sheik_bahae_1994#
M. Sheik-Bahae, J. Wang and E. W. Van Stryland, “Nondegenerate optical Kerr effect in semiconductors”, IEEE J. Quantum Electron. 30, 249-255 (1994)
Source record · Catalogue · Review register
DOI: 10.1109/3.283767
Source kind: measurement
Recorded technique: Theory and measurement. Theory: the two-parabolic-band model of sheik_bahae_1991 extended to the nondegenerate n2(omega1; omega2) (the index change at omega1 due to a beam at omega2) by a Kramers-Kronig transform of the nondegenerate absorption (2PA, electronic Raman, linear and quadratic optical Stark effects, the latter with two more terms than in 1991), with an Elliott-Loudon exciton enhancement in Section IV. Measurement: two-color Z-scan (Refs. [7], [16]) of polycrystalline ZnSe and ZnS; a Q-switched, mode-locked Nd:YAG laser gives single ‘tp ~ 83 ps (FWHM) pulses at lambda = 1.06 um as the excitation beam’ and its second harmonic at 532 nm ‘serves as the weak probe (tp ~ 58 ps)’ (p. 252).
Materials in structured entries: ZnS, ZnSe
Dataset quantities: band_gap, nonlinear_index, refractive_index, two_photon_absorption_coefficient
Contents: 11 datasets; 4 formulas.
Curation notes#
Journal pages 249-255 = PDF pages 1-7 (journal page = 248 + PDF page). Identity checked against the first page. The reference list is present in paper.md.
WHAT IS HERE. The unlabelled table (captioned ‘TABLE III’, tbl_e5d825fda3b1, p. 252; JSON only, merged cells; checked against the image): for polycrystalline ZnSe and ZnS the linear index at 0.532 and 1.06 um, E_g, the degenerate n2 and beta at 0.532 um and at 1.06 um, and the nondegenerate n2(2 omega; omega) and beta(2 omega; omega), omega the 1.064-um fundamental. Units (caption): n2 in 1e-14 cm2/W, beta in ‘10^-9 cm/w’ (1e-9 cm/W = cm/GW). Extracted by tools/extract/sheik_bahae_1994.py. Tables I and II (the nondegenerate F2(x1; x2) and G2(x1; x2)) and Eqs. (6), (8), (9), (11), (12) are formulas.
DEFINITIONS (p. 249, Eqs. 1a, 1b): Delta n(omega1; omega2) = 2 n2(omega1; omega2) I_omega2 and Delta alpha(omega1; omega2) = 2 alpha2(omega1; omega2) I_omega2, I_omega2 the irradiance of the excitation beam: ‘The factor of two in these expressions arises from the interference between the pump and the probe beams. In a self-modulation (single-beam) process this factor is unity [7]’. So the printed nondegenerate n2 is the probe’s index change per pump irradiance divided by two: normalized like a self coefficient (coefficient = ‘self’; the probe sees 2 n2). ‘The nondegenerate 2PA coefficient beta(omega1; omega2) is defined as equal to alpha2 when hbar omega1 + hbar omega2 > E_g’ (p. 250). The nondegenerate datasets have the probe wavelength (0.532 um, omega1) as abscissa and the pump (1064 nm) as parameters.pump_nm. Degenerate beta: dI/dz = -beta I^2 of a single beam (the Z-scan of [16]). Irradiances inside the sample are not discussed here. No uncertainties are given.
PROVENANCE. ‘THE DEGENERATE DATA FOR ZnSe ARE FROM [16]’ (caption): M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760-769 (1990) (not in the library): reproduced. The ZnS degenerate values are among ‘the earlier results’ (p. 252; ‘The degenerate n2 and 2PA coefficient (beta) of ZnSe and ZnS have been studied in the past [5], [9], [16]’), which of them is not said: reproduced, with the three citations. The nondegenerate values are new: primary. Comparison with sheik_bahae_1991 Table III (esu, converted with that table’s n0: gamma = 40 pi n2 / (c n0)): ZnSe at 1.06 um 170e-13 esu -> 2.87e-14 cm2/W (here 2.9: agree); ZnSe at 0.53 um -400e-13 esu -> -6.21e-14 cm2/W (here -6.8: 9 % larger); ZnS at 1.06 um 48e-13 esu -> 0.84e-14 cm2/W (here 0.76: 10 % smaller, so not the 1991 value).
