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Abstract
We analytically and numerically investigate surface second-harmonic generation (SHG) from a stack of dielectric layers. We develop a theoretical formalism based on the transfer matrix method for the calculation of the surface-driven second-harmonic radiation from multilayer structures and elaborate it for the case of ultrathin dielectric layers using a power series expansion to derive the effective surface nonlinear tensor for the whole stack. We show that for deeply subwavelength thicknesses of the layers the surface responses from all interfaces can efficiently sum up, leading to largely enhanced efficiency of SHG. As a result, such surface-driven nonlinearity can become comparable to the bulk nonlinearity in noncentrosymmetric semiconductors and can yield high performance for nonlinear nanophotonic applications.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Second-harmonic generation (SHG) is the simplest and most widely studied nonlinear optical process, for which the incident electromagnetic wave at the fundamental frequency $\omega$ is converted in a medium into a wave at the doubled frequency $2\omega$ [1]. Second-harmonic generation is known to be driven by two main sources, namely bulk and surface second-order nonlinearities. The bulk nonlinearity is related to the symmetry properties of the medium and is especially strong in noncentrosymmetric materials without inversion symmetry. In contrast, in centrosymmetric materials the second-order nonlinearity vanishes in the volume in the dipole approximation, and only higher-order multipole terms can contribute to the SHG.
Surface SHG arises inside the interfacial layer close to the medium interface due to the discontinuity of both the medium’s structure and the jump of the normal component of the optical field across the interface resulting in a large electric field gradient at the surface [2–4]. Even in centrosymmetric media the inversion symmetry gets broken nearby the medium’s surface, leading to a pronounced dipole second-order nonlinearity in the surface region. Due to this fact the surface SHG is dominating in the nonlinear response of centrosymmetric media, including plasmonic structures [5–7]. Besides that, such surface sensitivity makes surface SHG a powerful tool for probing interfaces [8].
The most standard approach to boost the conversion efficiency of SHG in nonlinear optics is to fulfill the phase-matching condition, resulting in energy transfer from the fundamental to the second-harmonic wave upon propagation. In nanophotonic applications the phase-matching condition ceases to play a major role, as the structures under consideration typically have subwavelength scales, which are much smaller than the nonlinear length required for energy transfer under the phase-matching condition. Therefore, the conversion efficiency in nanophotonic structures gets mainly enhanced through the excitation of strong resonances in all-dielectric nanopillars made of III-V semiconductor materials with bulk nonlinearity [9,10], alternatively combined with multiple-quantum-well structures to suppress the band-gap absorption [11].
SHG from nanostructures made of III-V semiconductors (GaAs, AlGaAs and others) is often well described with bulk nonlinearity only, while completely neglecting the effects of the surface nonlinearity [12–14]. However, even in III-V semiconductors with strong bulk nonlinearity the surface effects can play a significant role, in particular at frequencies close to the surface resonances [15] or in dielectric metasurfaces for SHG wavelengths in the opaque region above the bandgap [16–18]. At the same time, many materials that are widely used in all-dielectric nanophotonics, such as Si, TiO$_2$, amorphous SiO$_2$, Ge, and others are centrosymmetric, therefore SHG is primarily driven by the surface contribution. As a result, enhancement of SHG in structures made of these materials would mainly rely on a certain way to boost the surface nonlinear response. The surface SHG is enhanced with larger surface-to-volume ratio and high optical excitation field at the surface. These conditions can be naturally provided in nanostructures, having nanoscale dimensions and being able to concentrate the field by means of multiple resonances in the visible or infrared wavelength range [19].
In this paper we focus on another possible option to gain from the surface-driven SHG. Our idea is to combine multiple surfaces together so that the nonlinear responses from several interfaces are summed up. The most natural way to implement this idea is to take a stack composed of multiple dielectric layers. The enhancement of the SHG in multilayer structures with the bulk nonlinearities has already been studied before and different power scaling laws of the SHG conversion efficiency were obtained under certain resonant conditions [20–24]. If the surface nonlinearities are considered, the constructive interference of the surface-induced second-harmonic waves from the interfaces in the stack can be expected to significantly enhance the conversion efficiency as compared to other nanophotonic structures made of such materials, and could even yield comparable performance to nanostructures made of III-V semiconductor materials with strong bulk nonlinearity. Modern fabrication techniques, like the molecular beam epitaxy or atomic layer deposition, readily allow to grow such multilayer structures with layer thicknesses down to several nanometers [25–27]. In our consideration we do not explicitly account for effects like charge trapping and interface charging, which may arise at some semiconductor interfaces and lead to a constant electric field in the vicinity of interface and surface-like contribution to SHG from third-order bulk nonlinearity [28–30]. These contributions, however, can be formally added to the usual surface nonlinearity and thus are implicitly accounted for as well in our treatment.
The paper is organized as follows. In Section 2 we apply a transfer matrix method for modeling of SHG from multilayer structures, driven by the surface nonlinearities at the interfaces between the layers, and derive the effective nonlinear parameters of the whole stack for the case of ultrathin layers. In Section 3 we present the results of numerical simulations, where we compare the SHG efficiency of an exemplary multilayer structure with a single layer of similar thickness and bulk nonlinearity as well as with a single interface with effective surface nonlinearity of the whole stack. Finally, in Section 4 we give concluding remarks and summarize our findings.
