1. Introduction

Harmonic generation in structures with periodic modulation of the linear and/or nonlinear susceptibilities is of interest for increasing conversion efficiencies, which are usually limited by the phase mismatch. In these structures the phase mismatch between fundamental and harmonic waves can be reset periodically, thus allowing the generated signal to accumulate constructively over an extended propagation distance. This technique is known as quasi-phase-matching [1]. Periodic poling of ferroelectrics [2], leading to a periodic sign change of the nonlinear susceptibility, is a commonly used and well established technique. For a review on the development of structured media suitable for quasi-phase-matched interactions, see [3] and references therein.

Third-harmonic (TH) generation is a common technique to convert femtosecond laser pulses to shorter wavelengths. Since all materials have a nonzero cubic susceptibility, χ⁽³⁾, the TH can be directly generated in one step. A more common way to efficiently generate the TH is through a two-step process of second-harmonic (SH) generation, followed by sum frequency (SF) generation of the latter and the fundamental. Since this cascaded approach makes use of the much larger quadratic susceptibility, conversion efficiencies are usually larger. For high efficiencies, simultaneous phase-matching for both SH and SF generation is needed. This can be achieved by two crystals optimized for the SH and SF processes, respectively. Overall efficiencies typically do not exceed 10% – 15%.

With the advent of tabletop lasers capable of producing femtosecond pulses, with peak powers > 1 terawatt, efficient generation of phase-matched TH by use of the cubic susceptibility in a single nonlinear crystal has been demonstrated. It has been shown that in addition to the directly generated TH, non-phase-matched cascaded quadratic processes can also contribute to the total generated TH [4, 5]. Efficiencies of 11% have been demonstrated using a single BIBO crystal, for sub-100-fs pulses at 2.1 μm [5].

Another way to efficiently generate the TH in a single nonlinear material is through the use of quasi-phase-matched structures, where it is possible to achieve simultaneous phase-matching of more than one parametric process, otherwise hard to accomplish by the use of optical birefringence in a single crystal [6]. TH efficiencies of 10.8% have been achieved through the simultaneous phase matching of SH and SF generation, for 100-fs fundamental pulses at 1550 nm, using a MgO-doped multi-grating periodically poled crystal [7]. Other methods involving the periodic suppression of the cubic nonlinearity, for the direct generation of the TH, have been demonstrated. For instance, by clamping together silica-based structures consisting of 800-nm-thick dye-doped sol-gel silica films deposited on 300-micron-thick silica substrates, quasi-phase-matched TH generation was demonstrated, with efficiencies well below 1% [8].

Using high quality thin dielectric coatings (highly uniform with absorption and scattering losses well below 1%) for efficient TH generation is an attractive idea due to their wide bandgap and broad transmission spectrum. Because of the short interaction lengths, pulse broadening, and other effects associated with the propagation of short pulses, are greatly reduced. Custom stacks of films can easily be produced with high quality using well established coating methodologies. TH conversion efficiencies of ≈ 1% have been demonstrated from single dielectric layers of ZnO [9]. Efficient TH generation from stacks of films has also been proposed and showed promise [10].

To design film stacks for optimum TH generation a model is needed that takes into account interference effects and nonlinear frequency conversion simultaneously. We develop a matrix formalism that is a straightforward extension of the matrix algebra used in linear optics to, for example, design multilayer coatings of mirrors and filters, e.g. [11]. We derive explicit matrix expressions for the transmitted and reflected nonlinear signals, with both forward- and backward-generated TH taken into account. The results are equivalent to an approach that is based on matching boundary conditions, e.g. [12, 13]. We extend the model to include the contribution from a substrate for focused beams with cylindrically symmetric spatial beam profiles. We also generalize the approach for illumination with ultrashort light pulses. The model is then applied to design novel structures, consisting of an optimized sequence of films, which can lead to substantial TH generation by exploiting the interplay of local field enhancements and periodic resets of the phase mismatch. The model can also be applied to determine χ⁽³⁾ values of dielectric films in the presence of substrates and to analyze TH microscopy data [14].

2. Theoretical background

The geometry of the multilayered system is sketched in Fig. 1. The first step is to calculate the fundamental fields in each layer (index i) propagating to the right and left. This can be done using the well-known matrix formalism, connecting the input vector v⃗′₀ = (F′_0,r, F′_0,ℓ) to vectors of the type v⃗_i = (F_i,r, F_i,ℓ) within the stack.