CELLS NOT EXTRACTED: ZnS n2 at 0.532 um ‘N.A.’; beta at 1.06 um ‘0’ for both (below the two-photon edge, hbar omega < E_g / 2: none is possible); ZnS nondegenerate beta ‘<0.073’ (x 1e-9 cm/W), an upper limit (‘ZnS ... exhibits no nondegenerate 2PA’, p. 252). Band gaps: ‘ZnSe has E_g ~ 2.6 eV [17]’, ‘ZnS ... with E_g ~ 3.6 eV [17]’ (p. 252), J. I. Pankove, Optical Processes in Semiconductors (1971): an input, not measured.
THEORY AGAINST THE TABLE (text, p. 252-253; checked with the formulas, 2026-09-27). ‘The measured data are scaled by the predicted band-gap dependence ... where K = 4000 was used. The enhancement of the nondegenerate beta in ZnSe is seen to agree ... the theory underestimates n2 by a factor of ~1.5 when beta is quantitatively correct ... degenerate and nondegenerate n2 values can be fairly predicted by the theory provided that we use K ~ 6000 in (9) for n2.’ With the table’s n0 and E_g, E_p = 21 eV and hbar omega for 0.532 and 1.064 um: Eq. (6) with K = 4000 gives beta(2w; w) of ZnSe 15.2 cm/GW (table 15.3), beta(2w; 2w) 5.45 (ZnSe, table 5.8) and 3.0 (ZnS, table 3.4); Eq. (9) with K = 6000 and Table II (corrected, see formula g2_nondegenerate) gives n2 of ZnSe 2.94 (1.06 um, table 2.9), -4.53 (0.532 um, table -6.8), -3.06 (nondegenerate, table -5.1), and of ZnS 0.61 (1.06 um, table 0.76), 1.39 (nondegenerate, table 1.5), in 1e-14 cm2/W. With the exciton enhancement (E_r = 0.008) ‘using K = 2600 for both beta and n2, excellent agreement’ (p. 253): Figs. 5(a), (b), not reproduced here (a numerical KK integral).
K VALUES (cm GW^-1 eV^(5/2)): ~3100 ‘from the average degenerate 2PA coefficient of several semiconductors ... but varied by up to ~30 %’ [9]; ~4000 better for ZnSe and ZnS; 1940 the TPB calculation [5]; ~6000 for n2 (the factor of two of Eq. 1a, ‘ignored in calculating the n2 coefficient’ in [5], p. 251); 2600 with excitons. E_p ~ 21 eV; m_c / m0 ~ E_g / E_p (Eq. 7).
FIGURES (not digitised). Fig. 1: G2(x1; x2) against x1 for several x2; Fig. 2: scaled beta and n2 of ZnSe and ZnS (degenerate open, nondegenerate solid) with F2(2x; x), F2(x; x), G2(2x; x), G2(x; x); Fig. 3: exciton-enhanced G2 (E_r = 0.01); Fig. 4: the enhancement per mechanism; Fig. 5: as Fig. 2 with K = 2600 and E_r = 0.008. Footnote 1 (p. 253): Fig. 4 of sheik_bahae_1991 ‘in which the long-wavelength divergences of the individual contributions were removed incorrectly, thus suppressing the bandgap resonance of the electronic Raman effect’ is corrected here.
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
sheik_bahae_1994/n2_nondeg_ZnSe#
{
"id": "n2_nondeg_ZnSe",
"material": "ZnSe",
"file": "ZnSe/sheik_bahae_1994__n2_nondeg_ZnSe.csv",
"table": "tbl_e5d825fda3b1",
"quantity": "nonlinear_index",
"unit": "cm2/W",
"scale": 1e-14,
"abscissa_unit": "um",
"use": "primary",
"parameters": {
"definition": "Delta n(omega1; omega2) = 2 n2(omega1; omega2) I_omega2 (Eq. 1a), omega1 the probe (the abscissa), omega2 the pump",
"part": "total",
"coefficient": "self",
"pump_nm": 1064,
"technique": "two-color Z-scan",
"pulse_fs": 83000,
"form": "bulk"
}
}
Table III, ‘Nondegenerate (0.532 um, 1.06 um)’ n2 (p. 252): n2(2 omega; omega), the index change at the 532-nm probe per 1.064-um pump irradiance, Delta n = 2 n2 I_pump (Eq. 1a): the probe sees twice the printed value. Two-color Z-scan; pump 83 ps, probe 58 ps FWHM (p. 252). ‘For ZnSe ... n2(2 omega; omega) is negative and resonantly enhanced due primarily to the effects of 2PA’ (p. 253). Polycrystalline. With 58-83 ps pulses any nuclear response is included: part total (the paper treats it as bound-electronic).
