Abstract

We demonstrate experimentally a degenerate four-wave mixing (DFWM) geometry in waveguides that permits the simultaneous determination of the optical Kerr nonlinear refractive index, the nonlinear absorption, and the nonlinear response time of thin-film materials. The geometry consists of two counterpropagating guided pump beams in a planar waveguide. The probe beam simply passes through the thin film and is not guided. By measuring the DFWM signal energy and the guided pump energy at the waveguide output, one can observe several effects simultaneously. Each effect has a different sensitivity to the nonlinearity and, used together, these effects increase the accuracy in determination of the complex n2. This technique was tested on dialkylaminonitrostilbene films. Good agreement was found between the numerical simulations and experimental data. Therefore the technique appears to be a useful characterization technique that yields, in a single setup with basic laboratory equipment, all the relevant parameters of the optical Kerr nonlinearity of thin-film planar waveguides.

© 1998 Optical Society of America

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1996 (1)

1995 (2)

1994 (4)

M. Cha, W. E. Torruellas, G. I. Stegeman, W. H. G. Horsthuis, G. R. Möhlmann, and J. Meth, “Two-photon absorption of dialkylaminonitrostilbene side chain polymer,” Appl. Phys. Lett. 65, 2648–2650 (1994).
[CrossRef]

H. W. H. Lee and R. S. Hughes, Jr., “Antiresonant ring interferometric nonlinear spectroscopy for nonlinear-optical measurements,” Opt. Lett. 19, 1708–1710 (1994).
[CrossRef] [PubMed]

G. Vitrant, L. Mayollet, B. Vögele, A. Rameix, R. Reinisch, G. I. Stegeman, G. R. Möhlmann, W. H. G. Horsthuis, P. A. Chollet, and F. Kajzar, “Measurements of large nonresonant nonlinearities in doped polymers,” Nonlinear Opt. 8, 251–261 (1994).

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-hemyari, C. C. Yang, C.-H. Lin, H.-H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below half the band gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

1992 (1)

S. Guha and Conner, “Degenerate four-wave mixing in Kerr media in the presence of nonlinear refraction, pump depletion and linear absorption,” Opt. Commun. 89, 107–119 (1992).
[CrossRef]

1991 (3)

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B 44, 8214–8225 (1991).
[CrossRef]

G. Assanto, M. B. Marques, and G. I. Stegeman, “Grating coupling of light into third-order nonlinear waveguides,” J. Opt. Soc. Am. B 8, 553–561 (1991).
[CrossRef]

J. M. Nunzi and F. Charra, “Complex third-order phase conjugation nonlinearity of polymeric thin films,” Appl. Phys. Lett. 59, 13–15 (1991).
[CrossRef]

1990 (2)

1989 (2)

1988 (2)

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, and C. T. Seaton, “Degenerate four-wave mixing from a waveguide with guided wave pump beams,” J. Appl. Phys. 64, 4318–4322 (1988).
[CrossRef]

1987 (1)

A. Gabel, K. W. DeLong, C. T. Seaton, and G. I. Stegeman, “Efficient degenerate four-wave mixing in an ion-exchanged semiconductor-doped glass waveguide,” Appl. Phys. Lett. 51, 1682–1684 (1987).
[CrossRef]

1986 (1)

F. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040–3044 (1986).
[CrossRef]

1975 (1)

1974 (1)

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. QE-9, 919–933 (1973).
[CrossRef]

Aitchison, J. S.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-hemyari, C. C. Yang, C.-H. Lin, H.-H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below half the band gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Al-hemyari, K.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-hemyari, C. C. Yang, C.-H. Lin, H.-H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below half the band gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Andrejco, M. J.

Assanto, G.

Cha, M.

M. Cha, W. E. Torruellas, G. I. Stegeman, W. H. G. Horsthuis, G. R. Möhlmann, and J. Meth, “Two-photon absorption of dialkylaminonitrostilbene side chain polymer,” Appl. Phys. Lett. 65, 2648–2650 (1994).
[CrossRef]

Charra, F.

J. M. Nunzi and F. Charra, “Complex third-order phase conjugation nonlinearity of polymeric thin films,” Appl. Phys. Lett. 59, 13–15 (1991).
[CrossRef]

Chollet, P. A.

