Abstract

Herewith, we describe how intensity and phase of the ultrashort pulse retrieved with second-harmonic frequency-resolved optical gating (SHG FROG) can be utilized for measurement of the nonlinear refractive index (n 2). Through comparison with available literature, we show that our method surpasses Z-scan in terms of precision by a factor of two, and thus, constitutes an interesting alternative. We present results for various materials: fused silica, calcite, YVO 4, BiBO, CaF 2, and YAG at 1030 nm. Unlike the Z-scan, the use of this method is not restricted to free-space geometry, but due to its characteristics, it can be used in integrated waveguides or photonic crystal fibers as well.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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2018 (1)

2015 (2)

2014 (1)

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

2011 (1)

2008 (2)

D. J. Kane, “Principal components generalized projections: a review,” J. Opt. Soc. Am. B 25, A120–A132 (2008).
[Crossref]

S. Miller, F. Rotermund, G. Xu, F. Noack, V. Panyutin, and V. Petrov, “Polarization-dependent nonlinear refractive index of BiB 3O 6,” Opt. Mater. 30, 1469–1472 (2008).
[Crossref]

2006 (2)

A. G. Selivanov, I. A. Denisov, N. V. Kuleshov, and K. V. Yumashev, “Nonlinear refractive properties of Yb3+-doped KY(WO 4) 2,” Appl. Phys. B 83, 61–65 (2006).
[Crossref]

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

2003 (2)

Z. Wang, E. Zeek, R. Trebino, and P. Kvan, “Determining error bars in measurements of ultrashort laser pulses,” J. Opt. Soc. Am. B 20, 2400–2405 (2003).
[Crossref]

S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E. Koudoumas, and X. Michaut, “An experimental investigation of the nonlinear refractive index (n 2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369, 318–324 (2003).
[Crossref]

2002 (1)

1998 (2)

D. Milam, “Review and assessment of measured values of the nonlinear refractive-index coefficient of fused silica,” Appl. Opt. 37, 546–550 (1998).
[Crossref]

V. P. Kalosha, M. Müller, J. Herrmann, and S. Gatz, “Spatiotemporal model of femtosecond pulse generation in Kerr-lens mode-locked solid-state lasers,” JOSA B 15, 535–550 (1998).
[Crossref]

1996 (2)

A. J. Taylor, G. Rodriguez, and T. S. Clement, “Determination of n 2 by direct measurement of the optical phase,” Opt. Lett. 21, 1812–1814 (1996).
[Crossref] [PubMed]

K. W. DeLong, D. N. Fittinghoff, and R. Trebino, “Practical issues in ultrashort-laser-pulse measurement using frequency-resolved optical gating,” IEEE J. Quantum Electron. 32, 1253–1264 (1996).
[Crossref]

1994 (1)

1991 (1)

1989 (2)

1988 (1)

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39, 3337 (1988).

1987 (2)

S. R. Friberg and P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. 23, 2089–2094 (1987).
[Crossref]

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive-index measurements of glasses using three-wave frequency mixing,” J. Opt. Soc. Am. B 4, 875–881 (1987).
[Crossref]

1984 (1)

W. E. Williams, M. J. Soileau, and W. V. Stryland, “Optical switching and n 2 measurements in CS 2,” Opt. Commun. 50, 256–260 (1984).
[Crossref]

1978 (2)

M. J. Weber, D. Milam, and W. L. Smith, “Nonlinear refractive index of glasses and crystals,” Opt. Eng. 17, 463–469 (1978).
[Crossref]

D. Milam, M. J. Weber, and A. J. Glass, “Nonlinear refractive index of fluoride crystals,” Appl. Phys. Lett. 31, 822–825 (1978).
[Crossref]

1975 (1)

M. J. Moran, C. Y. She, and R. L. Carman, “Interferometric measurements of nonlinear refractive-index coefficient relative to CS 2 in laser-system-related materials,” IEEE J. Quantum Electron. 11, 259–263 (1975).
[Crossref]

1974 (2)

E. S. Bliss, D. R. Speck, and W. W. Simmons, “Direct interferometric measurements of the nonlinear refractive index coefficient n 2 in laser materials,” Appl. Phys. Lett. 25, 728–730 (1974).
[Crossref]

D. R. Barker and L. M. Diana, “Simple method for fitting data when both variables have uncertainties,” Am. J. Phys. 42, 224–227 (1974).
[Crossref]

1973 (1)

A. Owyoung, “Ellipse rotation studies in laser host materials,” IEEE J. Quantum Electron. 9, 1064–1069 (1973).
[Crossref]

1970 (2)

R. R. Alfano and S. L. Shapiro, “Direct distortion of electronic clouds of rare-gas atoms in intense electric fields,” Phys. Rev. Lett. 24, 1217 (1970).
[Crossref]

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses,” Phys. Rev. Lett. 24, 592 (1970).
[Crossref]

Abdolvand, A.