sheik_bahae_1994/beta_nondeg_ZnSe#
{
"id": "beta_nondeg_ZnSe",
"material": "ZnSe",
"file": "ZnSe/sheik_bahae_1994__beta_nondeg_ZnSe.csv",
"table": "tbl_e5d825fda3b1",
"quantity": "two_photon_absorption_coefficient",
"unit": "cm/W",
"scale": 1e-09,
"abscissa_unit": "um",
"use": "primary",
"parameters": {
"definition": "Delta alpha(omega1; omega2) = 2 alpha2(omega1; omega2) I_omega2 (Eq. 1b), omega1 the probe (the abscissa), omega2 the pump",
"pump_nm": 1064,
"technique": "two-color Z-scan",
"pulse_fs": 83000,
"form": "bulk"
}
}
Table III, ‘Nondegenerate (0.532 um, 1.06 um)’ beta (p. 252), printed in ‘10^-9 cm/w’: beta(2 omega; omega), the absorption change of the 532-nm probe per 1.064-um pump irradiance, Delta alpha = 2 beta I_pump (Eq. 1b; beta = alpha2 when hbar omega1 + hbar omega2 > E_g, p. 250): dI_probe/dz = -2 beta I_pump I_probe, not the single-beam -beta I^2. Two-color Z-scan; pump 83 ps, probe 58 ps FWHM (p. 252). Polycrystalline.
sheik_bahae_1994/n2_nondeg_ZnS#
{
"id": "n2_nondeg_ZnS",
"material": "ZnS",
"file": "ZnS/sheik_bahae_1994__n2_nondeg_ZnS.csv",
"table": "tbl_e5d825fda3b1",
"quantity": "nonlinear_index",
"unit": "cm2/W",
"scale": 1e-14,
"abscissa_unit": "um",
"use": "primary",
"parameters": {
"definition": "Delta n(omega1; omega2) = 2 n2(omega1; omega2) I_omega2 (Eq. 1a), omega1 the probe (the abscissa), omega2 the pump",
"part": "total",
"coefficient": "self",
"pump_nm": 1064,
"technique": "two-color Z-scan",
"pulse_fs": 83000,
"form": "bulk"
}
}
Table III, ‘Nondegenerate (0.532 um, 1.06 um)’ n2 (p. 252): n2(2 omega; omega), Delta n = 2 n2 I_pump (Eq. 1a). ‘ZnS ... has a positive and 2PA resonantly enhanced nondegenerate n2 at these wavelengths as predicted by the theory’ (p. 253); no nondegenerate 2PA (hbar omega1 + hbar omega2 ~ 3.5 eV < E_g ~ 3.6 eV). Two-color Z-scan; pump 83 ps, probe 58 ps FWHM. Polycrystalline (the caption; the cubic or hexagonal phase is not said). Part total (58-83 ps).
sheik_bahae_1994/n2_deg_ZnSe#
{
"id": "n2_deg_ZnSe",
"material": "ZnSe",
"file": "ZnSe/sheik_bahae_1994__n2_deg_ZnSe.csv",
"table": "tbl_e5d825fda3b1",
"quantity": "nonlinear_index",
"unit": "cm2/W",
"scale": 1e-14,
"abscissa_unit": "um",
"use": "reproduced",
"reproduced_from": [
"M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, \"Sensitive measurement of optical nonlinearities using a single beam,\" IEEE J. Quantum Electron. 26, 760-769 (1990) [16]"
],
"parameters": {
"part": "unknown",
"coefficient": "self",
"technique": "Z-scan"
}
}
Table III, ‘Degenerate’ n2 at 0.532 and 1.06 um (p. 252): ‘THE DEGENERATE DATA FOR ZnSe ARE FROM [16]’ (caption). Single-beam Z-scan (n = n0 + n2 I). sheik_bahae_1991 Table III prints the same measurements in esu: 170 and -400 x 1e-13 esu, which with its n0 (2.48, 2.70) are 2.87 and -6.21 x 1e-14 cm2/W (here 2.9 and -6.8: the 0.532-um value differs by 9 %). Pulses not stated here; part unknown.
