Abstract

A novel high-speed fiber-optic spectrometer has been demonstrated in our previous work. The high-speed spectrum measurement is enabled by translating the spectral-domain signal into a time-domain signal through a dispersion element. We present a mathematical model that accurately describes the relationship between the optical spectrum to be measured and the dispersed time-domain signal. Based on the model, the effects of the key parameters on the performance of the spectrometer are investigated in detail using numerical simulation. The analysis is useful for the design and application of such spectrometers.

© 2007 Optical Society of America

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References

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  1. B. Qi, G. Pickrell, J. Xu, P. Zhang, Y. Duan, W. Peng, Z. Huang, W. Huo, H. Xiao, R. G. May, and A. Wang, "Novel data processing techniques for dispersive white light interferometer," Opt. Eng. 42, 3165-3171 (2003).
    [CrossRef]
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2006

2005

2004

2003

B. Qi, G. Pickrell, J. Xu, P. Zhang, Y. Duan, W. Peng, Z. Huang, W. Huo, H. Xiao, R. G. May, and A. Wang, "Novel data processing techniques for dispersive white light interferometer," Opt. Eng. 42, 3165-3171 (2003).
[CrossRef]

1981

1980

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Figures (9)

Fig. 1
Fig. 1

Operational principle of the HSFOS.

Fig. 2
Fig. 2

Calculated spectrum of a FP sensor with a cavity length of 73 μ m .

Fig. 3
Fig. 3

Spectrometer output s λ ( λ ) corresponding to a spectrum with constant spectral density.

Fig. 4
Fig. 4

Simulation results of the error Δ 1 .

Fig. 5
Fig. 5

Simulation results of the error Δ 2 .

Fig. 6
Fig. 6

Definition of spectral resolution.

Fig. 7
Fig. 7

Resolution analysis at 1550   nm using Gaussian pulse.

Fig. 8
Fig. 8

Best spectral resolution at 1550   nm using Gaussian pulse.

Fig. 9
Fig. 9

Relationships between the maximum spectrum acquisition rate, the spectral width, and the spectral resolution when the best spectral resolution for each dispersion value is reached.

Tables (1)

Tables Icon

Table 1 Dispersion Properties of the Fibers Used in Simulation

Equations (44)

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E ( 0 , t ) = A ( t ) exp ( j ω 0 t ) ,
I ( 0 , t ) = | E ( 0 , t ) | 2 = E ( 0 , t ) E * ( 0 , t ) ,
P ( 0 , ω ) = 1 ( 2 π ) 2 + + E ( 0 , t ) E * ( 0 , t ) × exp [ j ω ( t t ) ] d t d t .
F 0 ( ω ) = 1 2 π + m 0 ( t ) exp ( j ω t ) d t .
s ( t ) = + P ( 0 , ω ) | + F 0 ( ω ω ) exp { j [ ( ω ω ) t ( β β ) L ] } d ω | 2 d ω .
β ( ω ) = m = 0 β m ( ω ) m ! ( ω ω ) m ,
β m = d m β d ω m ( m = 0 , 1 , 2 , ) .
β ( ω ) β ( ω ) = ( ω ω ) β 1 ( ω ) + γ ( ω , ω ) ,
γ ( ω , ω ) = m = 2 β m ( ω ) m ! ( ω ω ) m .
s ( t ) = + P ( 0 , ω ) | + F 0 ( ω ω ) × exp { j [ ( ω ω ) ( t β 1 ( ω ) L ) γ ( ω , ω ) L ] } d ω | 2 d ω .
γ ( ω , ω ) = Γ ( ω , ω ) = m = 2 β m ( ω ) m ! ω m .
F ( L , ω , ω ) = F 0 ( ω ) exp ( j Γ ( ω , ω ) L )
t d ( ω ) = β 1 ( ω ) L .
s ( t ) = + P ( 0 , ω ) | + F ( L , ω , ω ) × exp { j ω ( t t d ( ω ) ) } d ω | 2 d ω .
ω ( t d ) = t d 1 ( ω ) ,
d ω = d t d t ˙ d ( ω ) = d t d β 2 ( ω ) L ,
m ( L , t , t d ) = + F t ( L , ω , t d ) exp ( j ω ( t t d ) ) d ω ,
M ( L , t , t d ) = | m ( L , t , t d ) | 2 ,
s ( t ) = + P t ( 0 , t d ) β 2 t ( t d ) L M ( L , t , t d ) d t d ,
P n ( 0 , t d ) = P t ( 0 , t d ) β 2 t ( t d ) L = P ( 0 , ω ) β 2 ( ω ) L ,
s ( t ) = + P n ( 0 , t d ) M ( L , t , t d ) d t d .
Γ ω 0 ( ω ) = m = 2 β m ( ω 0 ) m ! ω m ,
F ( L , ω ) = F 0 ( ω ) exp ( j Γ ω 0 ( ω ) L ) ,
m ( L , t t d ) = + F ( L , ω ) exp ( j ω ( t t d ) ) d ω ,
M ( L , t t d ) = | m ( L , t t d ) | 2 ,
s ( t ) = + P n ( 0 , t d ) M ( L , t t d ) d t d ,
s ( t ) = P n ( 0 , t ) M ( L , t ) .
t = β 1 ( 2 π c λ ) L .
s λ ( λ ) = + P n ( 0 , t d ) M λ ( L , λ , t d ) d t d ,
s λ ( λ ) = P n λ ( 0 , λ ) M λ ( L , λ ) ,
D = 2 π c λ 2 β 2 ,
S = d D d λ ,
Δ 1 = ( + P n ( 0 , t d ) M λ ( L , λ , t d ) d t d ) n o r m a l i z e d P λ ( 0 , λ ) ,
Δ 2 = ( P n λ ( 0 , λ ) M λ ( L , λ ) ) n o r m a l i z e d ( + P n ( 0 , t d ) M λ ( L , λ , t d ) d t d ) n o r m a l i z e d .
m 0 ( t ) = exp ( t 2 2 T 0 2 ) .
M ( L , t ) = 1 1 + ( β 2 ( ω 0 ) L T 0 2 ) 2   exp ( t 2 T 0 2 ( 1 + ( β 2 ( ω 0 ) L T 0 2 ) 2 ) ) .
T 0 = T 0 1 + ( β 2 ( ω 0 ) L T 0 2 ) 2 .
r λ = 2 T 0 D L ( λ 0 ) = 2 ( T 0 D L ( λ 0 ) ) 2 + ( λ 0 2 2 π c T 0 ) 2 ,
D ( λ 0 ) = 2 π c λ 0 2 β 2 ( ω 0 ) ,
D L ( λ 0 ) = D ( λ 0 ) L ,
r λ m = 2 λ 0 2 π c | D L ( λ 0 ) | ,
T 0 = λ 0 2 2 π c | D L ( λ 0 ) | .
T s = Δ λ D L ( λ 0 ) + 6 T 0 ,
f m = 1 T s = ( ( Δ λ + 6 ( T 0 D L ( λ 0 ) ) 2 + ( λ 0 2 2 π c T 0 ) 2 ) D L ( λ 0 ) ) 1 .

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