Abstract

This work considers the estimation of dispersion in materials via an interferometric technique. At its core, the problem involves extracting the quadratic variation in phase over a range of wavelengths based on measured optical intensity. The estimation problem becomes extremely difficult for weakly dispersive materials where the quadratic nonlinearity is very small relative to the uncertainty inherent in experiment. This work provides a means of estimating dispersion in the face of such uncertainty. Specifically, we use a Markov Chain Monte Carlo implementation of Bayesian analysis to provide both the dispersion estimate and the associated confidence interval. The interplay between various system parameters and the size of the resulting confidence interval is discussed. The approach is then applied to several different experimental samples.

© 2010 Optical Society of America

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References

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  1. http://cvilaser.com/Common/PDFs/Dispersion_Equations.pdf.
  2. A. Brzsnyi, A. P. Kovcs, M. Grbe, and K. Osvay, “Advances and limitations of phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun. 281(11) 3051–3061 (2008).
    [Crossref]
  3. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, San Diego, Ca, 2006).
  4. F. K. Fatemi, T. F. Carruthers, and J. W. Lou, “Characterisation of telecommunications pulse trains by fourier-transform and dual-quadrature spectral interferometry,” Electron. Lett. 39(12), 921–922 (2003).
    [Crossref]
  5. M. Joffre, L. Lepetit, and G. Cheriaux, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12(12) 2467–2474 (1995).
    [Crossref]
  6. W. A. Link and R. J. Barker, Bayesian Inference with ecological examples (Academic Press, San Diego, CA, 2010).
  7. P. A. Merritt, R. P. Tatam, and D. A. Jackson, “Interferometric chromatic dispersion measurements on short lengths of monomode optical fiber,” J. Lightwave Technol. 7(4) 703–716 (1989).
    [Crossref]
  8. B. Tatian, “Fitting refractive-index data with the Sellmeier dispersion formula,” Appl. Opt. 23(24) 4477–4485 (1984).
    [Crossref] [PubMed]
  9. L. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Scie. Instrum. 72(1 I) 1–29 (2001).
    [Crossref]
  10. J. Wang and N. Zabaras, “Hierarchical Bayesian models for inverse problems in heat conduction,” Inverse Prob. 21, 183–206 (2005).
    [Crossref]

2008 (1)

A. Brzsnyi, A. P. Kovcs, M. Grbe, and K. Osvay, “Advances and limitations of phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun. 281(11) 3051–3061 (2008).
[Crossref]

2005 (1)

J. Wang and N. Zabaras, “Hierarchical Bayesian models for inverse problems in heat conduction,” Inverse Prob. 21, 183–206 (2005).
[Crossref]

2003 (1)

F. K. Fatemi, T. F. Carruthers, and J. W. Lou, “Characterisation of telecommunications pulse trains by fourier-transform and dual-quadrature spectral interferometry,” Electron. Lett. 39(12), 921–922 (2003).
[Crossref]

2001 (1)

L. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Scie. Instrum. 72(1 I) 1–29 (2001).
[Crossref]

1995 (1)

1989 (1)

P. A. Merritt, R. P. Tatam, and D. A. Jackson, “Interferometric chromatic dispersion measurements on short lengths of monomode optical fiber,” J. Lightwave Technol. 7(4) 703–716 (1989).
[Crossref]

1984 (1)

Barker, R. J.

W. A. Link and R. J. Barker, Bayesian Inference with ecological examples (Academic Press, San Diego, CA, 2010).

Brzsnyi, A.

A. Brzsnyi, A. P. Kovcs, M. Grbe, and K. Osvay, “Advances and limitations of phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun. 281(11) 3051–3061 (2008).
[Crossref]

Carruthers, T. F.

F. K. Fatemi, T. F. Carruthers, and J. W. Lou, “Characterisation of telecommunications pulse trains by fourier-transform and dual-quadrature spectral interferometry,” Electron. Lett. 39(12), 921–922 (2003).
[Crossref]

Cheriaux, G.

Diels, J.-C.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, San Diego, Ca, 2006).

Dorrer, C.

L. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Scie. Instrum. 72(1 I) 1–29 (2001).
[Crossref]

Fatemi, F. K.

F. K. Fatemi, T. F. Carruthers, and J. W. Lou, “Characterisation of telecommunications pulse trains by fourier-transform and dual-quadrature spectral interferometry,” Electron. Lett. 39(12), 921–922 (2003).
[Crossref]

Grbe, M.

