Dichroic Beamsplitter

A dichroic beamsplitter application has been worked out for the SRG under consid-eration. The geometry of two kinds of dichroic beamsplitters has been determined in the NIR range: one transmits light above 1430 nm and reflects below it (longpass), the other one does the contrary: reflects light above 1430 nm and transmits below it (shortpass).

The initial SRG structure was optimized for 45^◦ angle of incidence by varying the width and height of the top and bottom layers and also the period length of the SRGstructure.

An efficiency limit of 5% is defined, i.e. transmission is accepted above 95%, while effi-cient reflection is below 5% transmission. Using this definition the working ranges of the dichroic beamsplitters are reported.

The applied merit function for the longpass filter is:

M(P) = X

1130<λ0<1350

T(λ0, P) + X

1490<λ0<1700

R(λ0, P).

The optimization yielded the parameter set listed in Table 3.4 including the applied constraints. The obtained transmission characteristics is shown in Fig. 3.11, for which the corresponding source code is found in Appendix Aas PRG Fig. 3.11. The working range of this filter is 1110 - 1320 nm for reflection and 1520-1620 nm for transmission.

parameter initial value → longpass value shortpass value constrains

w_top 392 nm → 288 nm 308 nm 90. . .Λ nm

w_bottom 182 nm → 90 nm 108 nm 90. . .Λ nm

dtop 350 nm → 349 nm 268 nm 150. . . 600 nm

d_bottom 430 nm → 180 nm 440 nm 150. . . 600 nm

Λ 700 nm → 397 nm 521 nm 300. . . 600 nm

Table 3.4. Initial and optimized parameters for the long- and shortpass dichroic filter including the constrains for the optimization

Fig. 3.11. Transmission of the longpass dichroic beamsplitter.

The applied merit function for the shortpass filter is:

M(P) = X

1130<λ0<1350

R(λ₀, P) + X

1490<λ0<1700

T(λ₀, P).

The optimization yielded the parameter set listed in Table 3.4 including the applied constraints. The resulting transmission spectrum is plotted in Fig. 3.12, for which the corresponding source code is found in Appendix Aas PRG Fig. 3.12. The working range of this filter is 1150–1190 nm and 1240–1320 nm for reflection and 1540–1620 nm for transmission.

Both filters can be used to separate a bichromatic field consisting of 1310 nm and 1550 nm laser radiations. For instance they can be used in a Michelson interferometer, where the resonator is stabilized by piezo actuators to an auxiliary laser, therefore active laser will be locked and frequency stabilized as well. So by using the shortpass monolithic dichroic filter 1310 nm telecommunication wavelength can be locked for example to a commercially available 1.533 µm frequency-locked C₂H₂ laser, exploiting the advantage of the low thermal noise of monolithic filters. And the longpass version can be used for stabilization of 1550 nm telecommunication wavelength by an auxiliary laser locked to a preferred atomic transition in the range of 1100–1300 nm.

Fig. 3.12. Transmission of the shortpass dichroic beamsplitter.

Chapter 4

Numerical simulation of second harmonic wave generation by the FDFD method

In Chapter3the simulation of a Surface Relief Grating (SRG) was presented, which is a linear photonic structure. As discussed in Chapter1, spatially structured photonic devices are rarely described analytically so numerical simulations have to be used for accurate results. They are beyond the scope of standard diffraction theory, the polarization and the vector nature of the ElectroMagnetic (EM) fields have to be considered. In Chapter2linear simulational methods are thoroughly detailed. Four linear, differential equation based, structured grid numerical methods are presented: the Finite Difference Time Domain (FDTD) method, the Finite Difference Frequency Domain (FDFD) method, the Method of Lines (MoL), and the Rigorous Coupled Wave Analysis (RCWA) (for the categorization see Fig. 2.1). Although numerical simulation of structured photonic devices is already a challenging task, the situation becomes even more complicated if nonlinearity of the medium is also considered. Hence accurate numerical simulation of electromagnetic wave propagation in structured nonlinear dielectric media is one of the central problems of recent developments in nonlinear optics. Spatial structuring can affect both the linear and nonlinear parts of the susceptibility. The numerical solution of Maxwell’s equations is much more difficult in the presence of nonlinear polarization compared to linear media.