sheik_bahae_1994/beta_deg_ZnSe#
{
"id": "beta_deg_ZnSe",
"material": "ZnSe",
"file": "ZnSe/sheik_bahae_1994__beta_deg_ZnSe.csv",
"table": "tbl_e5d825fda3b1",
"quantity": "two_photon_absorption_coefficient",
"unit": "cm/W",
"scale": 1e-09,
"abscissa_unit": "um",
"use": "reproduced",
"reproduced_from": [
"M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, \"Sensitive measurement of optical nonlinearities using a single beam,\" IEEE J. Quantum Electron. 26, 760-769 (1990) [16]"
],
"parameters": {
"technique": "Z-scan"
}
}
Table III, ‘Degenerate’ beta at 0.532 um (p. 252), printed in ‘10^-9 cm/w’: from [16] (caption). The 1.06-um cell prints 0 (below the two-photon edge) and is not in the file. dI/dz = -beta I^2 of a single beam.
sheik_bahae_1994/n2_deg_ZnS#
{
"id": "n2_deg_ZnS",
"material": "ZnS",
"file": "ZnS/sheik_bahae_1994__n2_deg_ZnS.csv",
"table": "tbl_e5d825fda3b1",
"quantity": "nonlinear_index",
"unit": "cm2/W",
"scale": 1e-14,
"abscissa_unit": "um",
"use": "reproduced",
"reproduced_from": [
"sheik_bahae_1991",
"E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, et al., \"Two photon absorption, nonlinear refraction, and optical limiting,\" Opt. Eng. 24, 613-623 (1985) [9]",
"M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, \"Sensitive measurement of optical nonlinearities using a single beam,\" IEEE J. Quantum Electron. 26, 760-769 (1990) [16]"
],
"parameters": {
"part": "unknown",
"coefficient": "self"
}
}
Table III, ‘Degenerate’ n2 at 1.06 um (p. 252; 0.532 um ‘N.A.’). One of ‘the earlier results’ (‘studied in the past [5], [9], [16]’, p. 252); which is not said. Not sheik_bahae_1991’s value: its 48e-13 esu with n0 = 2.40 is 0.84e-14 cm2/W (0.88 with this table’s 2.3), against 0.76 here. Part unknown.
sheik_bahae_1994/beta_deg_ZnS#
{
"id": "beta_deg_ZnS",
"material": "ZnS",
"file": "ZnS/sheik_bahae_1994__beta_deg_ZnS.csv",
"table": "tbl_e5d825fda3b1",
"quantity": "two_photon_absorption_coefficient",
"unit": "cm/W",
"scale": 1e-09,
"abscissa_unit": "um",
"use": "reproduced",
"reproduced_from": [
"sheik_bahae_1991",
"E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, et al., \"Two photon absorption, nonlinear refraction, and optical limiting,\" Opt. Eng. 24, 613-623 (1985) [9]",
"M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, \"Sensitive measurement of optical nonlinearities using a single beam,\" IEEE J. Quantum Electron. 26, 760-769 (1990) [16]"
]
}
Table III, ‘Degenerate’ beta at 0.532 um (p. 252), printed in ‘10^-9 cm/w’. One of ‘the earlier results’ ([5], [9], [16]); which is not said (sheik_bahae_1991 tabulates no beta). The 1.06-um cell prints 0 and is not in the file.
sheik_bahae_1994/n0_ZnSe#
{
"id": "n0_ZnSe",
"material": "ZnSe",
"file": "ZnSe/sheik_bahae_1994__n0_ZnSe.csv",
"table": "tbl_e5d825fda3b1",
"quantity": "refractive_index",
"abscissa_unit": "um",
"use": "check",
"parameters": {
"assumed": true,
"index_medium": "not stated"
}
}
Table III, ‘Linear Index n0’ (p. 252), ‘2.7 (0.532 um)’ and ‘2.5 (1.06 um)’: the n01, n02 of Eqs. (6) and (9), to two figures; source not stated.
