F. P. Strohkendl, L. R. Dalton, R. W. Hellwarth, H. W. Sarkas, and Z. H. Kafafi, "Phase-mismatched degenerate four-wave mixing: complex third-order susceptibility tensor elements of C_{60} at 768 nm," J. Opt. Soc. Am. B 14, 92-98 (1997)

We describe and analyze an all-optical method to measure both the magnitude and the phase of each (complex) element of the χ^{(3)} tensor that determines degenerate four-wave mixing in a thin isotropic film on a thick substrate. We use the method to characterize the χ^{(3)} tensor of a 10-µm C_{60} film (on a CaF_{2} substrate) for 110-fs pulses at 768 nm. Using a fused-quartz plate as the nonlinear standard, we find the two independent Maker–Terhune elements of χ^{(3)} to have magnitudes $|{c}_{1221}(-\omega ,\omega ,\omega ,-\omega )|=(0.44\pm 0.03)\times {10}^{-12}$ esu and $|{c}_{1122}(-\omega ,\omega ,\omega ,-\omega )|=(0.50\pm 0.03)\times {10}^{-12}$ esu and phase angles ϕ_{1221} and ϕ_{1122}, whose magnitudes are 145°±17° and 139°±10°.

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Powers S_{4}, S_{5}, and S_{6} Generated in Beams 4, 5, and 6 of Fig. 1 for Various Polarizations of the Input Beams 1, 2, and 3a

a

b

c

d

Polarization Geometry

$\frac{{S}_{5}(f,s)}{{S}_{4}(q)}$

$\frac{{S}_{6}(f,s)}{{S}_{4}(q)}$

$\frac{{S}_{4}(s)}{{S}_{4}(q)}$

$\frac{{S}_{4}(f,s)}{{S}_{4}(q)}$

a

↑

→

$1.34\pm 3\%$

$1.14\pm 5\%$

$7.22\times {10}^{-2}\pm 8\%$

$1.00\pm 4\%$

↑

$\left[\frac{1122}{1122}\right]$

$\left[\frac{1221}{1122}\right]$

$\left[\frac{\mathit{xxyy}}{1122}\right]$

$\left[\frac{1122}{1122}\right]$

b

↑

↑

$1.05\pm 4\%$

$1.46\pm 6\%$

$9.93\times {10}^{-2}\pm 6\%$

$1.04\pm 6\%$

→

$\left[\frac{1221}{1122}\right]$

$\left[\frac{1122}{1122}\right]$

$\left[\frac{\mathit{xxyy}}{1122}\right]$

$\left[\frac{1122}{1122}\right]$

c

↑

↑

$1.31\pm 2\%$

$1.29\pm 3\%$

$0.235\pm 4\%$

$0.73\pm 2\%$

↑

$\left[\frac{1111}{1111}\right]$

$\left[\frac{1111}{1111}\right]$

$\left[\frac{1111}{1111}\right]$

$\left[\frac{1111}{1111}\right]$

d

→

↑

$1.60\pm 7\%$

$1.65\pm 7\%$

$0.132\pm 5\%$

$0.77\pm 8\%$

↑

$\left[\frac{1122}{1221}\right]$

$\left[\frac{1122}{1221}\right]$

$\left[\frac{\mathit{xyyx}}{1221}\right]$

$\left[\frac{1221}{1221}\right]$

The polarization geometries shown give the polarizations for beams 1, 2, and 3 in the same order as these beams are shown in the observation plane in Fig. 1. The powers are normalized to the power S_{4}(q) of beam 4 generated by the fused-quartz plate for the same polarizations. The arguments (f, s) and (s) indicate signals from film-on-substrate and substrate alone. The rows and columns of the table are each labeled by a, b, c, and d so that a single table entry can be referred to elsewhere by matrix notation; e.g., (c, b) refers to the entry $1.29\pm 3\%$. The brackets below each table entry contain the space indices of the χ^{(3)} tensor element governing the generated beam written over the space indices of the tensor element of fused quartz responsible for the normalization signal.

Table 2

Results for the Amplitudes and the Phases of the Third-Order Nonlinear Susceptibility Tensor ${\chi}^{f}(-\omega ,\omega ,\omega ,-\omega )$ of C_{60} Obtained by Use of the Data of Table 1 in Eqs. (12)–(16) for the Optical Fields Generated in Beams 4, 5, and 6 in Fig. 1a

esults are expressed in terms of the two independent tensor elements ${c}_{1221}^{f}$ and ${c}_{1122}^{f}$ defined in Eq. (10). The parallel-polarization element ${c}_{1111}^{f}$ must equal ${c}_{1221}^{f}+2{c}_{1122}^{f}.$ For the determination of the phases ${\varphi}_{\mathit{ijkl}}^{f}$ it is assumed that the nonlinear susceptibility in the CaF_{2} substrate has a zero imaginary part. In the other footnotes to this table the symbol (a, b) refers to the data in row a and column b of Table 1.
Average of the ratios of (a, a) to (a, b) and of (b, b) to (b, a), weighted by the inverse squares of their errors.
Average of data entries (c, a) and (c, b), weighted by the inverse squares of their errors and substituted in Eqs. (12), (13), and (16).
From data entry (c, d) interpreted by Eq. (14), with (c, c) giving the relative magnitudes of ${c}_{\mathit{xxxx}}^{\mathrm{s}}$, presumed real and positive, in Eq. (15).
Weighted average of values from entries (a, a) and (b, b).
Values predicted by values superscripted (a) and (b) above and assuming ${\varphi}_{1122}^{f}={\varphi}_{1221}^{f}.$
From data entries (a, d), (a, a), and (a, c).
From data entries (b, d), (b, b), and (b, c).
Weighted average of values from entries (a, b) and (b, a).
Value predicted by values superscripted (a) and (b) above.
Value obtained from entry (d, d) in which |${c}_{1221}^{f}$| is obtained from the average of entries (d, a) and (d, b) divided by the square of the ratio superscripted (a) above.