A. Brzsnyi, A. P. Kovcs, M. Grbe, and K. Osvay, “Advances and limitations of phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun. 281(11) 3051–3061 (2008).
[Crossref]

Jackson, D. A.

P. A. Merritt, R. P. Tatam, and D. A. Jackson, “Interferometric chromatic dispersion measurements on short lengths of monomode optical fiber,” J. Lightwave Technol. 7(4) 703–716 (1989).
[Crossref]

Joffre, M.

Kovcs, A. P.

A. Brzsnyi, A. P. Kovcs, M. Grbe, and K. Osvay, “Advances and limitations of phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun. 281(11) 3051–3061 (2008).
[Crossref]

Lepetit, L.

Link, W. A.

W. A. Link and R. J. Barker, Bayesian Inference with ecological examples (Academic Press, San Diego, CA, 2010).

Lou, J. W.

F. K. Fatemi, T. F. Carruthers, and J. W. Lou, “Characterisation of telecommunications pulse trains by fourier-transform and dual-quadrature spectral interferometry,” Electron. Lett. 39(12), 921–922 (2003).
[Crossref]

Merritt, P. A.

P. A. Merritt, R. P. Tatam, and D. A. Jackson, “Interferometric chromatic dispersion measurements on short lengths of monomode optical fiber,” J. Lightwave Technol. 7(4) 703–716 (1989).
[Crossref]

Osvay, K.

A. Brzsnyi, A. P. Kovcs, M. Grbe, and K. Osvay, “Advances and limitations of phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun. 281(11) 3051–3061 (2008).
[Crossref]

Rudolph, W.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, San Diego, Ca, 2006).

Tatam, R. P.

P. A. Merritt, R. P. Tatam, and D. A. Jackson, “Interferometric chromatic dispersion measurements on short lengths of monomode optical fiber,” J. Lightwave Technol. 7(4) 703–716 (1989).
[Crossref]

Tatian, B.

Walmsley, L.

L. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Scie. Instrum. 72(1 I) 1–29 (2001).
[Crossref]

Wang, J.

J. Wang and N. Zabaras, “Hierarchical Bayesian models for inverse problems in heat conduction,” Inverse Prob. 21, 183–206 (2005).
[Crossref]

Waxer, L.

L. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Scie. Instrum. 72(1 I) 1–29 (2001).
[Crossref]

Zabaras, N.

J. Wang and N. Zabaras, “Hierarchical Bayesian models for inverse problems in heat conduction,” Inverse Prob. 21, 183–206 (2005).
[Crossref]

Appl. Opt. (1)

Electron. Lett. (1)

F. K. Fatemi, T. F. Carruthers, and J. W. Lou, “Characterisation of telecommunications pulse trains by fourier-transform and dual-quadrature spectral interferometry,” Electron. Lett. 39(12), 921–922 (2003).
[Crossref]

Inverse Prob. (1)

J. Wang and N. Zabaras, “Hierarchical Bayesian models for inverse problems in heat conduction,” Inverse Prob. 21, 183–206 (2005).
[Crossref]

J. Lightwave Technol. (1)

P. A. Merritt, R. P. Tatam, and D. A. Jackson, “Interferometric chromatic dispersion measurements on short lengths of monomode optical fiber,” J. Lightwave Technol. 7(4) 703–716 (1989).
[Crossref]

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

Opt. Commun. (1)

A. Brzsnyi, A. P. Kovcs, M. Grbe, and K. Osvay, “Advances and limitations of phase dispersion measurement by spectrally and spatially resolved interferometry,” Opt. Commun. 281(11) 3051–3061 (2008).
[Crossref]

Rev. Scie. Instrum. (1)

L. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Scie. Instrum. 72(1 I) 1–29 (2001).
[Crossref]

Other (3)

http://cvilaser.com/Common/PDFs/Dispersion_Equations.pdf.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, San Diego, Ca, 2006).

W. A. Link and R. J. Barker, Bayesian Inference with ecological examples (Academic Press, San Diego, CA, 2010).

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Figures (9)

Fig. 1.
Fig. 1.

Experimental setup of the Shearing interferometer for a sample of thickness L. The laser source is reflected from the front and back surfaces of the material forming the interferometer.