sheik_bahae_1994/n0_ZnS#
{
"id": "n0_ZnS",
"material": "ZnS",
"file": "ZnS/sheik_bahae_1994__n0_ZnS.csv",
"table": "tbl_e5d825fda3b1",
"quantity": "refractive_index",
"abscissa_unit": "um",
"use": "check",
"parameters": {
"assumed": true,
"index_medium": "not stated"
}
}
Table III, ‘Linear Index n0’ (p. 252), ‘2.4 (0.532 um)’ and ‘2.3 (1.06 um)’: the n01, n02 of Eqs. (6) and (9), to two figures; source not stated.
sheik_bahae_1994/eg_ZnSe#
{
"id": "eg_ZnSe",
"material": "ZnSe",
"file": "ZnSe/sheik_bahae_1994__eg_ZnSe.csv",
"table": "tbl_e5d825fda3b1",
"quantity": "band_gap",
"unit": "eV",
"abscissa": "none",
"use": "check",
"reproduced_from": [
"J. I. Pankove, Optical Processes in Semiconductors (Prentice-Hall, Englewood Cliffs, NJ, 1971) [17]"
],
"parameters": {
"kind": "assumed",
"assumed": true
}
}
Table III, ‘Energy Gap E_g (eV)’ (p. 252): ‘ZnSe has E_g ~ 2.6 eV [17]’ (p. 252). The model’s input.
sheik_bahae_1994/eg_ZnS#
{
"id": "eg_ZnS",
"material": "ZnS",
"file": "ZnS/sheik_bahae_1994__eg_ZnS.csv",
"table": "tbl_e5d825fda3b1",
"quantity": "band_gap",
"unit": "eV",
"abscissa": "none",
"use": "check",
"reproduced_from": [
"J. I. Pankove, Optical Processes in Semiconductors (Prentice-Hall, Englewood Cliffs, NJ, 1971) [17]"
],
"parameters": {
"kind": "assumed",
"assumed": true
}
}
Table III, ‘Energy Gap E_g (eV)’ (p. 252): ‘ZnS, on the other hand, with E_g ~ 3.6 eV [17]’ (p. 252). The model’s input.
Formulas#
sheik_bahae_1994/alpha2_nondegenerate#
{
"id": "alpha2_nondegenerate",
"equation": "Eq. (6) and Table I, p. 250",
"printed": "alpha2(omega1; omega2) = K sqrt(E_p) / (n01 n02 E_g^3) F2(x1; x2); Table I: F2PA = (x1 + x2 - 1)^(3/2) / (2^7 x1 x2^2) (1/x1 + 1/x2)^2 for x1 + x2 > 1; F2RAM = (x1 - x2 - 1)^(3/2) / (2^7 x1 x2^2) (1/x1 - 1/x2)^2 for x1 - x2 > 1; F2LSE = -(x1 - 1)^(3/2) / (2^6 x1 x2^2) (1/x2^2) for x1 > 1; F2QSE = -1 / (2^10 x1 x2^2 (x1 - 1)^(1/2)) (1/(x1 - x2) + 1/(x1 + x2) - 2(x1 - 1)/(x1 - x2)^2 - 2(x1 - 1)/(x1 + x2)^2) for x1 > 1",
"variable": "x1 = hbar omega1 / E_g (probe), x2 = hbar omega2 / E_g (pump, below the gap: hbar omega2 < E_g); n01, n02 the linear indices at omega1, omega2; alpha2 in cm/GW with K in cm GW^-1 eV^(5/2) and E_g, E_p in eV; Delta alpha = 2 alpha2 I_omega2 (Eq. 1b)",
"coefficients": {
"K_average": "3100",
"K_ZnSe_ZnS": "4000",
"K_theory": "1940",
"K_exciton": "2600",
"E_p": "21"
},
"check": "Table I checked against its image (2026-09-27). The 2PA part (beta(omega1; omega2) = alpha2 for x1 + x2 > 1) with K = 4000 and the table's n0 and E_g gives beta(2w; w) = 15.2 cm/GW for ZnSe (Table III: 15.3) and the degenerate beta(532 nm) = 5.45 (ZnSe, 5.8) and 3.0 (ZnS, 3.4); its degenerate limit is sheik_bahae_1991's Eq. (25). The Kramers-Kronig transform of the 2PA, Raman and linear Stark terms, and separately of the quadratic Stark term, differs from Table II (corrected) by a pure c x2^-2 term, the removed divergence (numerical principal-value integrals at x1 = 0.3, 0.5, 0.8, residual < 1e-11): the table is internally consistent."