 

Fig. 1 TH generation in a stack of layers on a semi-infinite substrate. TH and fundamental fields are labeled E and F, respectively. Fields propagating to the right (r) and left (ℓ) are distinguished in each layer. Fields on the right interfaces are labeled with a prime.

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The overall transfer matrix ℳ of a stack of n layers (from 0 to n + 1) is obtained from an ordered product of 2 × 2 matrices,

Using this system matrix, the output vector in the substrate just after the last layer can be related to the input vector just before the first interface,

The fundamental field amplitude at the left interface of an arbitrary layer i of the stack can be written as

In the second step we calculate the TH field that is produced by the fundamental field in the stack. To this end we first formulate optical matrices for the TH field that relate an input vector w⃗′₀ = (E′_0,r, E′_0,ℓ) to the TH field in the substrate w⃗_n+1 = (E_r, E_ℓ) assuming no TH generation. The overall transfer matrix M for the stack,

To add TH generation we assume small signal conversion and neglect pump depletion. The TH field at the right boundary of layer i is related to the field on the left boundary through

Using the propagation matrix L_i for layer i the TH fields at opposite sides of the layer are related by

Repeating this procedure starting from the field on the left side of the stack, w⃗′₀, yields for the field in the final medium (substrate)

We will now derive the generated TH field Δ⃗_i from the known fundamental field v⃗_i, cf. Eq. (5). At each position z in the layer a fundamental field F propagating for example in the +z direction produces a third-order nonlinear polarization, proportional to the total fundamental field cubed, which acts as the source of field components at the TH frequency, propagating in both forward and backward directions. The generating wave equation for the TH field produced in layer i, and traveling in the +z direction Δ_i,re^{−iκ_i(z−z_i)}, is given by

Likewise we obtain the TH field produced in layer i, propagating to the left, at the left layer boundary,

Equations (10), (13), and (14) determine the TH field in the substrate and incident medium as a function of the incident fundamental and TH, if present.

The obtained TH fields, cf. Eq. (11), agree with [13] although we follow here a rather different line of thought. The approach used in [13] involves solving the generating second order differential wave equation, by finding solutions to the inhomogeneous equation (bound waves), and solutions to the homogeneous wave equation (free waves), which are needed to satisfy the boundary conditions. We make use of the slowly varying amplitude approximation to convert the generating wave equation from a second order differential equation to a first order differential equation, cf. Eq. (12), and solve the problem via integration. However, the physical implication of this approximation is in neglecting the oppositely propagating field generated by the nonlinear polarization. It can be shown [15] that solving the first order differential equation for both the right- and left-propagating fields, is equivalent to solving the second order differential equation, as done in [13]. Our approach is an intuitive extension of the optical matrices in linear optics found in textbooks and can easily be extended to include substrate effects and short pulse excitations, see below.

We now include the contribution from the substrate to the total TH detected in transmission (E_T) and reflection (E_R). If the substrate is thin we can just consider it as an additional layer and include it in the stack matrix. For thick substrates beam divergence and propagation effects cannot necessarily be neglected. For simplicity we will assume that the fields reflected at the substrate’s output face do not contribute to the total TH field. Experimentally this can be accomplished in several ways - (i) using a wedge-shaped substrate, (ii) having the output face of the substrate properly AR coated, or (iii) using index matching oil opaque for the TH for detection in reflection geometry [14]. Also, for thick enough substrates, because of beam divergence the coherent overlap of the reflected TH with the field in the stack is poor and can be neglected.

TH generation in the presence of diffraction can conveniently be done in the spatial frequency (ρ) domain [16]. The transmitted TH field E_T is the sum of the field produced in the film stack, E_r, multiplied by the spatial frequency spectrum and the Fresnel propagator, and the field produced in the substrate:

A similar procedure is used for the backward-generated TH in the substrate to obtain the total TH field E_R propagating to the left in the incident medium. The field E_R(ρ) is the sum of the TH produced in the stack of films and the backward-generated TH in the substrate, and transmitted through the film:

To extend this model to illumination with short laser pulses the finite pulse spectrum has to be taken into account in the TH generation process. To this end we start with the generating wave equation (e.g. [17]) for the fundamental and TH pulse envelopes propagating with the group velocities v_F and v_E, respectively:

A similar procedure is followed to obtain the TH field produced in layer i, propagating to the left. Equation (11) can then be used to determine the resultant TH field at frequency ν in the substrate (E_r) and incident medium (E′_0,ℓ), and ∫ |E(ν)|²dν yields the TH power.

