Abstract

We develop a series of analytical approximations allowing for rapid extraction of the nonlinear parameters from beam deflection measurements. We then apply these approximations to the analysis of cadmium silicon phosphide and compare the results against previously published parameter extraction methods and find good agreement for typical experimental conditions.

© 2017 Optical Society of America

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References

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  1. D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photonics 2(1), 60–200 (2010).
    [Crossref]
  2. C. B. de Araújo, A. S. L. Gomes, and G. Boudebs, “Techniques for nonlinear optical characterization of materials: a review,” Rep. Prog. Phys. 79(3), 036401 (2016).
    [Crossref] [PubMed]
  3. J. K. Wahlstrand, J. H. Odhner, E. T. McCole, Y. H. Cheng, J. P. Palastro, R. J. Levis, and H. M. Milchberg, “Effect of two-beam coupling in strong-field optical pump-probe experiments,” Phys. Rev. A 87(5), 053801 (2013).
    [Crossref]
  4. B. A. Ruzicka, S. Wang, J. Liu, K.-P. Loh, J. Z. Wu, and H. Zhao, “Spatially resolved pump-probe study of single-layer graphene produced by chemical vapor deposition [Invited],” Opt. Mater. Express 2(6), 708–716 (2012).
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    [Crossref] [PubMed]
  6. D. S. Kummli, H. M. Frey, and S. Leutwyler, “Femtosecond degenerate four-wave mixing of carbon disulfide: high-accuracy rotational constants,” J. Chem. Phys. 124(14), 144307 (2006).
    [Crossref] [PubMed]
  7. M. R. Ferdinandus, H. Hu, M. Reichert, D. J. Hagan, and E. W. Van Stryland, “Beam deflection measurement of time and polarization resolved ultrafast nonlinear refraction,” Opt. Lett. 38(18), 3518–3521 (2013).
    [Crossref] [PubMed]
  8. R. A. Negres, J. M. Hales, A. Kobyakov, D. J. Hagan, and E. W. Van Stryland, “Experiment and analysis of two-photon absorption spectroscopy using a white-light continuum probe,” Quantum Electronics, IEEE Journal of 38(9), 1205–1216 (2002).
    [Crossref]
  9. M. Reichert, H. Hu, M. R. Ferdinandus, M. Seidel, P. Zhao, T. R. Ensley, D. Peceli, J. M. Reed, D. A. Fishman, S. Webster, D. J. Hagan, and E. W. Van Stryland, “Temporal, spectral, and polarization dependence of the nonlinear optical response of carbon disulfide,” Optica 1(6), 436 (2014).
    [Crossref]
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  11. G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS2, WS2, and MoSe2 nanosheet dispersions [Invited],” Photonics Research 3(2), A51 (2015).
    [Crossref]
  12. K. J. A. Ooi, J. L. Cheng, J. E. Sipe, L. K. Ang, and D. T. H. Tan, “Ultrafast, broadband, and configurable midinfrared all-optical switching in nonlinear graphene plasmonic waveguides,” APL Photonics 1(4), 046101 (2016).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  19. K. Kato, N. Umemura, and V. Petrov, “Sellmeier and thermo-optic dispersion formulas for CdSiP2,” J. Appl. Phys. 109(11), 116104 (2011).
    [Crossref]
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    [Crossref] [PubMed]

2016 (3)

C. B. de Araújo, A. S. L. Gomes, and G. Boudebs, “Techniques for nonlinear optical characterization of materials: a review,” Rep. Prog. Phys. 79(3), 036401 (2016).
[Crossref] [PubMed]

E. Dremetsika, B. Dlubak, S. P. Gorza, C. Ciret, M. B. Martin, S. Hofmann, P. Seneor, D. Dolfi, S. Massar, P. Emplit, and P. Kockaert, “Measuring the nonlinear refractive index of graphene using the optical Kerr effect method,” Opt. Lett. 41(14), 3281–3284 (2016).
[Crossref] [PubMed]

K. J. A. Ooi, J. L. Cheng, J. E. Sipe, L. K. Ang, and D. T. H. Tan, “Ultrafast, broadband, and configurable midinfrared all-optical switching in nonlinear graphene plasmonic waveguides,” APL Photonics 1(4), 046101 (2016).
[Crossref]