G. Vitrant, L. Mayollet, B. Vögele, A. Rameix, R. Reinisch, G. I. Stegeman, G. R. Möhlmann, W. H. G. Horsthuis, P. A. Chollet, and F. Kajzar, “Measurements of large nonresonant nonlinearities in doped polymers,” Nonlinear Opt. 8, 251–261 (1994).

Conner,

S. Guha and Conner, “Degenerate four-wave mixing in Kerr media in the presence of nonlinear refraction, pump depletion and linear absorption,” Opt. Commun. 89, 107–119 (1992).
[CrossRef]

Dalgoutte, D. G.

Danckaert, J.

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B 44, 8214–8225 (1991).
[CrossRef]

DeLong, K. W.

V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. 14, 1140–1142 (1989).
[CrossRef] [PubMed]

A. Gabel, K. W. DeLong, C. T. Seaton, and G. I. Stegeman, “Efficient degenerate four-wave mixing in an ion-exchanged semiconductor-doped glass waveguide,” Appl. Phys. Lett. 51, 1682–1684 (1987).
[CrossRef]

Dühr, O.

Fick, J.

Finlayson, N.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

Fobelets, K.

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B 44, 8214–8225 (1991).
[CrossRef]

Gabel, A.

A. Gabel, K. W. DeLong, C. T. Seaton, and G. I. Stegeman, “Efficient degenerate four-wave mixing in an ion-exchanged semiconductor-doped glass waveguide,” Appl. Phys. Lett. 51, 1682–1684 (1987).
[CrossRef]

Goodhue, W. D.

Grant, R. S.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-hemyari, C. C. Yang, C.-H. Lin, H.-H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below half the band gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Guha, S.

S. Guha and Conner, “Degenerate four-wave mixing in Kerr media in the presence of nonlinear refraction, pump depletion and linear absorption,” Opt. Commun. 89, 107–119 (1992).
[CrossRef]

Haelterman, M.

Horsthuis, W. H. G.

M. Cha, W. E. Torruellas, G. I. Stegeman, W. H. G. Horsthuis, G. R. Möhlmann, and J. Meth, “Two-photon absorption of dialkylaminonitrostilbene side chain polymer,” Appl. Phys. Lett. 65, 2648–2650 (1994).
[CrossRef]

G. Vitrant, L. Mayollet, B. Vögele, A. Rameix, R. Reinisch, G. I. Stegeman, G. R. Möhlmann, W. H. G. Horsthuis, P. A. Chollet, and F. Kajzar, “Measurements of large nonresonant nonlinearities in doped polymers,” Nonlinear Opt. 8, 251–261 (1994).

Hughes, Jr., R. S.

Ironside, C. N.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-hemyari, C. C. Yang, C.-H. Lin, H.-H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below half the band gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Kajzar, F.

G. Vitrant, L. Mayollet, B. Vögele, A. Rameix, R. Reinisch, G. I. Stegeman, G. R. Möhlmann, W. H. G. Horsthuis, P. A. Chollet, and F. Kajzar, “Measurements of large nonresonant nonlinearities in doped polymers,” Nonlinear Opt. 8, 251–261 (1994).

F. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040–3044 (1986).
[CrossRef]

Kang, J.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-hemyari, C. C. Yang, C.-H. Lin, H.-H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below half the band gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Kennedy, G. T.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-hemyari, C. C. Yang, C.-H. Lin, H.-H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below half the band gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Kogelnik, H.

Le, H. Q.

Lee, H. W. H.

Lessard, R. A.

Lin, C.-H.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-hemyari, C. C. Yang, C.-H. Lin, H.-H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below half the band gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Lin, H.-H.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-hemyari, C. C. Yang, C.-H. Lin, H.-H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below half the band gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Malouin, C.

Marques, M. B.

Mayollet, L.

G. Vitrant, L. Mayollet, B. Vögele, A. Rameix, R. Reinisch, G. I. Stegeman, G. R. Möhlmann, W. H. G. Horsthuis, P. A. Chollet, and F. Kajzar, “Measurements of large nonresonant nonlinearities in doped polymers,” Nonlinear Opt. 8, 251–261 (1994).

