coefficients several orders of magnitude higher than those of KDP, ADP

and other materials used in frequency upconversion.

It is well known that bulk III-V compounds possess second order nonlinear optical coefficients several orders of magnitude higher than those of KDP, ADP and other materials used in frequency upconversion. However, the utilization of large nonlinear susceptibilities of III-V compounds has not been possible due to high dispersion and lack of birefringence necessary for phase matching [81].

As it was shown in the previous chapters, electrochemistry proved to be a powerful tool for introducing the necessary optical anisotropy in semiconductor materials. Anodically etched Si, for instance, was found to exhibit anisotropy in the infrared and visible regions [82, 83, 84]. The measured birefringence, defined as the difference in the effective refractive indices of the electric fields polarized parallel and perpendicular to the pore axis, reaches a maximum value of 0.366 at the wavelength λ=6.52 µm [82] which exceeds the birefringence of quartz by a factor of 43. Kashkarov el al [85] used the anodization-induced birefringence to achieve phase matching of SHG in porous Si films.

On the other hand, crystallographically oriented pores were used to fabricate semiconductor sieves in gallium phosphide, i.e., two dimensionally nanostructured membranes exhibiting a strongly enhanced optical second harmonic generation in comparison with the bulk material [86]. In this chapter it will be shown that enhanced optical nonlinearities accompanied by artificial anisotropy make electrochemically etched III-V compounds promising for advanced nonlinear optical applications.

**Theoretical predictions**

** **

In the framework of the effective medium theory a cubic semiconductor with randomly distributed air-filled pores all aligned in z-direction, is considered as a homogeneous uniaxial material with an effective dielectric tensor

(45)

and an effective non-linear optical tensor

In the case of a (111)-surface the cubic point group T_{d} is changed to C_{3V}, in which case cikl has three independent components χ_{111}, χ_{113}, and χ_{333} [81]. Following Hui and Stroud [87] χ_{111 }e.g. is given by

(47)

where <...>_{SV} is the average 1/V_{s}∫ V_{s}dV over the volume occupied by the semiconducting material.

The a_{1,2}(r) are given by

where E(r) is the electric field inside the sample (calculated in the approximation of the linear medium) and E_{0} is the average over E(r) taken over the total sample volume

The other non-vanishing components of χ_{ikl} have to be calculated from similar formulae, all containing combinations of third order fluctuation terms of the kind < E_{i}E_{k}E_{l} > divided by products of the kind < E_{i}^{0}E_{k}^{0}E_{l}^{0} >.

An enhanced non-linearity, i.e. χ_{ikl} components are much bigger compared to χ^{semic.}_{123}, requires strong third order field fluctuations. By common sense this seems to be unlikely, because fields in a semiconductor material with high dielectric constant are screened,

Likewise it can be proved exactly that . However, using the special isotropic model of Bruggeman [88] for a non-linear metal-insulator composite, Bergman [89] was able to prove that fourth order fluctuations even can diverge. Repeating this analysis for the dielectric case , the fourth order fluctuations diverge, too.

These divergences are connected with and due to a so-called percolation threshold, a relative concentration 0< f_{0}<1 at which the σ_{M}- respectively the ε_{2} -material forms connected paths through the sample. All simple structure models without such a percolation threshold do not result in large fourth order fluctuations. The Bruggeman model is based on spherical inclusions of both materials in a self-consistently calculated effective matrix. On the other hand, it is well known, that the fields near sharp edges are much larger than fields near spherical surfaces, therefore it is believed that non-spherical inclusions (non-cylindrical pores in our case) can result in large third order fluctuations, leading to strong nonlinear effects in such structures.

**Optical characterization **

Due to their non-spherical shape the pores growing perpendicular to the surface of (111)B oriented III-V sample in a good approximation can satisfy the conditions implied by the theoretical prediction discussed above. In order to perform polarized "second harmonic generation" (SHG) measurements in transmission geometries the porous layers have to be detached from the bulk substrate, i.e. free standing membranes have to be fabricated. Scanning electron microscope (SEM) images taken from such a porous InP(111) membrane are illustrated in Figure 4.14. One can see that most of the pores possess triangular-prism shapes and the lateral size of pores is between 50 and 100 nm. As a fundamental excitation beam for optical measurements, the 1064 nm output of a Q-switched Nd-YAG laser was used. To minimize the influence of the laser output fluctuations, the measured SHG intensity has to be normalized by monitoring the laser intensity in a reference channel.

Figure 4.14: SEM images taken from a porous InP membrane: a) top view; b) cross-section view.

Figure 4.15 curve 1, shows the transmission spectrum of a porous InP(111) membrane exhibiting parallel pores with triangular-prism like shape and transverse dimensions less than 100 nm. As one can see from Figure 4.15, the optical transmission spectrum shows pronounced interference fringes in the spectral interval corresponding to quantum energies lower than the band gap of bulk InP (hν

Figure 4.15: Transmission spectra of optically homogeneous (curve 1) and inhomogeneous (curve 2) porous InP membranes.

In optically homogeneous porous membranes the degree of porosity defines the optical anisotropy caused by the preferential orientation of pores along the <111> crystallographic direction. According to the effective medium theory [89], in the case of pores stretching perpendicular to the initial surface, the components of the dielectric tensor of the porous membrane can be written as follows:

(51)

(52)

where c is the concentration of semiconductor material, ε(ω) is the dielectric function of the III-V compound, and ε_{1} is the dielectric constant of air. Due to for all c, the porous semiconductor represents a positive uniaxial material.