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

Adair, R.

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39, 3337 (1988).

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive-index measurements of glasses using three-wave frequency mixing,” J. Opt. Soc. Am. B 4, 875–881 (1987).
[Crossref]

Agrawal, G.

G. Agrawal, Nonlinear fiber optics (Academic, 2001).

Ainslie, B. J.

Alfano, R. R.

R. R. Alfano and S. L. Shapiro, “Direct distortion of electronic clouds of rare-gas atoms in intense electric fields,” Phys. Rev. Lett. 24, 1217 (1970).
[Crossref]

R. R. Alfano and S. L. Shapiro, “Observation of self-phase modulation and small-scale filaments in crystals and glasses,” Phys. Rev. Lett. 24, 592 (1970).
[Crossref]

Auxier, J. M.

Barker, D. R.

D. R. Barker and L. M. Diana, “Simple method for fitting data when both variables have uncertainties,” Am. J. Phys. 42, 224–227 (1974).
[Crossref]

Bayya, S. S.

Beadie, G.

Bliss, E. S.

E. S. Bliss, D. R. Speck, and W. W. Simmons, “Direct interferometric measurements of the nonlinear refractive index coefficient n 2 in laser materials,” Appl. Phys. Lett. 25, 728–730 (1974).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Elsevier, 2003).

Carman, R. L.

M. J. Moran, C. Y. She, and R. L. Carman, “Interferometric measurements of nonlinear refractive-index coefficient relative to CS 2 in laser-system-related materials,” IEEE J. Quantum Electron. 11, 259–263 (1975).
[Crossref]

Chang, W.

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

Chase, L. L.

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39, 3337 (1988).

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive-index measurements of glasses using three-wave frequency mixing,” J. Opt. Soc. Am. B 4, 875–881 (1987).
[Crossref]

Chaux, R.

S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E. Koudoumas, and X. Michaut, “An experimental investigation of the nonlinear refractive index (n 2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369, 318–324 (2003).
[Crossref]

S. Xu, B. Lavorel, O. Faucher, and R. Chaux, “Characterization of self-phase modulated ultrashort optical pulses by spectral phase interferometry,” J. Opt. Soc. Am. B 19, 165–168 (2002).
[Crossref]

Clement, T. S.

Clifford, A. A.

A. A. Clifford, Multivariate Error Analysis: A Handbook of Error Propagation and Calculation in Many-Parameter Systems (John Wiley and Sons, 1973).

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Cotter, D.

Couris, S.

S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E. Koudoumas, and X. Michaut, “An experimental investigation of the nonlinear refractive index (n 2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369, 318–324 (2003).
[Crossref]

DeLong, K. W.

K. W. DeLong, D. N. Fittinghoff, and R. Trebino, “Practical issues in ultrashort-laser-pulse measurement using frequency-resolved optical gating,” IEEE J. Quantum Electron. 32, 1253–1264 (1996).
[Crossref]

K. W. DeLong, R. Trebino, J. Hunter, and W. E. White, “Frequency-resolved optical gating with the use of second-harmonic generation,” J. Opt. Soc. Am. B 11, 2206–2215 (1994).
[Crossref]

Denisov, I. A.

A. G. Selivanov, I. A. Denisov, N. V. Kuleshov, and K. V. Yumashev, “Nonlinear refractive properties of Yb3+-doped KY(WO 4) 2,” Appl. Phys. B 83, 61–65 (2006).
[Crossref]

Diana, L. M.

D. R. Barker and L. M. Diana, “Simple method for fitting data when both variables have uncertainties,” Am. J. Phys. 42, 224–227 (1974).
[Crossref]

Dudley, J. M.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Faucher, O.

S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E. Koudoumas, and X. Michaut, “An experimental investigation of the nonlinear refractive index (n 2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369, 318–324 (2003).
[Crossref]

S. Xu, B. Lavorel, O. Faucher, and R. Chaux, “Characterization of self-phase modulated ultrashort optical pulses by spectral phase interferometry,” J. Opt. Soc. Am. B 19, 165–168 (2002).
[Crossref]

Fittinghoff, D. N.

K. W. DeLong, D. N. Fittinghoff, and R. Trebino, “Practical issues in ultrashort-laser-pulse measurement using frequency-resolved optical gating,” IEEE J. Quantum Electron. 32, 1253–1264 (1996).
[Crossref]

Flom, S. R.

Friberg, S. R.

S. R. Friberg and P. W. Smith, “Nonlinear optical glasses for ultrafast optical switches,” IEEE J. Quantum Electron. 23, 2089–2094 (1987).
[Crossref]

Gatz, S.