}
The last two quadratic Stark terms ‘arise from the conservation of total number of states’ and ‘contribute less than 10% to the degenerate n2(omega) for hbar omega1 / E_g < 0.95, were ignored in [5] but are included here’ (p. 250). Only below-gap excitation (hbar omega2 < E_g) is treated; above it a stimulated-Raman-gain term would be needed (p. 250). Both beams have the same linear polarization; for an angle phi between them see formula polarization_factor.
sheik_bahae_1994/g2_nondegenerate#
{
"id": "g2_nondegenerate",
"equation": "Eqs. (9)-(10) and Table II, p. 251",
"printed": "n2(omega1; omega2) = (hbar c K / 2) sqrt(E_p) / (E_g^4 n01 n02) G2(x1; x2); G2(x1; x2) = (2/pi) int_0^inf F2(xi; x2) / (xi^2 - x1^2) dxi. Table II: G2PA + G2RAM = H(x1, x2) + H(-x1, x2) + H(x1, -x2) + H(-x1, -x2), H(x1, x2) = 1/(2^6 x2^4 x1^4) [(5/16) x2^3 x1^2 + (9/8) x1^2 x2^2 - (9/4) x1^2 x2 - (3/4) x2^3 + (1/2)(x1^2 + x1^2)[1 - (1 - x2)^(3/2)] + (1/2)(x1 + x2)^2 [(1 - x1 - x2)^(3/2) - (1 - x1)^(3/2)] - (3/16) x1^2 x2^2 [(1 - x1)^(-1/2) + (1 - x2)^(-1/2)] + (3/2) x1^2 x2 (1 - x2)^(1/2) + (3/4) x2 (x1 + x2)^2 (1 - x1)^(1/2) - (3/8) x2^3 x1 (1 - x1)^(-1/2) - (1/32) x1^2 x2^3 (1 - x1)^(-3/2)]; G2QSE (x1 != x2) = 1/(2^9 x1^2 x2^2) [-1/2 - 4/x1^2 + 4/x2^2 - (x2^2/x1) [(1 - x1)^(-1/2) - (1 + x1)^(-1/2)] / (x1^2 - x2^2) + 2 x1^2 (3 x2^2 - x1^2) / (x2^2 (x1^2 - x2^2)^2) [(1 - x2)^(1/2) + (1 + x2)^(1/2)] - 2 x2^2 (3 x1^2 - x2^2) / (x1^2 (x1^2 - x2^2)^2) [(1 - x1)^(1/2) + (1 + x1)^(1/2)]]; G2QSE (x1 = x2) = 1/(2^9 x1^4) [(3/4) [(1 - x1)^(-1/2) - (1 + x1)^(-1/2)] / x1 - [(1 - x1)^(-3/2) + (1 + x1)^(-3/2)] / 8 - 1/2]",
"variable": "x1 = hbar omega1 / E_g (probe), x2 = hbar omega2 / E_g (pump), both below 1; terms (...)^(3/2) are zero when the argument is negative (table note); G2 = G2PA + G2RAM + G2QSE, the infrared-divergent terms of each removed; n2 in cm2/GW with hbar c in eV cm, K in cm GW^-1 eV^(5/2), E_g and E_p in eV; Delta n = 2 n2 I_omega2 (Eq. 1a)",
"coefficients": {
"K_n2": "6000",
"K_ZnSe_ZnS": "4000",
"K_exciton": "2600",
"E_p": "21"
},
"check": "MISPRINT in H: the term printed (1/2)(x1^2 + x1^2)[1 - (1 - x2)^(3/2)] (the image shows x1^2 + x1^2 too) must be (1/2)(x1^2 + x2^2)[...]. Evidence (numerical, 2026-09-27): at x1 = 0.5 and x2 = 0.05-0.35 the principal-value KK integral of Table I's 2PA + Raman + linear Stark terms minus H-sum is, with x1^2 + x2^2, exactly c x2^-2 (c = 0.012814; the other Laurent terms and the constant < 4e-12; residual 2e-12), i.e. the removed divergence the text describes ('all divergences cancel except for the x2^-2 term', p. 251); with the printed x1^2 + x1^2 the difference has a constant -0.19 and does not fit a Laurent polynomial (residual 5e-5), and G2 diverges as x2 -> 0 (G2(0.3; 0.1) = -11.6 printed against 0.0167 corrected). The two agree at x1 = x2 only, so the degenerate G2 is unaffected. The quadratic Stark entries were checked the same way (KK of Table I's four-term F2QSE minus Table II = c x2^-2 at x1 = 0.3, 0.5, 0.8, residual < 1e-13), and the x1 = x2 entry is the limit of the x1 != x2 one (agreement to 1e-4 at x = 0.6, 0.9). With the correction and K = 6000: n2(2w; w) of ZnSe -3.06 and of ZnS 1.39 x 1e-14 cm2/W (Table III: -5.1, 1.5); degenerate ZnSe 2.94 at 1.064 um (2.9). The degenerate G2(x; x) differs from sheik_bahae_1991's by the two added quadratic Stark terms (G2(0.1; 0.1) = 0.01487 against 0.01438; G2(0.9; 0.9) = -0.0569 against -0.0642)."
