David J. Hagan

Dispersion of bound electron nonlinear refraction in solids

A two-hand model is used to calculate the scaling and spectrum of the nondegenerate nonlinear absorption. From this, the bound electronic nonlinear refractive index n/sub 2/ is obtained using a Kramers-Kronig transformation. The authors include the effects of two-photon and Raman transitions and the AC Stark shift (virtual band blocking). The theoretical calculation for n/sub 2/ shows excellent agreement with measured values for a five-order-of-magnitude variation in the modulus of n/sub 2/ in semiconductors and wide-gap optical solids. Beam distortion methods were used to measure n/sub 2/ in semiconductors. The observations result in a comprehensive theory that allows a prediction of n/sub 2/ at wavelengths beneath the band edge, given only the bandgap energy and the linear index of refraction. Some consequences for all-optical switching are discussed, and a wavelength criterion for the observation of switching is derived. >

Self-focusing and self-defocusing by cascaded second-order effects in KTP

We monitor the induced phase change produced by a cascaded chi((2)):chi((2)) process in KTP near the phase-matching angle on a picosecond 1.06-microm-wavelength beam using the Z-scan technique. This nonlinear refraction is observed to change sign as the crystal is rotated through the phase-match angle in accordance with theory. This theory predicts the maximum small-signal effective nonlinear refractive index of n(eff)(2) congruent with +/-2 x 10(-14) cm(2)/W (+/-1 x 10(-11) esu) for an angle detuning of +/-5 degrees from phase match for this 1-mm-thick crystal with a measured d(eff) of 3.1 pm/V. For a fixed phase mismatch, this n(eff)(2) scales linearly with length and as d(eff)(2) however, for the maximum n(eff)(2) the nonlinear phase distortion becomes sublinear with irradiance for phase shifts near pi/4.

χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons

Cascading is the process by which the exchange of energy between optical beams interacting via second order nonlinearities (χ(2)) leads to various effects such as nonlinear phase shifts, the generation of new beams, all-optical transistor action, the formation of soliton-like (solitary) waves, etc. Here we review the fundamentals of the processes and discuss experimental verification of the effects and various related applications.

Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe

We extend the application of the Z-scan experimental technique to determine free-carrier nonlinearities in the presence of bound electronic refraction and two-photon absorption. We employ this method, using picosecond pulses in CdTe, GaAs, and ZnTe at 1.06 μm and in ZnSe at 1.06 and 0.53 μm, to measure the refractive-index change induced by two-photon-excited free carriers (coefficient σr,), the two-photon absorption coefficient β, and the bound electronic nonlinear refractive index n2. The real and imaginary parts of the third-order susceptibility (i.e., n2 and β, respectively) are determined by Z scans with low inputs, and the refraction from carriers generated by two-photon absorption (an effecitve fifth-order nonlinearity) is determined from Z scans with higher input energies. We compare our experimental results with theoretical models and deduce that the three measured parameters are well predicted by simple two-band models. n2 changes from positive to negative as the photon energy approaches the band edge, in accordance with a recent theory of the dispersion of n2 in solids based on Kramers–Kronig transformations [ Phys. Rev. Lett.65, 96 ( 1990); IEEE J. Quantum Electron.27, 1296 ( 1991)]. We find that the values of σr are in agreement with simple band-filling models.

Direct measurements of nonlinear absorption and refraction in solutions of phthalocyanines

We report direct measurements of the excited singlet state absorption cross section and the associated nonlinear refractive cross section using picosecond pulses at 532 nm in solutions of phthalocyanine and naphthalocyanine dyes. By monitoring the transmittance and far field spatial beam distortion for different pulsewidths in the picosecond regime, we determine that both the nonlinear absorption and refraction are fluence (energy per unit area) rather than irradiance dependent. Thus, excited state absorption (ESA) is the dominant nonlinear absorption process, and the observed nonlinear refraction is also due to real population excitation.