Messier, J.

F. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040–3044 (1986).
[CrossRef]

Meth, J.

M. Cha, W. E. Torruellas, G. I. Stegeman, W. H. G. Horsthuis, G. R. Möhlmann, and J. Meth, “Two-photon absorption of dialkylaminonitrostilbene side chain polymer,” Appl. Phys. Lett. 65, 2648–2650 (1994).
[CrossRef]

Mizrahi, V.

Möhlmann, G. R.

G. Vitrant, L. Mayollet, B. Vögele, A. Rameix, R. Reinisch, G. I. Stegeman, G. R. Möhlmann, W. H. G. Horsthuis, P. A. Chollet, and F. Kajzar, “Measurements of large nonresonant nonlinearities in doped polymers,” Nonlinear Opt. 8, 251–261 (1994).

M. Cha, W. E. Torruellas, G. I. Stegeman, W. H. G. Horsthuis, G. R. Möhlmann, and J. Meth, “Two-photon absorption of dialkylaminonitrostilbene side chain polymer,” Appl. Phys. Lett. 65, 2648–2650 (1994).
[CrossRef]

Nunzi, J. M.

J. M. Nunzi and F. Charra, “Complex third-order phase conjugation nonlinearity of polymeric thin films,” Appl. Phys. Lett. 59, 13–15 (1991).
[CrossRef]

Petrov, V.

Ramaswamy, V.

Rameix, A.

G. Vitrant, L. Mayollet, B. Vögele, A. Rameix, R. Reinisch, G. I. Stegeman, G. R. Möhlmann, W. H. G. Horsthuis, P. A. Chollet, and F. Kajzar, “Measurements of large nonresonant nonlinearities in doped polymers,” Nonlinear Opt. 8, 251–261 (1994).

Rauschenbach, K.

Reinisch, R.

G. Vitrant, L. Mayollet, B. Vögele, A. Rameix, R. Reinisch, G. I. Stegeman, G. R. Möhlmann, W. H. G. Horsthuis, P. A. Chollet, and F. Kajzar, “Measurements of large nonresonant nonlinearities in doped polymers,” Nonlinear Opt. 8, 251–261 (1994).

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B 44, 8214–8225 (1991).
[CrossRef]

G. Vitrant, M. Haelterman, and R. Reinisch, “Transverse effects in nonlinear planar resonators. II. Modal analysis for normal and oblique incidence,” J. Opt. Soc. Am. B 7, 1319–1327 (1990).
[CrossRef]

Rosilio, C.

F. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040–3044 (1986).
[CrossRef]

Said, A. A.

Saifi, M. A.

Seaton, C. T.

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, and C. T. Seaton, “Degenerate four-wave mixing from a waveguide with guided wave pump beams,” J. Appl. Phys. 64, 4318–4322 (1988).
[CrossRef]

A. Gabel, K. W. DeLong, C. T. Seaton, and G. I. Stegeman, “Efficient degenerate four-wave mixing in an ion-exchanged semiconductor-doped glass waveguide,” Appl. Phys. Lett. 51, 1682–1684 (1987).
[CrossRef]

Seifert, F.

Sheik-Bahae, M.

Sibbett, W.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-hemyari, C. C. Yang, C.-H. Lin, H.-H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below half the band gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Stegeman, G. I.

G. Vitrant, L. Mayollet, B. Vögele, A. Rameix, R. Reinisch, G. I. Stegeman, G. R. Möhlmann, W. H. G. Horsthuis, P. A. Chollet, and F. Kajzar, “Measurements of large nonresonant nonlinearities in doped polymers,” Nonlinear Opt. 8, 251–261 (1994).

