Quasi Phase Matching

Another way to overcome the phase mismatch problem is the so-called Quasi Phase Matching (QPM) based on the periodical modulation of the sign of a selecteddijk compo-nent. In practice this can be achieved by the poling technique [33,34,36,37]. The method works in the following way: as seen in homogeneous crystals there is a constructive interfer-ence within the coherinterfer-ence length between the newly generated SHWand the pre-existing SHW which becomes destructive beyond the coherence length, leading to smaller SHW amplitude. By switching the sign of the d_ijk component at the coherence boundary the interference continues to be constructive.

As a basic example we consider the Periodically Poled Lithium Niobate (PPLN) crys-tal, where phase matching takes place collinearly. In an ideal PPLN material the period-icity of the poling is Λ = 2l_c, with 50% duty cycle, i.e. the length of each stack is equal

to the coherence length, which is in this case:

lc = π

|∆k| = λ₀

4(n^e_2ω−n^e_ω). (1.33)

Although this formula is derived for the negligible pump depletion case, it is also valid for pump depletion (see top of page 89 in [34]). Fig.1.8 shows the layout of such a structure.

Fig. 1.8. PPLN crystal for collinear quasi phase matching. Arrows indicate domain in-version.

The conversion efficiency is shown simultaneously in Fig. 1.9 in the homogeneous Lithium Niobate (LN) slab and in the PPLN slab, for the same input FW field strength.

Constructive interference of SHG can be seen as periodic poling is applied in accordance with the coherence length. In the homogeneous case (green curve) after one coherence length of propagation, the second harmonic field starts decreasing until it vanishes totally, and then the build up and drop down stages are periodically repeated. In the PPLNcase the sign of the nonlinear coefficient d33 gets inverted when the drop down is about to start, thus the second harmonic fields continue building up (blue curve).

If phase matching is applied the negligible pump depletion approximation is not valid anymore, so Eq. (1.27) cannot be applied for high conversion efficiencies (pump depletion).

In other words for negligible pump depletion only Eq. (1.20a) has to be considered, while in the case of pump depletion Eq. (1.20b) as well. The conversion efficiency for pump depletion can be calculated analytically (see [34], table 4, page 47):

η_2ω = tanh²(y/L_{N L}), (1.34)

In the case of QPM the effective nonlinear coefficient (dQ) is smaller than for perfect phase matching, the exact value is calculated as (see [33], Eq. (2.4.7), page 87):

d_Q = 2

πd_eff. (1.35)

Fig. 1.9. Conversion efficiency for PPLN crystal and homogeneous crystal.

So in Eq. (1.28)d_eff has to be replaced byd_Q = _π²d_eff = _π¹χ⁽²⁾, remarking the fact that the resulting “theoretical” curve will be just a smoothed approximation of the real conversion efficiency in the PPLNcase. In Fig.1.9 the red curve indicates the “theoretical” result for perfect phase matching when applying the effective value for d33.

The PPLN structure described above is the so-called first order QPMstructure since the change of the sign of the nonlinear coefficient occurs each time after a coherence length. Less efficient phase matching can be obtained when sign change occurs after an odd multiple of the coherence length, these types are called third order, etc. phase matching. Intensity amplification for different order QPMstructures and the ideal phase matching can be seen on Fig.1.10. This figure also indicates that even if we forced keeping constructive interference by sign change, the quasi phase matching cannot be as effective as “normal” phase matching if we use the same d_ijk component. But the big advantage of QPM is the freedom of selection of the d_ijk component, which can be then chosen to be the maximal one. Therefore, higher amplification can be reached for certain crystals by QPM than by normal phase matching.

If the second order dispersion is spatially structured like a grating, the phase matching can be realized in a non-collinear way. The phase matching condition is k_g =k_2ω−2k_ω. In Fig. 1.11, the dashed line is parallel to the interface between adjacent oppositely poled regions, k_g is perpendicular to them. The wave vector of the FW, k_ω makes an angle

Fig. 1.10. Intensity amplification for different orderQPMstructures and the ideal phase matching. The original figure can be found in [34] as Fig. 25.

φ with the grating, while the angle between k_2ω and k_ω is θ. For simplicity we neglect dispersion. In this case the wave vectors of the FW and SHW will point to symmetrical directions with respect to the grating, i. e. θ = 2φ, and the phase matching condition for the grating is

d= λ₀

4nsinφ, (1.36)

where λ₀ is the wavelength of the FW in vacuum, n is the refractive index of the crystal, and d is the period of modulation.

Fig. 1.11. Phase matching in tilted QPMgrating.

Even if (quasi) phase matching is applied, but either the intensity of incoming FW is small or the propagation length is short, the process can be considered to be in the low conversion regime, so pump depletion can be ignored. Thus the analytical conversion

effi-ciency can be obtained by considering the first term in the Taylor series of the conversion efficiency in Eq. (1.34) (see in [33]):

η_2ω = y

L_{N L} 2

, (1.37)

or it can be obtained simply by taking the limit of the sinc function in Eq. (1.27) when

∆k goes to zero.