Despite the fact that many results are derived by analytical handling of the prob-lem [32–34,103–107], in the case of complex geometry numerical simulations cannot be avoided. There are a number of numerical methods for solving such problems, all of them are applicable with some compromises. As simplest approach, in Ref. [103] two kinds of simulations of Quasi Phase Matched (QPM) structures are described, the Effective

layered dielectric structures the Transfer Matrix Method (TMM) proved to be an efficient tool, accordingly its extension to second harmonic generation seems to be reasonable, and was executed in [38,108]. This method is accurate, can handle pump depletion, but works well only in one-dimensional structures. The Beam Propagation Method (BPM) [109] is mostly used to solve the scalar wave equation in waveguides [78,110] for modeling in-tegrated optical devices. It has been extended to simulate Second Harmonic Generation (SHG) in optical fibers [111] and in periodic microstructured waveguides [112,113].

In two and three dimensions numerical simulation of SHG has been carried out by the modification/extension of other linear methods. The most commonly used one, the FDTD method (see Sec. 2.1) has been extended to simulate nonlinear processes, such as SHG, [114,115], where pump depletion is naturally incorporated. In principle the un-derlying dielectrics can have arbitrary spatial structure both in the linear and nonlin-ear susceptibility. The improved version of FDTD called PseudoSpectral Time Domain (PSTD) method was also extended for simulation of SHG [116,117]. The FDTD and PSTD methods are suitable for modeling the propagation of pulsed light fields as well.

The determination of frequency dependent transfer properties of a structure can be time consuming, because transients may take a long time to decay. Another well-established class of methods is the Finite Element Method (FEM) for Maxwell’s equations [118]. It can also be extended to model SHG in nanoparticles [119] without pump depletion. The Spectral Element Method is the higher order version of theFEM; it has been successfully used to model SHGin photonic crystals [120], including pump depletion. Finally we men-tion that in the case of nanoparticles, the Surface Integral Method has successfully been adopted to model SHG [121].

In Sec.2.2 the FDFDmethod has been detailed based on Refs. [83,84]:FDFDis sim-ilar to FDTD, however, instead of time domain propagation, the method determines the transmission of a single, sharp frequency field through a complex, linear dielectric struc-ture. In this chapter the FDFD method is extended to nonlinear cases resulting a new method called NonLinear Finite Difference Frequency Domain (NL-FDFD) method. The linear FDFD method works on a single frequency component: ω. In NL-FDFD method two meshes are considered with ω and 2ω fields and the nonlinear effects are handled as a coupling between them. First the two-dimensional case is examined followed by equa-tions derived for three dimensions. The effect of the nonlinear polarization appear as an additional source term in the FDFD equations. The source of the ω mesh contains 2ω components and vice versa, hence the FDFD equations for the two meshes are

cou-pled and nonlinear, therefore the solution can be obtained via iteration. As this method directly operates on fields, both the undepleted and depleted cases can be simulated.

Not only one-dimensional, periodically poled materials (e.g. Periodically Poled Lithium Niobate (PPLN)) but also two-dimensional, structured patterns can be examined. For demonstrating the efficiency of the method, three case studies are worked out:

1. SHG in a bulk nonlinear crystal;

2. SHG in a one-dimensional PPLN structure;

3. SHG in an array of Lithium Niobate (LN) cylinders.

It will be shown that only 2-3 iteration steps are needed for low conversion efficiency, while in the high conversion case (close to 100% conversion efficiency), about ten iteration steps are necessary to obtain the solution. Convergence can be checked via Cauchy criteria.