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-hemyari, C. C. Yang, C.-H. Lin, H.-H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below half the band gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

M. Cha, W. E. Torruellas, G. I. Stegeman, W. H. G. Horsthuis, G. R. Möhlmann, and J. Meth, “Two-photon absorption of dialkylaminonitrostilbene side chain polymer,” Appl. Phys. Lett. 65, 2648–2650 (1994).
[CrossRef]

G. Assanto, M. B. Marques, and G. I. Stegeman, “Grating coupling of light into third-order nonlinear waveguides,” J. Opt. Soc. Am. B 8, 553–561 (1991).
[CrossRef]

V. Mizrahi, K. W. DeLong, G. I. Stegeman, M. A. Saifi, and M. J. Andrejco, “Two-photon absorption as a limitation to all-optical switching,” Opt. Lett. 14, 1140–1142 (1989).
[CrossRef] [PubMed]

G. I. Stegeman, E. M. Wright, and C. T. Seaton, “Degenerate four-wave mixing from a waveguide with guided wave pump beams,” J. Appl. Phys. 64, 4318–4322 (1988).
[CrossRef]

G. I. Stegeman, E. M. Wright, N. Finlayson, R. Zanoni, and C. T. Seaton, “Third order nonlinear integrated optics,” J. Lightwave Technol. 6, 953–970 (1988).
[CrossRef]

A. Gabel, K. W. DeLong, C. T. Seaton, and G. I. Stegeman, “Efficient degenerate four-wave mixing in an ion-exchanged semiconductor-doped glass waveguide,” Appl. Phys. Lett. 51, 1682–1684 (1987).
[CrossRef]

Torruellas, W. E.

M. Cha, W. E. Torruellas, G. I. Stegeman, W. H. G. Horsthuis, G. R. Möhlmann, and J. Meth, “Two-photon absorption of dialkylaminonitrostilbene side chain polymer,” Appl. Phys. Lett. 65, 2648–2650 (1994).
[CrossRef]

Van Stryland, E. W.

Veretennicoff, I.

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B 44, 8214–8225 (1991).
[CrossRef]

Villeneuve, A.

C. Malouin, A. Villeneuve, G. Vitrant, and R. A. Lessard, “Degenerate four-wave mixing geometry in thin-film waveguides for nonlinear material characterization,” Opt. Lett. 21, 21–23 (1996).
[CrossRef] [PubMed]

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Al-hemyari, C. C. Yang, C.-H. Lin, H.-H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, “AlGaAs below half the band gap: the silicon of nonlinear optical materials,” Int. J. Nonlinear Opt. Phys. 3, 347–371 (1994).
[CrossRef]

Vitrant, G.

C. Malouin, A. Villeneuve, G. Vitrant, and R. A. Lessard, “Degenerate four-wave mixing geometry in thin-film waveguides for nonlinear material characterization,” Opt. Lett. 21, 21–23 (1996).
[CrossRef] [PubMed]

J. Fick and G. Vitrant, “Fast optical switching in nonlinear prism couplers,” Opt. Lett. 20, 1462–1464 (1995).
[CrossRef] [PubMed]

G. Vitrant, L. Mayollet, B. Vögele, A. Rameix, R. Reinisch, G. I. Stegeman, G. R. Möhlmann, W. H. G. Horsthuis, P. A. Chollet, and F. Kajzar, “Measurements of large nonresonant nonlinearities in doped polymers,” Nonlinear Opt. 8, 251–261 (1994).

J. Danckaert, K. Fobelets, I. Veretennicoff, G. Vitrant, and R. Reinisch, “Dispersive optical bistability in stratified structures,” Phys. Rev. B 44, 8214–8225 (1991).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

DFWM geometry. The two counterpropagating pump beams are guided, but the probe is not; it simply passes through the waveguide. The probe beam undergoes multiple reflections in the film that produce DFWM signals in reflection and in transmission. The probe beam is incident from the air side at angle θc with respect to the normal of the waveguide plane.

Fig. 2
Fig. 2

DFWM cross sections ηr and ηt as a function of probe angle θc for four film thicknesses. The parameters are nc=1, nf=1.623, ns=1.450, n2=7×10-18 m2/W, and λ =1.064 μm.

Fig. 3
Fig. 3

DFWM cross section for the reflected wave, ηr, as a function of film thickness h. The parameters are nc=1, nf =1.623, ns=1.450, n2=8×10-18 m2/W, λ=1.064 μm, and θc=20°.