2015 (4)

2014 (1)

2013 (3)

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref] [PubMed]

M. R. Ferdinandus, H. Hu, M. Reichert, D. J. Hagan, and E. W. Van Stryland, “Beam deflection measurement of time and polarization resolved ultrafast nonlinear refraction,” Opt. Lett. 38(18), 3518–3521 (2013).
[Crossref] [PubMed]

J. K. Wahlstrand, J. H. Odhner, E. T. McCole, Y. H. Cheng, J. P. Palastro, R. J. Levis, and H. M. Milchberg, “Effect of two-beam coupling in strong-field optical pump-probe experiments,” Phys. Rev. A 87(5), 053801 (2013).
[Crossref]

2012 (3)

2011 (2)

2010 (2)

D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photonics 2(1), 60–200 (2010).
[Crossref]

K. T. Zawilski, P. G. Schunemann, T. C. Pollak, D. E. Zelmon, N. C. Fernelius, and F. Kenneth Hopkins, “Growth and characterization of large CdSiP2 single crystals,” J. Cryst. Growth 312(8), 1127–1132 (2010).
[Crossref]

2006 (1)

D. S. Kummli, H. M. Frey, and S. Leutwyler, “Femtosecond degenerate four-wave mixing of carbon disulfide: high-accuracy rotational constants,” J. Chem. Phys. 124(14), 144307 (2006).
[Crossref] [PubMed]

2002 (1)

R. A. Negres, J. M. Hales, A. Kobyakov, D. J. Hagan, and E. W. Van Stryland, “Experiment and analysis of two-photon absorption spectroscopy using a white-light continuum probe,” Quantum Electronics, IEEE Journal of 38(9), 1205–1216 (2002).
[Crossref]

1998 (1)

Ang, L. K.

K. J. A. Ooi, J. L. Cheng, J. E. Sipe, L. K. Ang, and D. T. H. Tan, “Ultrafast, broadband, and configurable midinfrared all-optical switching in nonlinear graphene plasmonic waveguides,” APL Photonics 1(4), 046101 (2016).
[Crossref]

Baudisch, M.

Biegert, J.

Blau, W. J.

G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS2, WS2, and MoSe2 nanosheet dispersions [Invited],” Photonics Research 3(2), A51 (2015).
[Crossref]

Boltasseva, A.

Bonner, C. E.

Boudebs, G.

C. B. de Araújo, A. S. L. Gomes, and G. Boudebs, “Techniques for nonlinear optical characterization of materials: a review,” Rep. Prog. Phys. 79(3), 036401 (2016).
[Crossref] [PubMed]

Boulanger, B.

Chaitanya Kumar, S.

Cheng, J. L.

K. J. A. Ooi, J. L. Cheng, J. E. Sipe, L. K. Ang, and D. T. H. Tan, “Ultrafast, broadband, and configurable midinfrared all-optical switching in nonlinear graphene plasmonic waveguides,” APL Photonics 1(4), 046101 (2016).
[Crossref]

Cheng, Y.

G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS2, WS2, and MoSe2 nanosheet dispersions [Invited],” Photonics Research 3(2), A51 (2015).
[Crossref]

Cheng, Y. H.

J. K. Wahlstrand, J. H. Odhner, E. T. McCole, Y. H. Cheng, J. P. Palastro, R. J. Levis, and H. M. Milchberg, “Effect of two-beam coupling in strong-field optical pump-probe experiments,” Phys. Rev. A 87(5), 053801 (2013).
[Crossref]

Christodoulides, D. N.

D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photonics 2(1), 60–200 (2010).
[Crossref]

Ciret, C.

Coleman, J. N.

G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS2, WS2, and MoSe2 nanosheet dispersions [Invited],” Photonics Research 3(2), A51 (2015).
[Crossref]

Courtwright, D.

de Araújo, C. B.

C. B. de Araújo, A. S. L. Gomes, and G. Boudebs, “Techniques for nonlinear optical characterization of materials: a review,” Rep. Prog. Phys. 79(3), 036401 (2016).
[Crossref] [PubMed]

DeVault, C.

