ) ] R ( τ , θ ) * G ( τ ) } 2 R ( τ , θ ) * G ( τ ) | τ = 0 .
G ( τ ) = { 1 / Δ τ if τ 0 < τ < τ 0 + Δ τ , 0 otherwise .
Γ θ = T [ 1 Δ τ τ 0 τ 0 + Δ τ θ R ( τ , θ ) d τ ] 2 B / Δ τ .
Δ τ = [ 8 / ( 5 θ ) ] 1 / 2 .
n ( τ ) = T { R + M + [ R exp ( ½ θ τ 2 ) + M ] cos ω 0 τ } .
Γ θ = 1 8 R 2 τ 4 exp ( τ 2 θ ) ( 1 + cos 2 ω 0 τ ) R + M + ( R exp ( ½ θ τ 2 + M ) cos ω 0 τ .
τ [ τ 4 exp ( τ 2 θ ) 1 exp ( ½ θ τ 2 ) ] = 0 ,
τ 1 = 1.13 / θ
π ( a f a s ) 2 1 τ 0 Δ ω 1
π ( a f a s ) 2 a s τ 0 Δ ω Δ ω Δ ω p < 1
π ( a f a s ) 2 b τ 0 Δ ω < 1
0.12 ( R θ ) 2 T a D + B Δ ω
0.14 ( R θ ) 2 T a D + B Δ ω
0.06 ( R θ ) 2 T a D + B ( 2 π θ ) 1 / 2
0.12 ( R θ ) 2 T a 0.69 R + 2 B Δ ω + D
n = T [ S ( ω ) F ( ω ) d ω + D + 2 B Δ ω 1 ] ,
S ( ω ) = R ( 2 π θ ) 1 / 2 exp [ 1 2 ( ω ω 0 ) 2 θ ] ,
F ( ω ) = { 1 if | ω ω 0 | < Δ ω 0 , 1 if Δ ω 0 < | ω ω 0 | < Δ ω 1 , Δ ω 0 Δ ω 1 , 0 otherwise
Γ θ = ( n / θ ) 2 n .
n θ = T R ( 2 π ) 1 / 2 θ 3 / 2 2 Δ ω 0 exp [ ( Δ ω 0 ) 2 / 2 θ ] .
Γ θ = 2 π e T ( R θ ) 2 / ( D + 2 B Δ ω 1 ) .
n 1 = T a S ( ω ) F 1 ( ω ) d ω + D + B Δ ω 1 ,
n 2 = T a S ( ω ) F 2 ( ω ) d ω + D + B Δ ω 2 ,
F 1 ( ω ) = exp [ 1 2 ( ω ω 0 ) 2 θ 1 ] , ( 2 π θ 1 ) 1 / 2 = Δ ω 1 ,
F 2 ( ω ) = exp [ 1 2 ( ω ω 0 ) 2 θ 2 ] , ( 2 π θ 2 ) 1 / 2 = Δ ω 2 ,
n 1 = T a [ R ( θ 1 θ + θ 1 ) 1 / 2 + D + B Δ ω 1 ] ,
n 2 = T a ( R + D + B Δ ω 2 ) .
Γ θ = T 16 ( R / θ ) 2 D + B Δ ω ,
ρ θ , R = 1 1 + 4 ( θ 1 / θ 2 ) 1 / 2 ,
i ( t ) n ( t ) t = ½ [ R exp ( ½ τ 2 θ ) ( 1 + cos ω r t ) + D + B Δ ω ] .
i ( t ) ¯ = ½ ( R + D + B Δ ω ) , i ( t ) r ( t ) ¯ = 1 2 R exp ( ½ τ 2 θ ) ,
r ( t ) = 2 cos ω r t ,
n = 0 T n ( t ) d t
0 T n ( t ) t r ( t ) d t
Γ θ = T ( i ̅ / θ ) 2 + [ ( i r ¯ ) / θ ] 2 ½ ( D + 2 B Δ ω ) ,
Γ θ = 1 e 2 ( R / θ ) 2 D + 2 B Δ ω .
ρ θ , R = ( i ̅ / θ ) 2 ( i ̅ / θ ) 2 + [ ( i ̅ r ) / θ ] 2 | τ = ( 2 / θ ) 1 / 2 = 0.21 .
n 1 = T { R [ 1 exp ( ½ θ τ 0 2 ] + B 2 Δ ω + D } ,
n 2 = T ( R + B 2 Δ ω + D ) ,
Γ θ = T 4 R 2 τ 0 4 exp ( θ τ 0 2 ) R [ 1 exp ( ½ θ τ 0 2 ] + B 2 Δ ω + D ,
Γ θ = 0.12 T ( R / θ ) 2 0.69 R + 2 B Δ ω + D .
ζ = 0.27 R + 2 B Δ ω + D 0.69 R + 2 B Δ ω + D .

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