Fig. 4
Fig. 4

DFWM experimental setup. This setup permits simultaneous measurement of the DFWM signal energy and the outcoupled pump mode energy. The distance between the two gratings is 1.34 mm. The pump beam coupling angle is θi9.6°, and the probe angle θc=20°. The probe beam was focused slightly in front of the waveguide so the geometry could be used as a phase-conjugate mirror. bs, beam splitter.

Fig. 5
Fig. 5

Complete experimental setup of the DFWM geometry: PS, pulse selector; λ/2, half-wave plate; POL’s, Glan polarizers in the TE direction; M, M2, M, mirrors; SF, spatial filter; bs, 80–20% beam splitter; bs, BS, 50–50% beam splitters; bs, 100-μm-thick beam splitter; DL1, DL2, delay lines of pump beams 1 and 2, respectively; S, screen blocking the reflection of pump beam 2; BD, neutral glass filters used to redirect the three incident beams after they pass through the waveguide; A, D’s, angular and transverse alignments of mirror M2; L, lens that focuses the probe beam slightly in front of the waveguide; IF, interference filter centered at 1.064 μm; NG’s, neutral glass filters; PMT, photomultiplier tube (Philips Model 56CVP); BBO, 6-mm-thick β-barium borate crystal; P1, incident pump beam 1; P2, incident pump beam 2. Detector, reference beam at ω (ITT Model F4014), reference beam at 2ω (ITT Model FW-114), outcoupled beam (Hamamatsu Model R1193U-01).

Fig. 6
Fig. 6

Experimental and theoretical results for a fresh DANS sample. (a) DFWM reflectivity R versus incident pump energy. (b) Outcoupled energy versus incident energy pump energy during DFWM. (c) Outcoupled energy versus incident pump en ergy when one pump is blocked. For this set of measurements we obtained n2=(15+i6.8)×10-18 m2/W.

Fig. 7
Fig. 7

Experimental and theoretical results for an exposed DANS sample. (a) DFWM reflectivity R versus incident pump energy. (b) Outcoupled energy versus incident energy pump energy during DFWM. (c) Outcoupled energy versus incident pump energy when one pump is blocked. For this set of measurements we obtained n2=(4.6+i1.2)×10-18 m2/W. Reflectivity R has a more quadratic behavior than that in Fig. 6, and the outcoupled energies are less saturated and have higher values.

Fig. 8
Fig. 8

Effects of various combinations of the complex n2 on the three types of result. The outcoupled energies are kept the same.  

Fig. 9
Fig. 9

Effect of different combinations of the complex n2 on the three types of result. The DFWM reflectivity R is kept the same.

Fig. 10
Fig. 10

Measured R as a function of the pump beam delay and numerical results for several pulse durations. The filled and open circles are two different measurements made on an exposed DANS sample. The incident pump energy was 50 μJ. The best fit is obtained for τp=27 ps (FWHM). Note that in all other calculations we took τp=30 ps; here we adjusted n2 because we used a different pulse duration, τp.

Fig. 11
Fig. 11

Measured R as a function of the pump beam delay and numerical results for several n2 combinations. The experimental results are those of Fig. 10. Note that in all other calculations we took τp=30 ps; here we adjusted n2 because we used τp=27 ps.

Tables (1)

Tables Icon

Table 1 Summary of the Best-Fit Parameters Obtained from the Three Types of DFWM Data a

Equations (61)