Dlubak, B.

Dolfi, D.

Dremetsika, E.

Ebrahim-Zadeh, M.

Emplit, P.

Ensley, T. R.

Ferdinandus, M. R.

Fernelius, N. C.

K. T. Zawilski, P. G. Schunemann, T. C. Pollak, D. E. Zelmon, N. C. Fernelius, and F. Kenneth Hopkins, “Growth and characterization of large CdSiP2 single crystals,” J. Cryst. Growth 312(8), 1127–1132 (2010).
[Crossref]

Fishman, D. A.

Fox, D.

G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS2, WS2, and MoSe2 nanosheet dispersions [Invited],” Photonics Research 3(2), A51 (2015).
[Crossref]

Frey, H. M.

D. S. Kummli, H. M. Frey, and S. Leutwyler, “Femtosecond degenerate four-wave mixing of carbon disulfide: high-accuracy rotational constants,” J. Chem. Phys. 124(14), 144307 (2006).
[Crossref] [PubMed]

Gavrilenko, V. I.

Ghosh, G.

Giessen, H.

Gomes, A. S. L.

C. B. de Araújo, A. S. L. Gomes, and G. Boudebs, “Techniques for nonlinear optical characterization of materials: a review,” Rep. Prog. Phys. 79(3), 036401 (2016).
[Crossref] [PubMed]

Gorza, S. P.

Hagan, D. J.

Hales, J. M.

R. A. Negres, J. M. Hales, A. Kobyakov, D. J. Hagan, and E. W. Van Stryland, “Experiment and analysis of two-photon absorption spectroscopy using a white-light continuum probe,” Quantum Electronics, IEEE Journal of 38(9), 1205–1216 (2002).
[Crossref]

Hofmann, S.

Hu, H.

Jelínek, M.

Kato, K.

K. Kato, N. Umemura, and V. Petrov, “Sellmeier and thermo-optic dispersion formulas for CdSiP2,” J. Appl. Phys. 109(11), 116104 (2011).
[Crossref]

Kemlin, V.

Kenneth Hopkins, F.

K. T. Zawilski, P. G. Schunemann, T. C. Pollak, D. E. Zelmon, N. C. Fernelius, and F. Kenneth Hopkins, “Growth and characterization of large CdSiP2 single crystals,” J. Cryst. Growth 312(8), 1127–1132 (2010).
[Crossref]

Khoo, I. C.

D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photonics 2(1), 60–200 (2010).
[Crossref]

Kinsey, N.

Kobyakov, A.

R. A. Negres, J. M. Hales, A. Kobyakov, D. J. Hagan, and E. W. Van Stryland, “Experiment and analysis of two-photon absorption spectroscopy using a white-light continuum probe,” Quantum Electronics, IEEE Journal of 38(9), 1205–1216 (2002).
[Crossref]

Kockaert, P.

Krauth, J.

Kubecek, V.

Kumar, S. C.

Kummli, D. S.

D. S. Kummli, H. M. Frey, and S. Leutwyler, “Femtosecond degenerate four-wave mixing of carbon disulfide: high-accuracy rotational constants,” J. Chem. Phys. 124(14), 144307 (2006).
[Crossref] [PubMed]

Leutwyler, S.

D. S. Kummli, H. M. Frey, and S. Leutwyler, “Femtosecond degenerate four-wave mixing of carbon disulfide: high-accuracy rotational constants,” J. Chem. Phys. 124(14), 144307 (2006).
[Crossref] [PubMed]

Levis, R. J.

J. K. Wahlstrand, J. H. Odhner, E. T. McCole, Y. H. Cheng, J. P. Palastro, R. J. Levis, and H. M. Milchberg, “Effect of two-beam coupling in strong-field optical pump-probe experiments,” Phys. Rev. A 87(5), 053801 (2013).
[Crossref]

Liu, J.

Loh, K.-P.

Martin, M. B.

Massar, S.

McCole, E. T.