3. Efficient TH generation from optimized stacks of films

This model is applied to characterize TH generation from stacks of films. The general sample architecture is a stack of alternating layers of high and low (ideally vanishing) χ⁽³⁾ materials deposited on a substrate. The low-χ⁽³⁾ layers mainly act through their phase response controlling the relative phase of the fundamental wave and TH generated in the high-χ⁽³⁾ layers. In practice, this situation can be reached with a combination of hafnia (HfO₂) and silica (SiO₂) layers taking advantage of the fact that their χ⁽³⁾ ratio is ≈ 20 [14]. For simplicity we assume that the total thickness of the hafnia layers corresponds to the coherence length of TH generation for a fundamental wave at 800 nm, D = L_coh ≈ 627 nm. Note that for a single slab of hafnia L_coh is the thickness for maximum TH conversion.

The TH in reflection and transmission was calculated as a function of the number of silica layers. For each case a genetic algorithm was used to find the thickness and position of each layer for maximum total conversion efficiency. To assess the TH generation of stacks of films we neglected the substrate contribution. The results are shown in Fig. 2. For the data points without film (interference) effects we took into account single reflection at and transmission through interfaces but neglected multiple interference described by the matrix approach.

 

Fig. 2 (a) TH generation from stacks of films in which silica layers are progressively added, keeping the total thickness of hafnia constant, equal to one coherence length, D = L_coh ≈ 627 nm. TH signal in (b) transmission and (c) reflection with interference (crosses) and without interference (circles). Normalization is performed with respect to a single hafnia layer of D = L_coh. Dotted line shows expected TH from a single hafnia layer, of D = L_coh, assuming perfect phase matching. The nine-layer stack parameters (thicknesses) are substrate/[HfO₂/SiO₂]⁴HfO₂ = substrate/98/131/98/122/93/696/103/48/235 (layer thicknesses in nm).

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Without film effects the signal with additional silica layers increases relative to a single hafnia layer. This shows the effect of a periodic reduction of the phase mismatch introduced in the TH generating hafnia layers. Note that the theoretical maximum TH signal for a single slab is for zero phase mismatch (dotted line in Figs. 2(b) and 2(c)).

If we include in addition the interference effects in the film stack the signal increases further. There are two possible reasons - the stack optimization (i) maximizes the integral fundamental in the hafnia layers and (ii) leads to an optimum phase of the TH produced in the hafnia layers. According to our previous discussion, the TH signal is maximum if the fields produced in each layer and propagated through the remaining stack interfere constructively at the sample output (either in reflection or transmission).

For the frequency conversion of ultrashort laser pulses the bandwidth of the stack is important. The structures shown in Fig. 2 were optimized with respect to a single wavelength ν. If we use |E_R(ν)|² for a crude estimate we obtain bandwidths of the order of 20 nm, which would support pulses of ≈ 40 fs duration at 800 nm. In general stacks with fewer layers have larger bandwidths. In principle, stacks can be designed using nonlinear optimization where the figure of merit includes conversion efficiency, pulse chirp, and bandwidth. Chirp and bandwidth can be evaluated using Eq. (19).

Using this approach and input fluences well below the threshold of optical damage conversion efficiencies larger than ten percent seem possible. This would make such devices an attractive alternative to the configurations using crystals. Multiple stacks in sequence and the use of external Fabry-Perot enhancement could increase efficiencies even more. For enhanced reflectance metal layers at nodes of the standing field distribution can be implemented.

4. Summary

In summary, we present a matrix approach for TH generation in stacked materials, in both transmission and reflection geometries, that takes into account the contribution from the substrate to the total generated TH, interference of fundamental and nonlinear fields inside the stack, the nonlinear signal generation in forward and backward direction, the beam profile of the focused incident beam in the substrate, and the finite spectrum associated with short laser pulses. Considerable TH signal enhancement is predicted from optimized stacks of films, which is a consequence of favorable fundamental field enhancements and periodic corrections of the phase mismatch limiting TH conversion in single films.