Abstract

A fast Fourier-based measurement system to determine phase, group delay, and group delay dispersion during optical coating processes is proposed. The in situ method is based on a Michelson interferometer with a broad band light source and a very fast spectrometer. To our knowledge, group delay dispersion measurements directly on the moving substrates during a deposition process for complex interference coatings have been demonstrated for the first time. Especially for the very precise production of chirped mirrors it is advantageous to get information about the phase properties of the grown layer stack to react on errors and retrieve more information about the coating process.

© 2016 Optical Society of America

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References

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2015 (1)

S. Schlichting, T. Willemsen, H. Ehlers, U. Morgner, and D. Ristau, “Direct in situ GDD measurement in optical coating process,” Proc. SPIE 9627, 96271S (2015).

2013 (1)

2012 (1)

2011 (2)

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the error self-compensation effect associated with broadband optical monitoring,” Appl. Opt. 50(9), C111–C116 (2011).
[Crossref] [PubMed]

S. Schlichting, K. Heinrich, H. Ehlers, and D. Ristau, “Online re-optimization as a powerful part of enhanced strategies in optical broadband monitoring,” Proc. SPIE 8168, 81681E (2011).
[Crossref]

2010 (1)

2009 (2)

2008 (1)

A. Dursun, Z. Sarac, H. Sarac, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[Crossref]

2007 (2)

S. A. Fulop and K. Fitz, “Separation of components from impulses in reassigned spectrograms,” J. Acoust. Soc. Am. 121(3), 1510–1518 (2007).
[Crossref] [PubMed]

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45(12), 1186–1192 (2007).
[Crossref]

2006 (2)

S. A. Fulop and K. Fitz, “Algorithms for computing the time-corrected instantaneous frequency (reassigned) spectrogram, with applications,” J. Acoust. Soc. Am. 119(1), 360–371 (2006).
[Crossref] [PubMed]

A. V. Tikhonravov, M. K. Trubetskov, and T. V. Amotchkina, “Investigation of the effect of accumulation of thickness errors in optical coating production by broadband optical monitoring,” Appl. Opt. 45(27), 7026–7034 (2006).
[Crossref] [PubMed]

2004 (2)

M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12(11), 2404–2422 (2004).
[Crossref] [PubMed]

S. V. Narasimhan and S. Pavanalatha, “Estimation of evolutionary spectrum based on short time Fourier transform and modified group delay,” Signal Process. 84(11), 2139–2152 (2004).
[Crossref]

2003 (1)

C. R. Pinnegar and L. Mansinha, “The S -transform with windows of arbitrary and varying shape,” Geophysics 68(1), 381–385 (2003).
[Crossref]

2002 (2)

D. J. Nelson, “Instantaneous higher order phase derivatives,” Digit. Signal Process. 12(2-3), 416–428 (2002).
[Crossref]

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

2001 (2)

C. Dorrer and M. Joffre, “Characterization of the spectral phase of ultrashort light pulses,” C. R. Acad. Sci. IV, 1415–1426 (2001).

D. J. Nelson, “Cross-spectral methods for processing speech,” J. Acoust. Soc. Am. 110(5 Pt 1), 2575–2592 (2001).
[Crossref] [PubMed]

2000 (1)

1995 (1)

1990 (1)

1982 (1)

Amin, M. G.

M. G. Amin and K. D. Feng, “Short-time Fourier transform using cascade filter structures,” in Proceedings of IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. (IEEE, 1995), pp. 631–641.
[Crossref]

Amotchkina, T. V.

Arikan, O.

L. Durak and O. Arikan, “Short-Time Fourier Transform: Two Fundamental Properties and an Optimal Implementation,” in Proceedings of IEEE Trans. Signal Process. (IEEE, 2003), pp. 1231–1242.
[Crossref]

Auger, F.

F. Auger and P. Flandrin, “Improving the Readability of Time-Frequency and Time-Scale Representations by the Reassignment Method,” in Proceedings of IEEE Trans. Signal Process. (IEEE, 1995), pp. 1068–1089.
[Crossref]

Belabas, N.

Boudreaux-Bartels, G. F.

F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” in Proceedings of IEEE Signal Proc. Mag. (IEEE, 1992), pp. 21–67.

Cagigal, M. P.

Cagigas, M. A.

Chen, W.