V. P. Kalosha, M. Müller, J. Herrmann, and S. Gatz, “Spatiotemporal model of femtosecond pulse generation in Kerr-lens mode-locked solid-state lasers,” JOSA B 15, 535–550 (1998).
[Crossref]

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[Crossref]

Ghasemi, H.

Girdlestone, H. P.

Glass, A. J.

D. Milam, M. J. Weber, and A. J. Glass, “Nonlinear refractive index of fluoride crystals,” Appl. Phys. Lett. 31, 822–825 (1978).
[Crossref]

Herrmann, J.

V. P. Kalosha, M. Müller, J. Herrmann, and S. Gatz, “Spatiotemporal model of femtosecond pulse generation in Kerr-lens mode-locked solid-state lasers,” JOSA B 15, 535–550 (1998).
[Crossref]

Hölzer, P.

P. S. J. Russell, P. Hölzer, W. Chang, A. Abdolvand, and J. C. Travers, “Hollow-core photonic crystal fibres for gas-based nonlinear optics,” Nat. Photonics 8, 278–286 (2014).
[Crossref]

Hunter, J.

Ironside, C. N.

Kalosha, V. P.

V. P. Kalosha, M. Müller, J. Herrmann, and S. Gatz, “Spatiotemporal model of femtosecond pulse generation in Kerr-lens mode-locked solid-state lasers,” JOSA B 15, 535–550 (1998).
[Crossref]

Kane, D. J.

Kardas, T. M.

Kean, P. N.

Khalesifard, H. R.

Kirkpatrick, S.

R. L. Sutherland, D. G. McLean, and S. Kirkpatrick, Handbook of Nonlinear Optics (Marcel Dekker, 2003), 2nd ed.
[Crossref]

Koudoumas, E.

S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E. Koudoumas, and X. Michaut, “An experimental investigation of the nonlinear refractive index (n 2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369, 318–324 (2003).
[Crossref]

Kuleshov, N. V.

A. G. Selivanov, I. A. Denisov, N. V. Kuleshov, and K. V. Yumashev, “Nonlinear refractive properties of Yb3+-doped KY(WO 4) 2,” Appl. Phys. B 83, 61–65 (2006).
[Crossref]

Kvan, P.

Lavorel, B.

S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E. Koudoumas, and X. Michaut, “An experimental investigation of the nonlinear refractive index (n 2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369, 318–324 (2003).
[Crossref]

S. Xu, B. Lavorel, O. Faucher, and R. Chaux, “Characterization of self-phase modulated ultrashort optical pulses by spectral phase interferometry,” J. Opt. Soc. Am. B 19, 165–168 (2002).
[Crossref]

McLean, D. G.

R. L. Sutherland, D. G. McLean, and S. Kirkpatrick, Handbook of Nonlinear Optics (Marcel Dekker, 2003), 2nd ed.
[Crossref]

Michalska, M.

Michaut, X.

S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E. Koudoumas, and X. Michaut, “An experimental investigation of the nonlinear refractive index (n 2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369, 318–324 (2003).
[Crossref]

Milam, D.

D. Milam, “Review and assessment of measured values of the nonlinear refractive-index coefficient of fused silica,” Appl. Opt. 37, 546–550 (1998).
[Crossref]

D. Milam, M. J. Weber, and A. J. Glass, “Nonlinear refractive index of fluoride crystals,” Appl. Phys. Lett. 31, 822–825 (1978).
[Crossref]

M. J. Weber, D. Milam, and W. L. Smith, “Nonlinear refractive index of glasses and crystals,” Opt. Eng. 17, 463–469 (1978).
[Crossref]

Miller, S.

S. Miller, F. Rotermund, G. Xu, F. Noack, V. Panyutin, and V. Petrov, “Polarization-dependent nonlinear refractive index of BiB 3O 6,” Opt. Mater. 30, 1469–1472 (2008).
[Crossref]

Moran, M. J.

M. J. Moran, C. Y. She, and R. L. Carman, “Interferometric measurements of nonlinear refractive-index coefficient relative to CS 2 in laser-system-related materials,” IEEE J. Quantum Electron. 11, 259–263 (1975).
[Crossref]

Müller, M.

V. P. Kalosha, M. Müller, J. Herrmann, and S. Gatz, “Spatiotemporal model of femtosecond pulse generation in Kerr-lens mode-locked solid-state lasers,” JOSA B 15, 535–550 (1998).
[Crossref]

Noack, F.

S. Miller, F. Rotermund, G. Xu, F. Noack, V. Panyutin, and V. Petrov, “Polarization-dependent nonlinear refractive index of BiB 3O 6,” Opt. Mater. 30, 1469–1472 (2008).
[Crossref]

Owyoung, A.

A. Owyoung, “Ellipse rotation studies in laser host materials,” IEEE J. Quantum Electron. 9, 1064–1069 (1973).
[Crossref]

Panyutin, V.

