van_stryland_1997

E. W. Van Stryland and M. Sheik-Bahae, in Materials Characterization and Optical Probe Techniques, ed. R. A. Lessard and H. Franke, Proc. SPIE CR69, 501 (1997); reissued as Proc. SPIE 10291, 102910Q (2017)

Source record · Catalogue · Review register

DOI: 10.1117/12.279853

Source kind: theory

Recorded technique: A review of the Z-scan technique and its variants (closed- and open-aperture Z-scan, eclipsing Z-scan, two-color and time-resolved pump-probe Z-scans, top-hat beams, thick samples) and of the interpretation of nonlinear refraction and absorption.

Materials in structured entries: None recorded.

Dataset quantities: None recorded.

Contents: 0 datasets; 0 formulas.

Curation notes

No material values: the review has no tables and quotes no n2 or beta of any material (checked through all 24 pages). The record keeps the definitions and the working formulas, for later use in interpreting Z-scan data (pages are the PDF’s 1-24; the SPIE page numbers are 501-524).

Definitions (p. 5, Eq. 3): for third-order nonlinearities ‘Delta n = {(n2/2)|E|^2}_esu = {n2 I}_MKS and Delta alpha = beta I’, with E the peak electric field (cgs) and I the irradiance (MKS) of the beam within the sample. ‘n2(esu) and n2(MKS) are related through the conversion formula, n2(esu) = (c n0 / 40 pi) n2(MKS), where c (m/sec) is the speed of light in vacuum’; ‘in the remainder of this chapter n2 always refers to n2(MKS)’ (p. 6). beta ‘denotes the third-order nonlinear absorption coefficient, which for ultrafast NLA is equal to the two-photon absorption (2PA) coefficient’. Propagation in the thin-sample (external self-action) limit: d(Delta phi)/dz’ = (2 pi/lambda) Delta n(I) and dI/dz’ = -alpha(I) I (Eqs. 1-2); 2PA with residual linear absorption dI/dz’ = -alpha I - beta I^2 (Eq. 32). Excited-state absorption gives the same loss law in fluence, dF/dz’ = -alpha F - (alpha sigma / 2 hbar omega) F^2 (Eq. 33), so only the pulse-width dependence separates it from 2PA (pp. 18-19). Reflection (p. 12): ‘The reflection losses can be safely assumed to be linear and hence will be ignored in this formalism. In evaluating the nonlinear coefficients, however, one should account for reflection loss of the first surface by taking the irradiance (i.e. I0) to be that inside the sample.’

Thin sample: L < n0 Z0 (sufficient; p. 5) and L << Z0/Delta Phi0, with Z0 = pi w0^2 / lambda and w0 the 1/e^2 half-width of the irradiance (p. 3).

Closed aperture (p. 6): Delta T_pv ~= 0.406 (1 - S)^0.27 |Delta Phi0| (Eq. 5; ‘accurate to within +-3 % for Delta T_pv < 1’), Delta Phi0 = (2 pi/lambda) n2 I0 L_eff (Eq. 6), L_eff = (1 - exp(-alpha L))/alpha, with Delta Phi0 and I0 on axis, at the peak of the pulse, with the sample at focus; |Delta Z_pv| ~ 1.7 Z0 (Eq. 7), peak and valley at ~+-0.856 Z0. Temporal averaging reduces Delta T_pv by A_tau = int f^2 dt / int f dt (Eq. 8): 1/sqrt(2) for a Gaussian pulse, 2/3 for sech^2, 1/2 for a response much slower than the pulse. Fifth order (Delta n = n4 I^2): Delta T_pv ~= 0.21 (1 - S)^0.27 |Delta Phi0|, |Delta Z_pv| ~ 1.2 Z0, Delta Phi0 = k n4 I0^2 L’_eff, L’_eff = [1 - exp(-2 alpha L)]/(2 alpha) (Eqs. 9-10). Eclipsing Z-scan: Delta T_pv ~= 0.68 (1 - S)^-0.44 |Delta Phi0| (Eq. 11; +-3 % for |Delta Phi0| <= 0.2 and 0.98 < S < 0.995 as printed [‘0.98>S> 0.995’]), Delta Z_pv ~ 0.9-1.0 Z0. Top-hat beam (Zhao and Palffy-Muhoray): Delta T_pv ~ 2.8 (1 - S)^1.14 tanh(0.37 Delta Phi0) (Eq. 13), ~1.036 Delta Phi0 for S ~ 0.

Open aperture (p. 9): for q0 = beta I0 L_eff < 1 and a Gaussian pulse, Delta T(Z) ~ -q0 / (2 sqrt 2) / (1 + Z^2/Z0^2) (Eq. 12); the factor 2 sqrt 2 becomes 2 for a response much slower than the pulse. Exactly (Eqs. 25-26), T(Z, S = 1) = sum_m (-q0)^m / [(1 + Z^2/Z0^2)^m (m + 1)^(3/2)] for a Gaussian pulse, |q0| < 1. The coupling factor lambda beta / (2 pi n2) = q0 / Delta Phi0 is twice Im chi(3) / Re chi(3) (p. 14). Field at the exit face: E_e = E exp(-alpha L/2) (1 + q)^(i k n2/beta - 1/2) (Eq. 16), and the Gaussian-decomposition sum of Eqs. 20-23.

Thick samples (pp. 15-17): T(x) ~ 1 + Delta Phi_z0 F(x, l) with Delta Phi_z0 = (2 pi/lambda) n2 I0 Z0, x = Z/Z0, l = L/(n0 Z0) (Eq. 27), F of Eq. 28 (Hermann and McDuff), peak-valley positions of Eq. 29, and the effective length l_eff = F(|X_pv|, l)/(2 x 0.406) (Eq. 30) with the empirical fit l_eff = 2.706 [(l + 1)^1.44 - 1] / (l^1.44 + 3.924) (Eq. 31).

Two copies. The library holds this SPIE copy (a scan with a doubled OCR text layer; running head ‘Materials Characterization and Optical Probe Techniques’, pp. 501-524, reissued in the SPIE Digital Library in 2017 with the footer ‘Proc. of SPIE Vol. 10291 (CR 69), 102910Q, (c) (2002) 2017 SPIE’; the library and Crossref date it 1997, and its references cite work ‘to be published ... (1997)’) and Unverified_94700d0ad99f, a clean born-digital PDF (Ghostscript, 2022) titled ‘Z-Scan Technique for Nonlinear Materials Characterization’, 24 pages, without the SPIE running heads. Compared on 2026-09-26: the same title, authors and affiliations, abstract and keywords, the same sections, figures 1-8 and captions, equations 1-35 (the unverified copy’s extraction lost the display forms of Eqs. 7 and 31, which its PDF has), the same 46 references, and the same page breaks (e.g. Eq. 30, Eq. 31 and Fig. 8 on page 17 of both). It is the same work and version (probably the authors’ manuscript file of this paper); no second record is made.

Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets

No datasets are recorded.

Formulas

No formulas are recorded.

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.