}
The corrected H (x1^2 + x2^2) is what a model must use; the printed string above keeps the misprint as printed (no coefficient to correct, so this is recorded here rather than as [[formulas.corrections]]). ‘In [5], the interference factor of two, as appears in (1a), was ignored in calculating the n2 coefficient. In correction, the value of K ~ 6000 gives a better fit to the degenerate n2 data for semiconductors, although variations of a factor of ~2 from one material to another occur’ (p. 251). ‘In summing the 2PA and Raman contributions, all the odd terms in x2, arising from the individual divergences, cancel’ (p. 251). The linear Stark term contributes only divergent terms and is absent from Table II.
sheik_bahae_1994/polarization_factor#
{
"id": "polarization_factor",
"equation": "Eq. (8), p. 250",
"printed": "K(phi) = K (1 - (2/3) sin^2 phi)",
"variable": "phi: the angle between the two linear polarizations",
"coefficients": {},
"check": "Not checkable against a printed value (no cross-polarized data here). phi = 90 deg gives K/3, the isotropic ratio chi1122 / chi1111 = 1/3."
}
‘If we assume that the valence band couples isotropically to all the k states in the conduction band’ (p = parallel to k, characteristic of the light-hole band); the heavy-hole band gives ‘a more complex polarization dependence that varies with wavelength and is different for each mechanism [14]’ (p. 250).
sheik_bahae_1994/exciton_enhancement#
{
"id": "exciton_enhancement",
"equation": "Eqs. (11)-(12), p. 253",
"printed": "|U(eta)|^2 = pi eta exp(pi eta) / sinh(pi eta); eta^2 = E_r / (x1 +- x2 - 1); E_r = E_b / E_g = R / (2 E_p eps_s^2)",
"variable": "+ for 2PA, - for Raman; for the optical Stark effect the denominator is x1 - 1; E_b the exciton binding energy, R = 13.6 eV, eps_s the static dielectric constant; F2 of Table I is multiplied by |U|^2 and G2 follows by a numerical KK integral",
"coefficients": {
"R": "13.6",
"E_p": "21",
"E_r_typical": "0.01",
"E_r_ZnSe": "0.008",
"E_r_ZnS": "0.010",
"K": "2600"
},
"check": "R / (2 E_p) = 13.6 / 42 = 0.324, the printed 'E_r ~ 0.33 / eps_s^2'; eps_s = 5-10 gives 0.013-0.0032 (printed '0.013 to 0.003'). The enhanced G2 (Figs. 3-5) is a numerical integral: not reproduced."
}
Loudon’s envelope-function approach after Elliott; ‘ZnSe and ZnS, for instance, have an E_r of 0.008 and 0.010, respectively [9]’ (p. 253). With E_r = 0.008 and K = 2600 ‘for both beta and n2, excellent agreement between theory and experiment is achieved’ (Fig. 5).
Corrections#
No dataset errata are recorded in this source record.
Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the review register for the checks and decisions.