Figure 4.16: Transmission spectra of optically homogeneous (curve 1) and inhomogeneous (curve 2) porous GaP membranes.

Figure 4.17 shows the transmission of light with λ=1064 nm by a porous membrane with the thickness 8.2 ?m as a function of the incident angle of the laser beam. The position of the maxima displayed by the interference patterns depends upon the direction of light polarization. For the ordinary beam, the maxima occur at incidence angles of 17 and 43 degrees, while for the extraordinary beam the maxima occur at 21 and 49 degrees. The analysis of the interference conditions for the two beams taking into account Eqs. 50 and 52 allows one to calculate the refractive indices for ordinary and extraordinary beams: n_{o}=2.43 and n_{e}=2.67. Thus, it is obvious that the porous membranes exhibit pronounced birefringence necessary for phase matching in optical second harmonic generation.

Figure 4.17: Transmission of light with λ=1064 nm by a porous GaP membrane as a function of the incident angle of the laser beam measured in q-s and q-p polarization geometries.

Figure 4.18 illustrates the transmitted s-polarized second harmonic signals (λ_{2w}=532 nm) from both bulk (111)-oriented GaP and a porous membrane as a function of the incident angle of the s-polarized fundamental beam. Despite of the short coherence length (L_{coh}~1 µm) it is not possible to see Maker fringes in bulk GaP because of a sufficiently high absorption at the SHG frequency [90]. The pronounced absorption at 2? is caused by the fact that the corresponding energy is higher than the indirect band gap of GaP (2hw>Eg=2.24 eV).

As one can see from Figure 4.18, under identical conditions the porous membranes exhibit a SHG intensity of at least two orders of magnitude higher than that inherent to bulk GaP. This enhancement can be attributed to giant electric field fluctuations expected, according to theoretical estimations, for some porous structures. Another interesting feature is that the fundamental incident angle dependence of the SHG intensity for porous membranes measured in s-s polarization geometry shows pronounced shoulders at -35 and +35_{o}.

Figure 4.18: Measured s-polarized SH intensity as a function of the incident angle of the s-polarized fundamental beam for bulk and porous GaP.

Figure 4.19 (solid squares) illustrates the rotational dependence of the second harmonic intensity for an optically homogeneous porous GaP(111) membrane possessing triangular-prism like pores. It reflects perfectly the crystallographic features of (111)-oriented GaP demonstrating the high crystalline quality of the porous skeleton. On the contrary, optically inhomogeneous GaP membranes reflect no crystallographic features of the semiconductor compound (Figure 4.19, solid triangles) since in the case of strong diffuse scattering any dependence of the SHG signal upon the rotation angle of the porous membrane about the surface normal is removed.

In optical SHG type I phase matching is achieved if the condition n_{o}(2ω)=n_{eo}(2ω) is satisfied, where

with

Figure 4.19: SH intensity induced by a 1064 nm polarized pump beam at normal incidence as a function of the azimuthal rotation angle of the optically homogeneous (solid squares) and inhomogeneous (solid triangles) porous GaP membranes measured in parallel polarization. The solid line is a fit.

where θ is the angle between the optical axis and the exciting laser beam inside the membrane. Taking into account that ε_{1}=1, ε(ω)=3.1192^{2} and ε(2ω)=3.4595^{2} [81], one can solve the equation n_{0}(2&omega)=n_{e0}(ω) in order to determine θ as a function of c. A solution exists for all c<0.696, it means for the degree of porosity (1-c) > 30 %.

The dependence of the phase matching angle as a function of the GaP concentration is illustrated in Figure 4.20. Note that membranes with 30 % porosity fulfill the phase matching conditions provided that the fundamental and SHG beams propagate in directions that are nearly perpendicular to the pores.

In conclusion, porosity-based technological approaches prove to be important for elaborating new nonlinear optical elements ready to be integrated in optoelectronic circuits. First of all, the formation of pores leads to symmetry breaking. In particular, pores parallel to the <111> direction in III-V compounds change the cubic crystal symmetry (point group T_{d}) to the uniaxial trigonal one (point group C_{3v}). The porosity-induced artificial birefringence opens the possibility to meet the phase matching conditions for the second harmonic generation in III-V materials. For gallium phosphide, in particular, the phase matching conditions can be fulfilled for degrees of porosity higher than 30 %.

Secondly, porous structures represent heterogeneous media where the electric field undergoes large spatial variations. Especially strong local field fluctuations are expected in structures containing pores with sharp edges, e.g., triangular-prism like pores. Porous GaP membranes with triangular-prism like pores exhibit a SHG efficiency two orders of magnitude higher than that of bulk material. Thus, in spite of the electric field screening in the semiconductor, the third order field fluctuations responsible for the SHG enhancement seem to be giant in such kind of structures. Taking into account existing theories, an important role in the SHG enhancement may be attributed to the material percolation which is responsible also for the mechanical stability and good thermal conductivity of porous membranes.

It is interesting to note that new nonlinear optical media can be created just by filling in the pores in porous III-V compounds with other materials. In this case the semiconductor skeleton can be designed to provide phase matching while the material filling the pores will contribute mainly to the SHG. Note that porous III-V compounds as phase-matching matrices are much more promising than elementary semiconductors due to their larger band gap and their more pronounced anisotropy when subjected to electrochemical etching.

Figure 4.20: Dependence of phase matching angle for optical SHG in porous membranes upon the GaP concentration.