Fig. 2.
Fig. 2.

Interferograms of BK-7 (a, c, e) and Silicon (b, d, f) for transmission (a) and (b), reflection (c) and (d), and reflection divided by transmission (e) and (f).

Fig. 3.
Fig. 3.

(a-e) Estimated probability distributions associated with the five model parameters Idc,IA ,ϕ 0,ϕ 1,ϕ 2 along with (f) the estimated distribution for the noise variance. The true parameter values are marked by a vertical line.

Fig. 4.
Fig. 4.

(a) Estimated probability density function for the ϕ 2 parameter where the true value was taken as ϕ 2 = 1e -3 [ps2] and (b) the stationary Markov chain that produced this distribution

Fig. 5.
Fig. 5.

95% Confidence interval as a function of (a) bandwidth, (b) number of data, (c) the linear frequency term ϕ 1 and (d) the signal-to-noise ratio

Fig. 6.
Fig. 6.

Identification of material dispersion for Si glass. (a) Stationary Markov chain for coefficient ϕ 2 (b) resulting estimated probability density function and (c) Sampled data along with the model fit

Fig. 7.
Fig. 7.

Identification of material dispersion for BK-7. (a) Stationary Markov chain for coefficient ϕ 2 (b) resulting estimated probability density function and (c) Sampled data along with the model fit

Fig. 8.
Fig. 8.

Estimated PDFs of material dispersion for BK-7 glass. a) Data collected at 1550nm over a 40nm bandwidth (2x larger than that used in generating Fig. 7b) and b) Data collected at 800nm over a 100nm bandwidth

Algorithm 1.
Algorithm 1.

The MCMC algorithm using Metropolis-Hastings with Gibbs sampling for Gaussian likelihood and Uniform “transition” distributiong(ψ * j , ψj (i - 1)) = 1/2Aj .

Equations (23)

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

ϕ ̂ 2 = E [ p π ( ϕ 2 ) ] ,
I R ( ω ) = a ( 1 γ cos ϕ ( ω ) ) u + v cos ϕ ( ω )
I T ( ω ) = b u + v cos ϕ ( ω )
I ( ω ) = I R ( ω ) I T ( ω )
I ( ω ) = I dc + I A cos ϕ ( ω )
n ( ω ) = n ( ω 0 ) + n ( ω 0 ) ( ω ω 0 ) + ( 1 / 2 ) n ( ω 0 ) ( ω ω 0 ) 2 +
I ( ω ) = I dc + I A cos ( ϕ 0 + ϕ 1 Δ ω + 1 2 ϕ 2 Δ ω 2 + )
ϕ 0 = 2 L c ω 0 n ( ω 0 )
ϕ 1 = 2 L c ( n ( ω 0 ) + ω 0 n ( ω 0 ) )
ϕ 2 = 2 L c ( 2 n ( ω 0 ) + ω 0 n ( ω 0 ) ) .
y n = I n ( I dc , I A , ϕ 0 , ϕ 1 , ϕ 2 ) + η n
p L ( η ) = 1 ( 2 π σ G 2 ) N / 2 exp [ 1 2 σ G 2 n = 1 N η n 2 ] .
p L ( y ψ , σ G 2 ) = 1 ( 2 π σ G 2 ) N / 2 exp [ 1 2 σ G 2 n = 1 N ( y n I n ( ψ ) ) 2 ]
p π ( θ y ) = p L ( y θ ) p π ( θ ) / p D ( y ) .
p π j ( θ j y ) = p 1 p π ( θ y ) d θ j p 1 p L ( y θ ) p π ( θ ) d θ j
r = p π ( θ * ) g ( θ ( i 1 ) θ * ) p π ( θ ( i 1 ) ) g ( θ * θ ( i 1 ) ) ;
g ( θ * θ ( i 1 ) ) = 1 2 A , θ * θ ( i 1 ) < A ,
θ j = ( θ 1 , θ 2 , , θ j 1 , θ j + 1 , , θ P )
p π ( θ j ( i ) y , θ j = θ j ( i 1 ) ) ,
SNR = I A 2 2 σ G 2
Q y ψ = n ( y n I n ( ψ ) ) 2
IG a b = 1 / G ( a , 1 / b )
σ G 2 ( i ) ~ 1.0 / [ G ( N / 2 + 1,2 / Q ( y , ψ ( i ) ) ] .

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