J. K. Wahlstrand, J. H. Odhner, E. T. McCole, Y. H. Cheng, J. P. Palastro, R. J. Levis, and H. M. Milchberg, “Effect of two-beam coupling in strong-field optical pump-probe experiments,” Phys. Rev. A 87(5), 053801 (2013).
[Crossref]

Ménaert, B.

Milchberg, H. M.

J. K. Wahlstrand, J. H. Odhner, E. T. McCole, Y. H. Cheng, J. P. Palastro, R. J. Levis, and H. M. Milchberg, “Effect of two-beam coupling in strong-field optical pump-probe experiments,” Phys. Rev. A 87(5), 053801 (2013).
[Crossref]

Naik, G. V.

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref] [PubMed]

Negres, R. A.

R. A. Negres, J. M. Hales, A. Kobyakov, D. J. Hagan, and E. W. Van Stryland, “Experiment and analysis of two-photon absorption spectroscopy using a white-light continuum probe,” Quantum Electronics, IEEE Journal of 38(9), 1205–1216 (2002).
[Crossref]

Odhner, J. H.

J. K. Wahlstrand, J. H. Odhner, E. T. McCole, Y. H. Cheng, J. P. Palastro, R. J. Levis, and H. M. Milchberg, “Effect of two-beam coupling in strong-field optical pump-probe experiments,” Phys. Rev. A 87(5), 053801 (2013).
[Crossref]

Ooi, K. J. A.

K. J. A. Ooi, J. L. Cheng, J. E. Sipe, L. K. Ang, and D. T. H. Tan, “Ultrafast, broadband, and configurable midinfrared all-optical switching in nonlinear graphene plasmonic waveguides,” APL Photonics 1(4), 046101 (2016).
[Crossref]

Palastro, J. P.

J. K. Wahlstrand, J. H. Odhner, E. T. McCole, Y. H. Cheng, J. P. Palastro, R. J. Levis, and H. M. Milchberg, “Effect of two-beam coupling in strong-field optical pump-probe experiments,” Phys. Rev. A 87(5), 053801 (2013).
[Crossref]

Peceli, D.

Petrov, V.

Pollak, T. C.

K. T. Zawilski, P. G. Schunemann, T. C. Pollak, D. E. Zelmon, N. C. Fernelius, and F. Kenneth Hopkins, “Growth and characterization of large CdSiP2 single crystals,” J. Cryst. Growth 312(8), 1127–1132 (2010).
[Crossref]

Reed, J. M.

Reichert, M.

Ruzicka, B. A.

Salamo, G. J.

D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photonics 2(1), 60–200 (2010).
[Crossref]

Schunemann, P. G.

Schunneman, P. G.

Segonds, P.

Seidel, M.

Seneor, P.

Shalaev, V. M.

Sipe, J. E.

K. J. A. Ooi, J. L. Cheng, J. E. Sipe, L. K. Ang, and D. T. H. Tan, “Ultrafast, broadband, and configurable midinfrared all-optical switching in nonlinear graphene plasmonic waveguides,” APL Photonics 1(4), 046101 (2016).
[Crossref]

Stegeman, G. I.

D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photonics 2(1), 60–200 (2010).
[Crossref]

Steinmann, A.

Syed, A. A.

Tan, D. T. H.

K. J. A. Ooi, J. L. Cheng, J. E. Sipe, L. K. Ang, and D. T. H. Tan, “Ultrafast, broadband, and configurable midinfrared all-optical switching in nonlinear graphene plasmonic waveguides,” APL Photonics 1(4), 046101 (2016).
[Crossref]

Umemura, N.

K. Kato, N. Umemura, and V. Petrov, “Sellmeier and thermo-optic dispersion formulas for CdSiP2,” J. Appl. Phys. 109(11), 116104 (2011).
[Crossref]

Van Stryland, E. W.