M. Zhong, W. Chen, and M. Jiang, “Application of S-transform profilometry in eliminating nonlinearity in fringe pattern,” Appl. Opt. 51(5), 577–587 (2012).
[Crossref] [PubMed]

W. Chen, N. Kehtarnavaz, and T. W. Spencer, “An efficient recursive algorithm for time-varying Fourier transform., ” in Proceedings of IEEE Trans. Signal Process. (IEEE, 1993), pp. 2488–2490.
[Crossref]

Chériaux, G.

Deng, Y.

Dorrer, C.

C. Dorrer and M. Joffre, “Characterization of the spectral phase of ultrashort light pulses,” C. R. Acad. Sci. IV, 1415–1426 (2001).

C. Dorrer, N. Belabas, J.-P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17(10), 1795–1802 (2000).
[Crossref]

Duker, J.

Durak, L.

L. Durak and O. Arikan, “Short-Time Fourier Transform: Two Fundamental Properties and an Optimal Implementation,” in Proceedings of IEEE Trans. Signal Process. (IEEE, 2003), pp. 1231–1242.
[Crossref]

Dursun, A.

A. Dursun, Z. Sarac, H. Sarac, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[Crossref]

Ecevit, F. N.

A. Dursun, Z. Sarac, H. Sarac, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[Crossref]

Ehlers, H.

S. Schlichting, T. Willemsen, H. Ehlers, U. Morgner, and D. Ristau, “Direct in situ GDD measurement in optical coating process,” Proc. SPIE 9627, 96271S (2015).

S. Schlichting, K. Heinrich, H. Ehlers, and D. Ristau, “Online re-optimization as a powerful part of enhanced strategies in optical broadband monitoring,” Proc. SPIE 8168, 81681E (2011).
[Crossref]

Feng, K. D.

M. G. Amin and K. D. Feng, “Short-time Fourier transform using cascade filter structures,” in Proceedings of IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. (IEEE, 1995), pp. 631–641.
[Crossref]

Fercher, A. F.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

Fitz, K.

S. A. Fulop and K. Fitz, “Separation of components from impulses in reassigned spectrograms,” J. Acoust. Soc. Am. 121(3), 1510–1518 (2007).
[Crossref] [PubMed]

S. A. Fulop and K. Fitz, “Algorithms for computing the time-corrected instantaneous frequency (reassigned) spectrogram, with applications,” J. Acoust. Soc. Am. 119(1), 360–371 (2006).
[Crossref] [PubMed]

Flandrin, P.

F. Auger and P. Flandrin, “Improving the Readability of Time-Frequency and Time-Scale Representations by the Reassignment Method,” in Proceedings of IEEE Trans. Signal Process. (IEEE, 1995), pp. 1068–1089.
[Crossref]

Fujimoto, J.

Fulop, S. A.

S. A. Fulop and K. Fitz, “Separation of components from impulses in reassigned spectrograms,” J. Acoust. Soc. Am. 121(3), 1510–1518 (2007).
[Crossref] [PubMed]

S. A. Fulop and K. Fitz, “Algorithms for computing the time-corrected instantaneous frequency (reassigned) spectrogram, with applications,” J. Acoust. Soc. Am. 119(1), 360–371 (2006).
[Crossref] [PubMed]

Gelb, A.

R. B. Platte and A. Gelb, “A hybrid Fourier–Chebyshev method for partial differential equations,” J. Sci. Comput. 39(2), 244–264 (2009).
[Crossref]

Heinrich, K.

S. Schlichting, K. Heinrich, H. Ehlers, and D. Ristau, “Online re-optimization as a powerful part of enhanced strategies in optical broadband monitoring,” Proc. SPIE 8168, 81681E (2011).
[Crossref]

Hitzenberger, C. K.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

Hlawatsch, F.

F. Hlawatsch and G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” in Proceedings of IEEE Signal Proc. Mag. (IEEE, 1992), pp. 21–67.

Ina, H.

Jiang, M.

Joffre, M.

Karamata, B.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

Kehtarnavaz, N.

W. Chen, N. Kehtarnavaz, and T. W. Spencer, “An efficient recursive algorithm for time-varying Fourier transform., ” in Proceedings of IEEE Trans. Signal Process. (IEEE, 1993), pp. 2488–2490.
[Crossref]

Kemao, Q.

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45(12), 1186–1192 (2007).
[Crossref]

Ko, T.

Kobayashi, S.

Kong, W.

Kowalczyk, A.

Lasser, T.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

Lepetit, L.

Likforman, J.-P.

Liu, K. J. R.

K. J. R. Liu, “Novel parallel architecture for short-time Fourier transform,” in Proceedings of IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process. (IEEE, 1993), pp. 786–790.
[Crossref]

Liu, X.

