Resonance-forbidden second-harmonic generation in nonlinear photonic crystals

2.1 Resonance-forbidden SHG by symmetry

We start by showing how SHG can be forbidden by the symmetry-mismatch between the SH dipoles and free-space modes. As shown in Figure 2a, a 1D PhC slab is uniform along y and periodic along x. Air gaps of width w = 300 nm and depth d = 100 nm are etched into the GaAs slab of thickness h = 400 nm at a periodicity of a = 600 nm. In the linear regime, resonances in such a structure can be well-understood using symmetry. Specifically, this structure is invariant under the 180° rotation around the z-axis, C 2 z , which changes x ̂ to − x ̂ and y ̂ to − y ̂ . At the center of the momentum space, Γ, modes that are even under C 2 z are BICs with diverging Q.

Figure 2:

Symmetry-forbidden SHG through nonlinear resonances. (a) Schematic drawing of a 1D GaAs PhC slab unit cell. (b) The band structure and quality factor (Q) of a TE band. Mode profiles of E _x and E _z components at the Γ point are both odd under the C 2 z operation. (c) Through the quadratic nonlinearity of GaAs, the resonance at Γ induces nonlinear polarization P y ( 2 ) and electric field E y ( 2 ) at frequency (2ω), both of which are even under C 2 z . (d) Due to symmetry mismatch, the induced SH dipoles cannot radiate, leading to a divergence of Q ̃ at Γ, which is inversely proportional to SHG efficiency η. a = 600 nm. h = 400 nm. w = 300 nm. d = 100 nm. The pump field is set to be 1 GV/m.

Instead, here we focus on a transverse-electric (TE) mode that is odd under C 2 z with a finite Q of about 100. Specifically, this TE mode has field components of H _y , E _x , and E _z . The eigenfrequencies ω and quality factors Q of the second lowest TE band is shown in Figure 1b, together with its mode profile at Γ. As shown, the mode profile is indeed odd under C 2 z . We now compute the SHG efficiency through this resonance via the quadratic nonlinear susceptibilities of GaAs, of which the non-zero terms are χ x y z ( 2 ) and its permutations. Accordingly, this TE mode of interest can only create SH dipoles along y:

Here, d ₂₅ = 119 pm/V is the quadratic nonlinearity of GaAs. A simple calculation reveals that the nonlinear dipole P ⁽²⁾ is even under C 2 z :

This symmetry analysis is confirmed by the distribution of the SH dipoles P y ( 2 ) and the SH field E y ( 2 ) as shown in Figure 2c. By satisfying the sub-wavelength condition, the SH field is below the diffraction limit, allowing only one radiation channel available to the SH dipoles. Meanwhile, due to symmetry-mismatch, the SH dipoles cannot radiate into free-space; accordingly, no energy is transferred to the SH frequency and the SHG efficiency remains 0, despite the existence of optical nonlinearity. Again, we emphasize that neither the fundamental resonance nor the second harmonic dipoles correspond to a BIC: the fundamental resonance has a finite Q of about 100. Meanwhile, the second harmonic dipoles do not correspond to a resonance of the underlying structure.

To quantitatively evaluate the radiation efficiency of SH dipoles, we define parameter Q ̃ of a nonlinear resonance in a similar way as the quality factor Q of a linear resonance:

Here, E is the energy stored in SH dipoles inside the slab. P is the radiation power of the SH field in the far field. f is the SH frequency, which is doubled from the fundamental frequency. We note that the nonlinear Q ̃ -factor is different from the standard Q-factors of resonances, in that the nonlinear dipoles do not correspond to resonances or eigenmodes of the underlying structure. Naturally, Q ̃ diverges when “resonance-forbidden SHG” occurs, which is observed at the Γ point as shown in Figure 2d. The energy conversion efficiency η = 1 − (T + R) is calculated by first computing how much of the pump energy is transmitted (T) or reflected (R). It can be seen that no pump energy is converted into the second harmonic frequency at the k = 0. When the momentum deviates from k _x = 0, the C 2 z symmetry is broken and Q ̃ becomes finite. In these cases, the SH dipoles can radiate to the far field, and the system returns to the “resonance-enhanced SHG” scenario instead.