N. Kinsey, A. A. Syed, D. Courtwright, C. DeVault, C. E. Bonner, V. I. Gavrilenko, V. M. Shalaev, D. J. Hagan, E. W. Van Stryland, and A. Boltasseva, “Effective third-order nonlinearities in metallic refractory titanium nitride thin films,” Opt. Mater. Express 5(11), 2395 (2015).
[Crossref]

M. Reichert, P. Zhao, J. M. Reed, T. R. Ensley, D. J. Hagan, and E. W. Van Stryland, “Beam deflection measurement of bound-electronic and rotational nonlinear refraction in molecular gases,” Opt. Express 23(17), 22224–22237 (2015).
[Crossref] [PubMed]

M. Reichert, H. Hu, M. R. Ferdinandus, M. Seidel, P. Zhao, T. R. Ensley, D. Peceli, J. M. Reed, D. A. Fishman, S. Webster, D. J. Hagan, and E. W. Van Stryland, “Temporal, spectral, and polarization dependence of the nonlinear optical response of carbon disulfide,” Optica 1(6), 436 (2014).
[Crossref]

M. R. Ferdinandus, H. Hu, M. Reichert, D. J. Hagan, and E. W. Van Stryland, “Beam deflection measurement of time and polarization resolved ultrafast nonlinear refraction,” Opt. Lett. 38(18), 3518–3521 (2013).
[Crossref] [PubMed]

D. N. Christodoulides, I. C. Khoo, G. J. Salamo, G. I. Stegeman, and E. W. Van Stryland, “Nonlinear refraction and absorption: mechanisms and magnitudes,” Adv. Opt. Photonics 2(1), 60–200 (2010).
[Crossref]

R. A. Negres, J. M. Hales, A. Kobyakov, D. J. Hagan, and E. W. Van Stryland, “Experiment and analysis of two-photon absorption spectroscopy using a white-light continuum probe,” Quantum Electronics, IEEE Journal of 38(9), 1205–1216 (2002).
[Crossref]

Wahlstrand, J. K.

J. K. Wahlstrand, J. H. Odhner, E. T. McCole, Y. H. Cheng, J. P. Palastro, R. J. Levis, and H. M. Milchberg, “Effect of two-beam coupling in strong-field optical pump-probe experiments,” Phys. Rev. A 87(5), 053801 (2013).
[Crossref]

Wang, G.

G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS2, WS2, and MoSe2 nanosheet dispersions [Invited],” Photonics Research 3(2), A51 (2015).
[Crossref]

Wang, J.

G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS2, WS2, and MoSe2 nanosheet dispersions [Invited],” Photonics Research 3(2), A51 (2015).
[Crossref]

Wang, S.

Webster, S.

Wu, J. Z.

Zawilski, K. T.

Zelmon, D. E.

K. T. Zawilski, P. G. Schunemann, T. C. Pollak, D. E. Zelmon, N. C. Fernelius, and F. Kenneth Hopkins, “Growth and characterization of large CdSiP2 single crystals,” J. Cryst. Growth 312(8), 1127–1132 (2010).
[Crossref]

Zhang, H.

G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS2, WS2, and MoSe2 nanosheet dispersions [Invited],” Photonics Research 3(2), A51 (2015).
[Crossref]

Zhang, L.

G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS2, WS2, and MoSe2 nanosheet dispersions [Invited],” Photonics Research 3(2), A51 (2015).
[Crossref]

Zhang, S.

G. Wang, S. Zhang, X. Zhang, L. Zhang, Y. Cheng, D. Fox, H. Zhang, J. N. Coleman, W. J. Blau, and J. Wang, “Tunable nonlinear refractive index of two-dimensional MoS2, WS2, and MoSe2 nanosheet dispersions [Invited],” Photonics Research 3(2), A51 (2015).
[Crossref]

Zhang, X.

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M. R. Ferdinandus, H. Hu, M. Reichert, Z. Wang, D. J. Hagan, and E. Van Stryland, “Beam deflection measurement of time and polarization resolved nonlinear refraction,” in Frontiers in Optics 2013, I. R. D. A. N. Kang, and D. Hagan, eds. (Optical Society of America, Orlando, Florida, 2013), p. FTh4C.2.