Luo, Z.

Mansinha, L.

C. R. Pinnegar and L. Mansinha, “The S -transform with windows of arbitrary and varying shape,” Geophysics 68(1), 381–385 (2003).
[Crossref]

Mogi, K.

Morgner, U.

S. Schlichting, T. Willemsen, H. Ehlers, U. Morgner, and D. Ristau, “Direct in situ GDD measurement in optical coating process,” Proc. SPIE 9627, 96271S (2015).

Naganuma, K.

Narasimhan, S. V.

S. V. Narasimhan and S. Pavanalatha, “Estimation of evolutionary spectrum based on short time Fourier transform and modified group delay,” Signal Process. 84(11), 2139–2152 (2004).
[Crossref]

Nelson, D. J.

D. J. Nelson, “Instantaneous higher order phase derivatives,” Digit. Signal Process. 12(2-3), 416–428 (2002).
[Crossref]

D. J. Nelson, “Cross-spectral methods for processing speech,” J. Acoust. Soc. Am. 110(5 Pt 1), 2575–2592 (2001).
[Crossref] [PubMed]

Özder, S.

A. Dursun, Z. Sarac, H. Sarac, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[Crossref]

Pan, C.

C. Pan, “Gibbs Phenomenon Removal and Digital Filtering Directly through the Fast Fourier Transform,” in Proceedings of IEEE Trans. Signal Process. (IEEE, 2001), pp. 444–448.

Pavanalatha, S.

S. V. Narasimhan and S. Pavanalatha, “Estimation of evolutionary spectrum based on short time Fourier transform and modified group delay,” Signal Process. 84(11), 2139–2152 (2004).
[Crossref]

Pinnegar, C. R.

C. R. Pinnegar and L. Mansinha, “The S -transform with windows of arbitrary and varying shape,” Geophysics 68(1), 381–385 (2003).
[Crossref]

Platte, R. B.

R. B. Platte and A. Gelb, “A hybrid Fourier–Chebyshev method for partial differential equations,” J. Sci. Comput. 39(2), 244–264 (2009).
[Crossref]

Ristau, D.

S. Schlichting, T. Willemsen, H. Ehlers, U. Morgner, and D. Ristau, “Direct in situ GDD measurement in optical coating process,” Proc. SPIE 9627, 96271S (2015).

S. Schlichting, K. Heinrich, H. Ehlers, and D. Ristau, “Online re-optimization as a powerful part of enhanced strategies in optical broadband monitoring,” Proc. SPIE 8168, 81681E (2011).
[Crossref]

Sarac, H.

A. Dursun, Z. Sarac, H. Sarac, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[Crossref]

Sarac, Z.

A. Dursun, Z. Sarac, H. Sarac, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[Crossref]

Schlichting, S.

S. Schlichting, T. Willemsen, H. Ehlers, U. Morgner, and D. Ristau, “Direct in situ GDD measurement in optical coating process,” Proc. SPIE 9627, 96271S (2015).

S. Schlichting, K. Heinrich, H. Ehlers, and D. Ristau, “Online re-optimization as a powerful part of enhanced strategies in optical broadband monitoring,” Proc. SPIE 8168, 81681E (2011).
[Crossref]

Shen, W.

Spencer, T. W.

W. Chen, N. Kehtarnavaz, and T. W. Spencer, “An efficient recursive algorithm for time-varying Fourier transform., ” in Proceedings of IEEE Trans. Signal Process. (IEEE, 1993), pp. 2488–2490.
[Crossref]

Srinivasan, V.

Sticker, M.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

Takeda, M.

Tao, J.

Tikhonravov, A. V.

Trubetskov, M. K.

Valle, P. J.

Wang, X.

Willemsen, T.

S. Schlichting, T. Willemsen, H. Ehlers, U. Morgner, and D. Ristau, “Direct in situ GDD measurement in optical coating process,” Proc. SPIE 9627, 96271S (2015).

Wojtkowski, M.

Xia, C.

Yamada, H.

Yang, W.

Yin, Y.

Zawadzki, R.

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

Zhang, S.

Zhang, Z.

Zhong, M.

Zhou, C.

Appl. Opt. (3)

C. R. Acad. Sci. (1)

C. Dorrer and M. Joffre, “Characterization of the spectral phase of ultrashort light pulses,” C. R. Acad. Sci. IV, 1415–1426 (2001).