Infrared to ultraviolet measurements of two-photon absorption and n/sub 2/ in wide bandgap solids

The bound electronic nonlinear refractive index, n/sub 2/, and two-photon absorption (2PA) coefficient, /spl beta/, are measured in a variety of inorganic dielectric solids at the four harmonics of the Nd:YAG laser using Z scan. The specific materials studied are: barium fluoride (BaF/sub 2/), calcite (CaCO/sub 3/), potassium bromide (KBr), lithium fluoride (LiF), magnesium fluoride (MgF/sub 2/), sapphire (Al/sub 2/O/sub 3/), a tellurite glass (75%TeO/sub 2/+20%ZnO+5%Na/sub 2/O) and fused silica (SiO/sub 2/). We also report n/sub 2/ and /spl beta/ in three second-order, /spl chi//sup (2)/, nonlinear crystals: potassium titanyl phosphate (KTiOPO/sub 4/ or KTP), lithium niobate (LiNbO/sub 3/), and /spl beta/-barium berate (/spl beta/-BaB/sub 2/O/sub 4/ or BBO). Nonlinear absorption or refraction can alter the wavelength conversion efficiency in these materials. The results of this study are compared to a simple two-parabolic band model originally developed to describe zincblende semiconductors. This model gives the bandgap energy (E/sub g/) scaling and spectrum of the change in absorption. The dispersion of nl as obtained from a Kramers-Kronig transformation of this absorption change scales as E/sub g//sup -1/. The agreement of this theory to data for semiconductors was excellent. However, as could be expected, the agreement for these wide bandgap materials is not as good, although general trends such as increasing nonlinearity with decreasing bandgap energy can be seen.

Kramers-Kronig relations in nonlinear optics

We review dispersion relations, which relate the real part of the optical susceptibility (refraction) to the imaginary part (absorption). We derive and discuss these relations as applied to nonlinear optical systems. It is shown that in the nonlinear case, for self-action effects the correct form for such dispersion relations is nondegenerate, i.e. it is necessary to use multiple frequency arguments. Nonlinear dispersion relations have been shown to be very useful as they usually only require integration over a limited frequency range (corresponding to frequencies at which the absorption changes), unlike the conventional linear Kramers-Krönig relation which requires integration over all absorbing frequencies. Furthermore, calculation of refractive index changes using dispersion relations is easier than a direct calculation of the susceptibility, as transition rates (which give absorption coefficients) are, in general, far easier to calculate than the expectation value of the optical polarization. Both resonant (generation of some excitation that is long lived compared with an optical period) and nonresonant ‘instantaneous’ optical nonlinearities are discussed, and it is shown that the nonlinear dispersion relation has a common form and can be understood in terms of the linear Kramers-Krönig relation applied to a new system consisting of the material plus some ‘perturbation’. We present several examples of the form of this external perturbation, which can be viewed as the pump in a pump-probe experiment. We discuss the two-level saturated atom model and bandfilling in semiconductors among others for the resonant case. For the nonresonant case some recent work is included where the electronic nonlinear refractive coefficient,n2, is determined from the nonlinear absorption processes of two-photon absorption, Raman transitions and the a.c. Stark effect. We also review how the dispersion relations can be extended to give alternative forms for frequency summation which, for example, allows the real and imaginary parts ofχ(2) to be related.

Eclipsing Z-scan measurement of λ/10 4 wave-front distortion

We introduce a simple modification to the Z-scan technique that results in a sensitivity enhancement that permits measurement of nonlinearly induced wave-front distortion of ≃λ/104. This sensitivity was achieved with 10-Hz repetition-rate pulsed laser sources. Sensitivity to nonlinear absorption is also enhanced by a factor of ≃3. This method permits characterization of nonlinear thin films without the need for waveguiding.

Time-resolved Z-scan measurements of optical nonlinearities

We introduce a temporal delay in one beam of the two-color Z-scan apparatus, which measures nondegenerate nonlinear absorption and nondegenerate nonlinear refraction. This technique allows us to time resolve separately the sign and the magnitude of the nonlinear absorption and refraction at frequency ωp that are due to the presence of a strong excitation at frequency ωe. For example, in semiconductors we specifically measure the bound electronic, nondegenerate nonlinear refraction and nondegenerate two-photon absorption, as well as the two-photon-generated free-carrier refraction and absorption as functions of time. We demonstrate this technique on ZnSe, ZnS, and CS2, using picosecond pulses at 1.06 and 0.532 μm.

Measurement of nondegenerate nonlinearities using a two-color Z scan

A simple dual-wavelength (two-color) Z-scan geometry is demonstrated for measuring nonlinearities at frequency omega(p) owing to the presence of light at omega(e). This technique gives the nondegenerate two-photon absorption (2PA) coefficient beta(omega(p); omega(e)) and the nondegenerate nonlinear refractive index n(2)(omega(p); omega(e)), i.e., cross-phase modulation. We demonstrate this technique on CS(2) for n(2) and on ZnSe where 2PA and n(2) are present simultaneously.