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Figures (5)

Fig. 1
Fig. 1 Schematic of the beam deflection experiment with the probe displaced at ∆x = we / 2. Diagram of the irradiance profiles of the excitation and probe pulses. When the probe is centered on the quad-segmented photodiode the signal ∆Ep / Ep = 0, when deflected ∆Ep / Ep > 0.
Fig. 2
Fig. 2 a) Effect of magnitude transmission gradient on probe beam. For W << 1 the transmission gradient (black) makes it appear as if probe (blue) has been reduced in magnitude and translated with minimal distortion (red). b) Plot of contamination C for vs. normalized sample-detector distance D. For typical experimental geometries (D > 15), C << 1, so that the NLA and NLR signals are essentially independent. b inset) For small D the probe is small on the detector, so that a translation has a large effect of ΔEp. For D >> 1, ΔEp due to the same translation is much smaller due to the expansion of the beam over the increased propagation distance.
Fig. 3
Fig. 3 Comparison of Fresnel propagation (open circles) and analytical method (solid lines) for various values of ρ with σe = 0 for the a) transmission and b) deflection. The analytical method shows excellent agreement, with an error of less than 2.0% at the peak of the signal. The simulation parameters are W = 0.175, T = 1.09, η = 0.118, Γ = 0, D = 16.78 with ρ variable.
Fig. 4
Fig. 4 a) Transmission and b) refraction signals for various values of linear absorption σe with ρ = 10. The simulation parameters are W = 0.175, T = 1.09, η = 0.118, Γ = 0, D = 16.78 with σe variable. Increasing σe reduces the signal at negative delay due to the excitation catching up to the probe at the back of sample, after it has been attenuated by propagation through the sample.
Fig. 5
Fig. 5 Fit of CSP data using analytic approximation for a) transmission and b) deflection. Note that the reduction in the signal between τd = 5 and τd = 25 is due to the depletion of the excitation. The difference in the slope of the rise and fall of the signal is due to GVD, which is not accounted for. Inset) absorption coefficient αe vs. peak excitation irradiance I0,e. Plots have been shifted vertically by 2.5% for clarity.

Equations (13)

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a p [ r,Z,τ ] Z +ρ a p [ r,Z,τ ] τ =( σ p +iG[ r,Z,τ ] ) a p [ r,Z,τ ]
a e [ r,Z,τ ]= e ( ( X X 0 ) 2 + Y 2 ) e τ 2 /2 e Z σ e
a p [ r,0,τ ]=A e ( X 2 + Y 2 W 2 ) e ( τ τ d ) 2 2 T 2
a p [ r,1,τ ]=AExp[ ( X 2 + Y 2 W 2 ) ( ρτ+ τ d ) 2 2 T 2 σ p e 2 ( X X 0 ) 2 2 Y 2 H[ τ ] ]
H[ τ ]= 1 0 ( iG[ 0,Z',τ+ρ( 1+Z' ) ] )dZ'= H [ τ ]+iH''[ τ ]
Q[ r,τ ]= e 2 e 2( ( X X 0 ) 2 + Y 2 ) H [ τ ] 0 2 n ( e 2 X 0 2 H [ τ ] ) n ( 18 e 2 X 0 2 X X 0 H [ τ ] )/n!
0 2 n ( e 2 X 0 2 H [ τ ] ) n ( 18 e 2 X 0 2 X X 0 H [ τ ] )/n!
a d [ r,τ, τ d ]= n=0 2 n A 2 e 2 Y 2 D 2 W 2 2 ( XΔS[ τ ] ) 2 D 2 W 2 ( ρτ+ τ d ) 2 T 2 ( e 2 X 0 2 H [ τ ] ) n ( 1+8 e 2 X 0 2 X 0 ( X+ΔS[ τ ] ) H [ τ ] ) D 2 n!
ΔP[ τ, τ d ]=2 A 2 e 2 X 0 2 ( ρτ+ τ d ) 2 T 2 2 e 2 X 0 2 H [ τ ] 2π W 3 X 0 ( R H '' [ τ ]+ H [ τ ]/D )
P[ τ, τ d ]= 1 2 A 2 e ( ρτ+ τ d ) 2 T 2 2 H [ τ ] e π W 2
Δ E p / E p [ τ d ]= 2 e 1 2 τ d 2 1+ T 2 2 π W( Γ/D+Rη ) T 2 +1 F
F= 1+ e 2 σ e 2 σ e
Q[ τ d ]= 1 π T   e ( ρτ+ τ d ) 2 T 2 2 H [ τ ] dτ

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