S. Miller, F. Rotermund, G. Xu, F. Noack, V. Panyutin, and V. Petrov, “Polarization-dependent nonlinear refractive index of BiB 3O 6,” Opt. Mater. 30, 1469–1472 (2008).
[Crossref]

Payne, S. A.

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index of optical crystals,” Phys. Rev. B 39, 3337 (1988).

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive-index measurements of glasses using three-wave frequency mixing,” J. Opt. Soc. Am. B 4, 875–881 (1987).
[Crossref]

Petrov, V.

S. Miller, F. Rotermund, G. Xu, F. Noack, V. Panyutin, and V. Petrov, “Polarization-dependent nonlinear refractive index of BiB 3O 6,” Opt. Mater. 30, 1469–1472 (2008).
[Crossref]

Radzewicz, C.

Rasouli, S.

Renard, M.

S. Couris, M. Renard, O. Faucher, B. Lavorel, R. Chaux, E. Koudoumas, and X. Michaut, “An experimental investigation of the nonlinear refractive index (n 2) of carbon disulfide and toluene by spectral shearing interferometry and z-scan techniques,” Chem. Phys. Lett. 369, 318–324 (2003).
[Crossref]

Rodriguez, G.

Rotermund, F.

S. Miller, F. Rotermund, G. Xu, F. Noack, V. Panyutin, and V. Petrov, “Polarization-dependent nonlinear refractive index of BiB 3O 6,” Opt. Mater. 30, 1469–1472 (2008).
[Crossref]

Russell, P. S. J.

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Figures (6)

Fig. 1
Fig. 1 Schematic description of the method of nonlinear index of refraction measurement. (a) First, the known ultrashort pulse is propagated through the sample to be measured, which modifies its phase via self-phase modulation, whilst having little effect on the intensity envelope. (b) Subtraction of the phase of the initial pulse from the phase measured after propagation through the sample yields SPM-induced phase component. (c) Intensity and the SPM-induced phase are fitted together and according to Eq. (1) the value of n   2 is obtained.
Fig. 2
Fig. 2 Measurement setup used in the experiment. Femtosecond pulses are weakly focused onto the sample and after attenuation of the beam are characterized via the FROG apparatus.
Fig. 3
Fig. 3 Procedure of fitting SPM-induced phase to intensity profile for measuring n   2 in fused silica for 160 μJ. (a) Visual comparison of intensity profiles for a pulse propagated through the sample and a reference pulse. (b) Subtraction of the reference phase from the phase measured after propagation through the sample. Their difference is the SPM-induced component of the phase. (c) Phase predicted from intensity profile is fitted to the measured SPM-induced phase. Best fit line is described by Eq. (3). Slope of the purple line presented here is equal to n   2 value for measured material. (d) Intensity profile scaled according to Eq. (3) is compared to the measured SPM-induced phase, showing almost perfect agreement.
Fig. 4
Fig. 4 SPM-induced phases (dashed red) fitted to intensity profiles (blue) for different incident pulse energies for fused silica, CaF   2, YAG and BiBO.
Fig. 5
Fig. 5 SPM-induced phases (dashed red) fitted to intensity profiles (blue) for different incident pulse energies for birefringent materials: Calcite and YVO   4.
Fig. 6
Fig. 6 All measured n2 values for varying pulse energy for different samples. Straight lines are weighted averages of values for given sample.

Tables (1)

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Table 1 Nonlinear Refractive Index Measured in This Work at 1030 nm Compared with Values Known from Literaturea

Equations (9)

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Δ ϕ ( t ) = n 2 2 π d λ I ( t ) ,
Δ ϕ ( t ) = ϕ s ( t ) ϕ i ( t )
Δ ϕ ( t ) = n 2 2 π d λ I ( t ) + a t + b ,
ϕ t = 2 π d λ I s = 2 π d λ E π w 0 2 I s I s d t
ϕ t e r r = p ( d ϕ t d p p e r r ) 2 = ϕ t ( I s e r r I s ) 2 + ( λ e r r λ ) 2 + ( d e r r d ) 2 + ( E e r r E ) 2 + 4 ( w 0 e r r w 0 ) 2
W 1 = 1 ( ϕ S P M e r r ) 2
W j = 1 ( ϕ S P M e r r ) 2 + ( n 2 , j 1 ϕ t e r r ) 2
n ¯ 2 = i k n 2 , i / n 2 , i e r r i k 1 / n 2 , i e r r , n ¯ 2 e r r = max  ( σ i n t 2 , σ e x t 2 )
σ i n t 2 = 1 i k 1 / n 2 , i e r r and σ e x t 2 = σ i n t 2 k 1 i k ( n 2 , i n ¯ 2 n 2 , i e r r ) 2

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