Chin. Opt. Lett. (1)

Digit. Signal Process. (1)

D. J. Nelson, “Instantaneous higher order phase derivatives,” Digit. Signal Process. 12(2-3), 416–428 (2002).
[Crossref]

Geophysics (1)

C. R. Pinnegar and L. Mansinha, “The S -transform with windows of arbitrary and varying shape,” Geophysics 68(1), 381–385 (2003).
[Crossref]

J. Acoust. Soc. Am. (3)

S. A. Fulop and K. Fitz, “Separation of components from impulses in reassigned spectrograms,” J. Acoust. Soc. Am. 121(3), 1510–1518 (2007).
[Crossref] [PubMed]

S. A. Fulop and K. Fitz, “Algorithms for computing the time-corrected instantaneous frequency (reassigned) spectrogram, with applications,” J. Acoust. Soc. Am. 119(1), 360–371 (2006).
[Crossref] [PubMed]

D. J. Nelson, “Cross-spectral methods for processing speech,” J. Acoust. Soc. Am. 110(5 Pt 1), 2575–2592 (2001).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

J. Sci. Comput. (1)

R. B. Platte and A. Gelb, “A hybrid Fourier–Chebyshev method for partial differential equations,” J. Sci. Comput. 39(2), 244–264 (2009).
[Crossref]

Measurement (1)

A. Dursun, Z. Sarac, H. Sarac, S. Özder, and F. N. Ecevit, “Phase recovery from interference fringes by using S-transform,” Measurement 41(4), 403–411 (2008).
[Crossref]

Opt. Commun. (1)

A. F. Fercher, C. K. Hitzenberger, M. Sticker, R. Zawadzki, B. Karamata, and T. Lasser, “Dispersion compensation for optical coherence tomography depth-scan signals by a numerical technique,” Opt. Commun. 204(1-6), 67–74 (2002).
[Crossref]

Opt. Express (3)

Opt. Lasers Eng. (1)

Q. Kemao, “On window size selection in the windowed Fourier ridges algorithm,” Opt. Lasers Eng. 45(12), 1186–1192 (2007).
[Crossref]

Opt. Lett. (1)

Proc. SPIE (2)

S. Schlichting, K. Heinrich, H. Ehlers, and D. Ristau, “Online re-optimization as a powerful part of enhanced strategies in optical broadband monitoring,” Proc. SPIE 8168, 81681E (2011).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Schematic of measurement equipment.
Fig. 2
Fig. 2 In situ interferometric measurement of a chirped mirror.
Fig. 3
Fig. 3 Real part of Fourier transformed interferometric measurement data S(ω) in time domain and super Gaussian separation function with σ=240 fs and λ=8 .
Fig. 4
Fig. 4 Real part of the separated Fourier transformed signal shifted to zero to eliminate time delay τ.
Fig. 5
Fig. 5 STFT result of separated and shifted Fourier transformed signal, flat-top sampling window function with width 144 fs.
Fig. 6
Fig. 6 Comparison of GD measurements of a chirped mirror −600 fs2 in the wavelength range of 830 to 855 nm wavelength, offset corrected.
Fig. 7
Fig. 7 GDD measurement of a chirped mirror −600 fs2 from 830 to 855 nm wavelength (design with 54 layers).
Fig. 8
Fig. 8 In situ GDD measurements during deposition of last layer of a chirped mirror −600 fs2 from 830 to 855 nm wavelength (design with 58 layers).

Equations (9)

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Δφ=arg[ E S ( ω ) ]arg[ E R ( ω ) ].
S( ω )= | E R ( ω ) | 2 + | E S ( ω ) | 2 +2Re{ E R * ( ω ) E S ( ω ) } =  | E R ( ω ) | 2 + | E S ( ω ) | 2 +  E R * ( ω ) E S ( ω )exp[ i( ωτ+Δφ ) ]+c.c. .
F T 1 S( ω )= E R * ( T ) E R ( T )+ E S * ( T ) E S ( T )+S( Tτ )+S ( Tτ ) * .
w( T )=exp[ ( ( Tτ ) σ ) 2λ ].
X( T,ω )= - h[ t ]S[ t+T ]exp( -iωt )dt
GD( T,ω )= ω arg[ X( T,ω ) ].
GD( ω )= T X( T,ω ) dT X( T,ω ) dT .
GDD( ω )= ( GD ) ω
h[ t ]= a 0 a 1 cos( 2πt n )+ a 2 cos( 4πt n ) a 3 cos( 6πt n )+ a 4 cos( 8πt n )

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