2.2 Resonance-forbidden SHG through parameter-tuning

In this section, we present another example of “resonance-forbidden SHG” that cannot be explained by symmetry; instead, it relies on a different mechanism of parameter-tuning. The nonlinear PhC slab is schematically shown in Figure 3a. Air gaps of width w = 30 nm are etched all the way through a GaSe membrane with thickness of h = 325.3 nm at a periodicity of a = 600 nm. The band structure and Q of transverse-magnetic (TM) modes are plotted in Figure 3b, which only have out-of-plane electric field components E _y .

Figure 3:

Forbidding SHG through parametric tuning. (a) Schematic drawing of a 1D GaSe PhC unit cell. (b) Calculated band structure and Qs of the TM bands. One band of interest is shown in blue, along with the frequency and momentum of its second-harmonic fields shown in red. The mode of interest at k _x = 0.01 × 2π/a is marked by a cross. Its mode profile (E _y ) is shown in colormap. (c) Distribution of the induced nonlinear polarization P y ( 2 ) and second harmonic field E y 2 . (d) Q ̃ diverges at k _x = 0.01 × 2π/a, confirming that the resonance of interest forbids SHG. a = 600 nm. h = 325.3 nm. w = 30 nm.

The band of interest is highlighted in blue. The dispersion of the its SH field (red) is calculated by doubling the frequency and crystal momentum of the fundamental mode: ω _SHG = 2ω ₀ and k _SHG = 2k ₀. As shown, in the range of our interest, the SH field stays above the light line, but below the folded light line; accordingly, only one radiation channel is allowed for the SH field, which is beneficial in designing resonance-forbidden SHG. We further note that the SH fields (red curve) do not correspond to a resonance of the underlying structure.

In GaSe, the non-zero quadratic nonlinearity susceptibilities are χ y y y ( 2 ) = − χ x x y ( 2 ) = − χ y x x ( 2 ) . Accordingly, only y-polarized SH dipoles can be created through TM modes:

Based on the distribution of P y ( 2 ) , the SH fields E y ( 2 ) are computed for resonances at different k _x . See Methods for more details of the calculation.

Interestingly, “resonance-forbidden SHG” occurs at k _x = 0.01 × 2π/a in this structure, which is marked by a blue cross in Figure 3b. The SH dipole P y ( 2 ) and SH field E y ( 2 ) of the corresponding mode are shown in Figure 3c. As shown, the SH field only has an evanescent component in the free space with no far-field radiation. This is quantitatively confirmed in the Q ̃ plot, which diverges at k _x = 0.01 × 2π/a (Figure 3d). As the C 2 z symmetry is broken by the non-zero k _x , this “resonance-forbidden SHG” cannot be explained by symmetry alone and is based on a different mechanism of parameter-tuning.

2.3 Topological nature of resonance-forbidden SHG

In this section, we discuss the robustness of “resonance-forbidden SHG” and elucidate the fundamental nature of this phenomenon as topological defects in one dimension.

Following the Bloch’s theorem, the SH field generated by a resonance in the setup of Figure 3 can be written as E y ( 2 ) ( k ) = e i 2 k x v k ( x ) and v _k is a periodic function in x. Since the SH field is below the diffraction limit, there is only one radiation channel with an amplitude of c 0 ( k ) = v k . Here, ⋅ denotes spatial averaging of the given function over a unit cell, which is the same as the zeroth-order Fourier coefficient of this given function.

For structures with up–down mirror symmetry, which invariant under the operation of changing z ̂ into − z ̂ (e.g. Figure 3), SH fields at the top and bottom surfaces are always related. c ₀ for the structure in Figure 3 is plotted in Figure 4a. The sign of c ₀ is flipped from positive to negative when k _x increases from 0 to 0.02 × 2π/a. Accordingly, the phenomenon of “resonance-forbidden SHG” at k _x a/2π = 0.01 is fundamentally a topological defect of parameter c ₀, defined in the one dimensional space of k _x . Naturally, it exists in a robust manner when structural parameters are continuously varied. For example, as shown in Figure 4b, as the thickness of the GaSe slab h is continuously varied between 320 and 326 nm, the phenomenon of “resonance-forbidden SHG” can always be observed along the k _x axis.