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Ey(x, z, t)=½E(x, z)exp(-iωt)+c.c.,
E+=Pgf(x)exp(ineffk0z)
E-=Pgf(x)exp(-ineffk0z)
f(x)=CTE[Af+ exp(iqfk0x)+Af- exp(-iqfk0x)]
Af+=12 1-i qcqf=Af-*.
qf2=nf2-neff2,
qc2=neff2-nc2,
qs2=neff2-ns2,
Einc=g(x)exp(iβck0z),
g(x)=Bf+ exp(iκfk0x)+Bf- exp(-iκfk0x).
Bf+
=2κc(κf+κs)exp(-2iW)(κf+κc)(κf+κs)exp(-2iW)+(κf-κs)(κc-κf),
Bf-
=2κc(κf-κs)(κf+κc)(κf+κs)exp(-2iW)+(κf-κs)(κc-κf),
W=κfk0h,
κi=ni cos θi,
βc=nc sin θc,
PNL=2nfneff02cn2E+E-Einc*,
PNL=pP(p)exp(-iβck0z)exp(ipk0x),
P(p=+2qf+κf)=2nfneff02cn2PgAf+2Bf-*,
P(p=+2qf-κf)=2nfneff02cn2PgAf+2Bf+*,
P(p=-κf)=4nfneff02cn2PgAf+Af-Bf+*,
P(p=+κf)=4nfneff02cn2PgAf+Af-Bf-*,
P(p=-2qf-κf)=2nfneff02cn2PgAf-2Bf+*,
P(p=-2qf+κf)=2nfneff02cn2PgAf-2Bf-*.
1k02 d2Encdx2+κf2Enc=-10 pP(p)exp(ipk0x).
Er=Dc exp(-iκck0x),
Et=Ds exp[-iκsk0(x-h)],
Ef=Df+ exp(iκfk0x)+Df- exp(-iκfk0x).
R=IrIi=½nc0c|Er|2½nc0c|Ei|2=|Dc|2,
T=ItIi=½ns0c|Et|2½nc0c|Ei|2=nsnc |Ds|2,
R=|Dc|2=ηrPg2,
T=nsnc |Ds|2=ηtPg2,
Rbulk=(2k0)2|n2|2L2I2,
IPg/heff,
Ras=TF(2k0)2|n2|2Pg2,
E(x, z, t)=Pg(z, t)f(x)exp[-i(neff+ΔneffNL)k0z],
Pg(z, t)z=iΔneffNLk0Pg(z, t),
Re(ΔneffNL)=nfn2rneff Pg(z, t)heffNL,
Im(ΔneffNL)=nfn2ineff Pg(z, t)heffNL,
heffNL=-|f(x)|2dx20h|f(x)|4dx
ddz I+=-α2(I++2I-)I+,
ddz I-=+α2(I-+2I+)I-,
Pg(z, t)z=i-Δneffl+ik0lc+ΔneffNL×k0Pg(z, t)+ξAi(z, t),
Δneffl=neff-sin θi-m(λ/Λ),
y+ay+by=r(x),
Enc=10 p±κf P(p)p2-κf2 exp(ipk0x)+ik0x20κf P(+κf)exp(+iκfk0x)-ik0x20κf P(-κf)exp(-iκfk0x)
Hnc=iωμ0 dEncdx.
Enc(x=0)=Enc(0)=10 p±κf P(p)p2-κf2,
Enc(x=h)=Enc(h)=10 p±κf P(p)p2-κf2 exp(ipk0h)+ik0h20κf P(κf)exp(iκfk0h)-ik0h20κf P(-κf)exp(-iκfk0h)
μ0cHnc(x=0)=(Hnc)(0)=-10 p±κf pP(p)p2-κf2+120κf P(κf)-120κf P(-κf),
μ0cHnc(x=h)=(Hnc)(h)=-10 p±κf pP(p)p2-κf2+120κf [P(κf)exp(iκfk0h)×(1+iκfk0h)+P(-κf)×exp(-iκfk0h)(-1+iκfk0h)]
Dc=Enc(0)+Df++Df-
Ds=Enc(h)+Df+ exp(iW)+Df- exp(-iW)
κcDc=μ0cHnc(0)-κfDf++κfDf-
-κsDs=μ0cHnc(h)-κfDf+ exp(iW)+κfDf- exp(-iW)
R=IrIi=½nc0c|Er|2½nc0c|Ei|2=|Dc|2,
T=ItIi=½ns0c|Et|2½nc0c|Ei|2=nsnc |Ds|2,
Dcn=κf(κs cos W-iκf sin W)Enc(0)-κfκsEnc(h)+κfHnc(h)-(κf cos W-iκs sin W)Hnc(0),
Dsn=-κfκcEnc(0)+κf(κc cos W-iκf sin W)Enc(h)+(κf cos W-iκc sin W)Hnc(h)-κfHnc(0),
Dd=κf(κc+κs)cos W-i(κf2+κcκs)sin W.

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