Chapter 2

Simulational methods

Computational ElectroMagnetic (CEM) methods can be classified in several ways [47]. A summary chart can be seen in Fig. 2.1, which categorizes most types of CEM methods. High frequency methods are implemented by most commercial optical design softwares (e.g. Zemax [48]): standard ray tracing and diffraction simulation using either the Fourier or Huygens integral method are well-known examples. These methods can only be applied if structure parameters are much larger than the wavelength (d λ).

When structure parameters are comparable to the wavelength (d ∼λ), “true” numerical methods are needed, where the polarization and the vector nature of the ElectroMagnetic (EM) fields become important. Two methods are usually distinguished. The plane wave expansion method [49–54] describes the band structure of the photonic crystal, and the Transfer Matrix Method (TMM) [55–61] calculates the transmission and reflection of a one-dimensional layered structure.

Numerical methods can be classified further based on the criterion whether the differ-ential or integral form of Maxwell’s equations are discretized. The most popular methods derived from the integral form are the Method of Moments [62–64], and the Fast Multipole Method [65,66]. Methods derived from the differential form require a grid on which theEM field values are defined. Depending on whether the grid is structured or non-structured the methods can be further classified. If the geometry of the structure is complex and contains many curved surfaces the usage of a non-structured grid seems to be the right choice, this method is called the Finite Element Method (FEM) [67–69], which is widely used in commercial softwares.

The methods, which will be used in this work belong to the structured grid cate-gory. Two main classes are included depending on whether the method is defined in the time or in the frequency domain. The Finite Difference Time Domain (FDTD) method (see Sec. 2.1) is very popular recently, due to its versatile application for wide range of

problems, like active devices, and much more. The PseudoSpectral Time Domain (PSTD) method [70–73] is an advanced form ofFDTDwhere the discrete differential operators are replaced by derivation using Fourier transformations. In the frequency domain the meth-ods can be further divided into real space and Fourier space type. Among the real space methods we consider first the Finite Difference Frequency Domain (FDFD) method (see Sec.2.2). It starts from Maxwell’s equations in the frequency space, but only one frequency is regarded (monochromatic simulation). By rearranging the two- or three-dimensional field matrices into a vector, the final field distribution is obtained via a matrix inversion.

The Beam Propagation Method (BPM) [74–76] is basically a technique for simulating the propagation of light in slowly varying approximation, it is a one-way model, multiple reflections (like in fiber Bragg grating structures) are eliminated, however, a stable bidi-rectional beam propagation method has already been worked out [77,78]. The Method of Lines (MoL) (see Sec. 2.3) leaves one axis analytical and discretization is present only in the transversal plane. Accordingly the structure is divided into “lines”, fields are matched between the layers, reflection and transmission are calculated by connecting the first layer to the last one. Finally the Rigorously Coupled Wave Analysis (RCWA) (see Sec. 2.4)

is mentioned, which operates on the Fourier components of the fields. This method is widely used for optimization of subwavelength structures both in academic and industrial applications (optical critical dimension calculations).

The various methods are optimal for different problems. Fig. 2.2 details the strength and weaknesses of the four selected methods (FDTD, FDFD, MoL, RCWA) indicating their optimal applications. This summary table is based on lecture notes in [79]. The FDTD method is an early discretized representation of Maxwell’s equations, hence its presentation embeds the concepts for the other methods. Necessary tools for numerical electromagnetic simulations such as the implementation of the source and the absorbing boundary condition are introduced by the FDTD method. The FDFD method is in the main focus of this Thesis since this method will be extended for second harmonic nonlin-ear simulations. The FDFD method uses the same formulation for spatial derivatives as FDTD, the main difference is that the FDFD method operates in the frequency domain.

MoL plays the role of a natural bridge between the FDFD and RCWA methods. It will be detailed not only for the sake of completeness but MoL also fits better for metals or structures complex in longitudinal dimension than the RCWA method. The RCWA method is used for the optimization of structures in Chapter 3. It is very fast and used in commercial environments as well.

Fig. 2.2. Advantages and disadvantages of the four selected methods.

FDTD and RCWA are the most used and widespread methods on structured grids in CEM. Fig. 2.3 shows the step-by-step evolution of the FDTD method leading to the RCWA method, giving a comprehensive picture of numerical simulational methods. One of the reasons of dealing with four different methods is the possibility of direct comparison

of the results, another is finding - by scrutinizing their implementation - the method best fitted for generalization in second order nonlinear simulations.

Fig. 2.3. Route fromFDTD to RCWA via FDFDand MoL methods.

In the rest of this chapter we elaborate the details of these methods. All of them will be used for simulation of linear dielectric structures in Chapter 3, where the RCWA method will also be used for the optimization of the structures. The FDFD method will be developed further for second harmonic nonlinearity in Chapter4. In Chapter5a novel method called PSFDwill be detailed, which is based on the FDFD method.