Figure 4:

Robustness and topological origin of resonance-forbidden SHG. (a) For the structure shown in Figure 3, the radiation component of the SH field c ₀ reduces to 0 at k _x = 0.01 × 2π/a, where “resonance-forbidden SHG” occurs. This point, marked by a red circle, is fundamentally a topological defect between positive (red) and negative c ₀ (blue). (b) “Resonance-forbidden SHG” can be robustly found at different k _x points when a structural parameter h is varied.

2.4 Finite-size effect on resonance-forbidden SHG

In this section, we discuss the practical question of how SHG efficiency scales with the size of the nonlinear PhC slab. When the PhC slab is of a finite size, momentum k is no-longer perfectly conserved, and the generated nonlinear dipoles can always radiate into free space. To investigate how sample sizes affect the pump depletion rate, we simulate a finite-size PhC, consisting of N unit cells, which is always pumped on-resonance with the mode of interest at k _x a/2π = 0.01. The finite-size PhC in Figure 5a is surrounded by perfect matching layers (PMLs). The number of unit cells N is varied from 400 to 1600. Normalized conversion efficiency is calculated for each N, which is defined as η 0 = I SHG / ( I pump ) 2 . Here, I is the energy flux with a unit of W/m². Our simulation results (Figure 5b) show that the normalized conversion efficiency η ₀ decreases as the sample size increases. This can be intuitively understood: as the structure approaches infinitely large, the structure recovers the periodic boundary condition as in Figure 3 and the normalized conversion efficiency is expected to reduce to 0. In experiment, we expect a large sample illuminated by a de-focused Gaussian beam to be a good enough approximation of an infinite structure and to observe this phenomenon of resonance-forbidden SHG.

Figure 5:

Finite-size effect on resonance-forbidden SHG. (a) A finite-size PhC slab of N unit cells with the same parameters as in Figure 3 is excited by a plane wave. Normalized SHG efficiency η ₀ is simulated for different values of N. (b) The normalized SHG efficiency η ₀ decreases as the sample size N increases.

2.5 Effects of lateral excitation through a waveguide

In this section, we discuss what happens when the resonance-forbidden SHG structure is excited on the side, such as through a waveguide. Different from the vertical excitation as before, the side coupling breaks the translation symmetry; as a result, the nonlinear dipoles will couple to eigenmodes of the underlying structure at the same frequency of 2ω, even if their in-plane momenta are quite different.

In simulation, a finite-size PhC slab consisting of N = 400 unit cells is excited through a waveguide (Figure 6a). The structural parameters of the PhC are the same with those in Figure 3. The excitation frequency ω is set to be on-resonance with the mode of interest at k _x a/2π = 0.01. This fundamental mode propagates along the slab and decays exponentially due to its own radiation loss (Figure 6b). Meanwhile, the SH polarization generated by the fundamental mode can couple with all slab resonances at 2ω, as momentum is no longer conserved. As a result, the SH field oscillates along the x direction as shown in Figure 6c, which is similar to SHG in ordinary waveguides without a perfect phase-matching condition. Meanwhile, the coupling to slab resonances of finite quality factors also introduces vertical radiation loss, which is also observed in Figure 6c.

Figure 6:

Effects of lateral excitation on resonance-forbidden SHG. (a) A PhC slab with the same design as in Figure 3 but of a finite length is excited through a wave-guide. The structure is surrounded with perfectly matched layers (PML). (b) The electric field distribution at the fundamental frequency ω within the first 13 unit cells is plotted. (c) The electric field generated at the SH-frequency 2ω shows an oscillation along the slab. The periodicity of the oscillation is defined by the in-plane momentum mismatch between the slab resonances at 2ω and the nonlinear dipoles.