Tables (2)

Table 1

Powers S_{4}, S_{5}, and S_{6} Generated in Beams 4, 5, and 6 of Fig. 1 for Various Polarizations of the Input Beams 1, 2, and 3a

a

b

c

d

Polarization Geometry

$\frac{{S}_{5}(f,s)}{{S}_{4}(q)}$

$\frac{{S}_{6}(f,s)}{{S}_{4}(q)}$

$\frac{{S}_{4}(s)}{{S}_{4}(q)}$

$\frac{{S}_{4}(f,s)}{{S}_{4}(q)}$

a

↑

→

$1.34\pm 3\%$

$1.14\pm 5\%$

$7.22\times {10}^{-2}\pm 8\%$

$1.00\pm 4\%$

↑

$\left[\frac{1122}{1122}\right]$

$\left[\frac{1221}{1122}\right]$

$\left[\frac{\mathit{xxyy}}{1122}\right]$

$\left[\frac{1122}{1122}\right]$

b

↑

↑

$1.05\pm 4\%$

$1.46\pm 6\%$

$9.93\times {10}^{-2}\pm 6\%$

$1.04\pm 6\%$

→

$\left[\frac{1221}{1122}\right]$

$\left[\frac{1122}{1122}\right]$

$\left[\frac{\mathit{xxyy}}{1122}\right]$

$\left[\frac{1122}{1122}\right]$

c

↑

↑

$1.31\pm 2\%$

$1.29\pm 3\%$

$0.235\pm 4\%$

$0.73\pm 2\%$

↑

$\left[\frac{1111}{1111}\right]$

$\left[\frac{1111}{1111}\right]$

$\left[\frac{1111}{1111}\right]$

$\left[\frac{1111}{1111}\right]$

d

→

↑

$1.60\pm 7\%$

$1.65\pm 7\%$

$0.132\pm 5\%$

$0.77\pm 8\%$

↑

$\left[\frac{1122}{1221}\right]$

$\left[\frac{1122}{1221}\right]$

$\left[\frac{\mathit{xyyx}}{1221}\right]$

$\left[\frac{1221}{1221}\right]$

The polarization geometries shown give the polarizations for beams 1, 2, and 3 in the same order as these beams are shown in the observation plane in Fig. 1. The powers are normalized to the power S_{4}(q) of beam 4 generated by the fused-quartz plate for the same polarizations. The arguments (f, s) and (s) indicate signals from film-on-substrate and substrate alone. The rows and columns of the table are each labeled by a, b, c, and d so that a single table entry can be referred to elsewhere by matrix notation; e.g., (c, b) refers to the entry $1.29\pm 3\%$. The brackets below each table entry contain the space indices of the χ^{(3)} tensor element governing the generated beam written over the space indices of the tensor element of fused quartz responsible for the normalization signal.

Table 2

Results for the Amplitudes and the Phases of the Third-Order Nonlinear Susceptibility Tensor ${\chi}^{f}(-\omega ,\omega ,\omega ,-\omega )$ of C_{60} Obtained by Use of the Data of Table 1 in Eqs. (12)–(16) for the Optical Fields Generated in Beams 4, 5, and 6 in Fig. 1a

esults are expressed in terms of the two independent tensor elements ${c}_{1221}^{f}$ and ${c}_{1122}^{f}$ defined in Eq. (10). The parallel-polarization element ${c}_{1111}^{f}$ must equal ${c}_{1221}^{f}+2{c}_{1122}^{f}.$ For the determination of the phases ${\varphi}_{\mathit{ijkl}}^{f}$ it is assumed that the nonlinear susceptibility in the CaF_{2} substrate has a zero imaginary part. In the other footnotes to this table the symbol (a, b) refers to the data in row a and column b of Table 1.
Average of the ratios of (a, a) to (a, b) and of (b, b) to (b, a), weighted by the inverse squares of their errors.
Average of data entries (c, a) and (c, b), weighted by the inverse squares of their errors and substituted in Eqs. (12), (13), and (16).
From data entry (c, d) interpreted by Eq. (14), with (c, c) giving the relative magnitudes of ${c}_{\mathit{xxxx}}^{\mathrm{s}}$, presumed real and positive, in Eq. (15).
Weighted average of values from entries (a, a) and (b, b).
Values predicted by values superscripted (a) and (b) above and assuming ${\varphi}_{1122}^{f}={\varphi}_{1221}^{f}.$
From data entries (a, d), (a, a), and (a, c).
From data entries (b, d), (b, b), and (b, c).
Weighted average of values from entries (a, b) and (b, a).
Value predicted by values superscripted (a) and (b) above.
Value obtained from entry (d, d) in which |${c}_{1221}^{f}$| is obtained from the average of entries (d, a) and (d, b) divided by the square of the ratio superscripted (a) above.