Nonlinear coefficient

The usual practice in these appHcations is to concentrate on model development and computation rather than on statistical aspects. In general, nonlinear regression should be appHed only to problems in which there is a weU-defined, clear association between the independent and dependent variables. The generalization of statistics to the associated confidence intervals for nonlinear coefficients is not well developed. 

The complex amplitude of the signal field at the output of the crystal is amplified by a factor g with respect to the input field, g is always larger than one so that there is always coherent amplification of the signal field one speaks of parametric amplification. Nonlinear optics provides an amplification different from the amplification occurring in a medium which present population inversion. For a t)q)ical value of the nonlinear coefficient of 1 pm/V, a pump of... 

MAP has a non-centrosymmetric structure (24) P2- leading to four non-vanishing crystalline nonlinear coefficients. According to the previously defined two-level model, there are also four non-vanishing molecular nonlinearity coefficients which are linearly related to the crystalline coefficients. (3)... 

Figure 6. Classification of the noncentrosymmetric crystal point groups by decreasing value of the maximal efficient phase-matchable nonlinear coefficient per molecule...

One dimensional conjugated carbon polymers can occur in many configurations as depicted in Figure 2 where also we included some chains with nitrogen and sulfur for later reference. Also included there are inorganic one dimensional semiconductors, like SbSI and SbSBr for later comparison. Besides the depicted one-dimensional system others like TCNQ- and KCP-salts could be included here as well but rough measurements of their nonlinear coefficients gave deceptively small values which combined with their ill-characterisation make them poor candidates for nonlinear optical devices. 

Table II. Nonlinear coefficients of some inorganic semiconductors (reference 13)...

The interest in efficient optical frequency doubling has stimulated a search for new nonlinear materials. Kurtz 316) has reported a systematic approach for finding nonlinear crystalline solids, based on the use of the anharmonic oscillator model in conjunction with Miller s rule to estimate the SHG and electro optic coefficients of a material. This empirical rule states that the ratio of the nonlinear optical susceptibility to the product of the linear susceptibilities is a parameter which is nearly constant for a wide variety of inorganic solids. Using this empirical fact, one can arrive at an expression for the nonlinear coefficients that involves only the linear susceptibilities and known material constants. 

Absolute measurements of nonlinear coefficients in GaAs in the millimeter region 10 —10 Hz have been determined 31 ) by mixing adjacent vibrational-rotational lines of a CO2 laser in a GaAs-filled waveguide which was matched to the difference frequencies. The experiments yielded the astonishing result that the nonlinear coefficient and the dielectric constant show opposite dispersions. 

Thin-film dielectrics (Ba0 92Ca108)(Ti0 92Zr0 08)O3 for the thin-film capacitors were prepared using Ba, Ca, and Zr ethoxides and Ti isopropoxide in refluxed methoxyethanol solutions as precursors. Films were deposited on a usual platinized Si substrate. Crystalline thin films after heat treatment at 800°C demonstrated dielectric permittivity of 1200, dielectric loss of0.5%, nonlinear coefficient a = 0.92, and break-down voltage of980 V [1595],... 

Crystals. In the discussion of equation 11, it was pointed out that the macroscopic nonlinear coefficients could be related to the microscopic ones in a relatively straightforward manner. For the hypothetical crystal shown in Figure 5, the relationship is (5)... 

An effective second-order nonlinear coefficient can be defined as... 

Our theoretical understanding of third-order optical nonlinearity at the microscopic level is really in its infancy. Currently no theoretical method exists which can be reliably used to predict, with reasonable computational time, molecular and polymeric structures with enhanced optical nonlinearities. The two important approaches used are the derivative method and the sum-over-states method (7,24). The derivative method is based on the power expansion of the dipole moment or energy given by Equations 3 and 4. The third-order nonlinear coefficient Y is, therefore, simply given by the fourth derivative of the energy or the third derivative of the induced dipole moment with respect to the applied field. These... 

From Equations 6, it is clear that there are a number of key factors governing efficient conversion. These factors are 1) phase-matching 2) optimization of the overlap integral 3) large nonlinear coefficients consistent with phase-matching and 4) low waveguide losses (large L). 

Table II lists the nonlinear coefficient, damage threshold and transparency cutoff for the most well known of the inorganic materials. Urea is included for comparison. As in Table I, materials are listed alphabetically. The data in Table II is taken from a tabulation by Bierlein (personal communication) and from one by Lin and Chen.( 103 ...

Tg = 125-140°C, and was stable in a N2 atmosphere to 220°C. The 13C NMR spectrum revealed the presence of 7t-interaction between those chromophoric groups that are in isotactic relationship to each other ( 30-35% of total). The orientation of the chromophores in a PPNA sample (Tg = 125°C) was achieved in a thin film by the corona poling technique at temperatures above Tg. The subsequent freezing process resulted in a polymeric film that exhibited an initial high second-order nonlinear coefficient, d33=31 pm/V, as measured by Maker-fringe technique with 1.06 im fundamental. 

The increasing use of optical fibre in the telecommunications network will, ultimately, require all-optical signal processing to exploit the full bandwidth available. This has led to a search for materials with fast, large third order optical nonlinearities. Most of the current materials either respond in the nanosecond regime or the nonlinearity is too small (1-3). Organic materials are attractive because of their ultra-fast, broadband responses and low absorption. However the main problem in the materials studied to date, e.g. polydiacetylenes (4) and aromatic main chain polymers (5), has been the small nonlinear coefficients. 

MW/cm, typical units for the nonlinear coefficients are cm/MW for / and cm/MW for n2. In general however, CGS units are used to describe nonlinear coefficients and values for nonlinear optics, so [x ] itself is often reported in esu (electrostatic units). Although x interactions are usually smaller by orders of... 

Fragment or Parameter Freqa Linear Coefficient Nonlinear Coefficient Freqa Primary Coefficient Ultimate Coefficient... 

In centrosymmetric media the P-E function must have odd symmetry, so that the reversal of E results in the reversal of P without any other change. The second-order nonlinear coefficient d must then vanish, and the lowest order nonlinearity is of third order. 

Typical values of the second-order nonlinear coefficient d for dielectric crystals, semiconductors, and organic materials used in photonics applications lie in the range d = 10 24 to 10 - (mks units, As/V2). Typical values of the third-order nonlinear coefficient x(3> for glasses, crystals, semiconductors, semiconductor-doped glasses, and organic materials of interest in photonics are x -3 = 10 34 to 10 29 (mks units). 

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