2. Transfer matrix method for modeling surface SHG
2.1 Description of SHG and boundary conditions
We consider a structure representing a stack of $N$ isotropic dielectric layers deposited on a substrate, as schematically shown in Fig. 1. The $m$-th layer in the stack has the thickness $h_m$ and the complex relative permittivity $\varepsilon _m$. The imaginary parts of the permittivities $\varepsilon _m$ therefore describe the losses inside the stack. Furthermore, we assume the losses originating at the interfaces inside the stack to be also effectively included into the imaginary parts of the complex relative permittivities of the materials. The stack is located on the top of a semi-infinite isotropic substrate with dielectric permittivity $\varepsilon _{\textrm {sub}}$, the medium in the upper space (superstrate) is assumed to be air. We take the coordinate $z$-axis directed normal to the stack interfaces and the $xy$-plane is correspondingly assumed to coincide with the plane of the stack interfaces. The structure is illuminated from the top side by a wave at the fundamental frequency. In the simplest case, which we mainly examine here, the incident field is a plane wave with the wavevector $\vec k^{\textrm {FW}}$ and the incidence angle $\theta$.
Fig. 1. Sketch of a multilayer structure under consideration, consisting of $N$ layers with thicknesses $h_m$ and dielectric permittivities $\varepsilon _m$, located on the top of a semi-infinite substrate with dielectric permittivity $\varepsilon _{\textrm {sub}}$.
The transfer matrix method (TMM) is widely used for modeling of light propagation in one-dimensional structures, including photonic crystals (PC), defect structures or Bragg reflectors [31]. The transfer matrix method is also used for studying nonlinear optical phenomena in layered media. In Ref. [32] third-harmonic generation in PC structures was analyzed using the transfer matrix method under the undepleted-wave approximation. Later this approach was generalized by a number of authors to account for pump depletion, both for second- and third-order nonlinear processes [33–39]. However, to the best of our knowledge, the modeling of SHG in multilayer structure so far has been limited to bulk nonlinearities only. It is also worth to mention works [40–43] where an alternative theoretical approach for modeling second- and third-harmonic generation from dielectric and metallodielectric multilayer structures was presented, based on solving dynamical equations in the time domain for coupled Drude-Lorentz oscillators corresponding to free and bound charges. This approach is expected to inherently incorporate both surface and bulk contributions simultaneously without their explicit separation. Therefore, based on this approach the enhanced conversion efficiency from a stack was demonstrated thanks to the increased number of active surfaces and pump field localization.
Here we develop an analytical formalism based on the transfer matrix method, assuming that the surface nonlinearity is dominating the nonlinear response of the stack. In the treatment below we neglect the bulk nonlinear contribution and limit the investigation to the surface nonlinearity only. This assumptions is well justified in the case of centrosymmetric dielectrics even for thick films, since the bulk SHG provided by higher-order multipoles is much smaller than the surface SHG. However, this assumption can be also reasonable for thin enough layers of noncentrosymmetric materials. In the latter case one would need the corresponding thickness $h_m$ to satisfy the relation:
where $\chi ^{(2)}_{\textrm {bulk}}$ and $\chi ^{(2)}_{\textrm {surf}}$ are the bulk and surface second-order nonlinear tensors, respectively. For example, it was experimentally shown [44,45], that in GaP nanopillars the surface contribution becomes dominating for nanopillar diameters below $\sim$ 170 nm. In Ref. [46] the surface contribution was experimentally found to prevail in the SHG response from 50-nm and 100-nm-thick GaAs films. These values could thereby serve as an order-of-magnitude estimate of the threshold thickness in Eq. (1). One can also expect these estimates to hold for other noncentrosymmetric materials, first of all for other III-V semiconductors.In the presence of the surface nonlinear polarization the standard linear boundary conditions for the components of the electric and magnetic fields across an interface are not applicable, since the tangential field components exhibit a jump across an interface. Instead, the following generalized boundary conditions at each interface should be imposed [47]:
For the important case of plane-wave illumination certain conclusions can be derived from the general form of the boundary conditions stated in Eq. (2). In particular, the normal component of the surface nonlinear polarization at every interface would satisfy
where $\vec k^{\textrm {FW}}_{||}$ is the tangential component of the wavevector of the incident plane wave. Since $\vec k^{\textrm {FW}}_{||}$ is constant throughout the whole stack according to the standard boundary equations for electromagnetic waves (which still hold for the fundamental frequency), it follows thatEquations (4) and (5) together impose certain restrictions on the second-harmonic plane waves, which could be radiated. Let us define the wave polarization with the electric field vector lying in the plane of the stack interfaces, i.e. $xy$-plane, as the TE-polarization and the wave polarization with the magnetic field vector lying in the $xy$-plane of the stack interfaces as the TM-polarization. Then, for instance, a TE-polarized plane wave with an electric field in the direction of $\Delta \vec E_{||}$ can not be generated, because in this case one would get a plane wave with $\vec E \cdot \vec k^{\textrm {SH}} \ne 0$. For the same reason a TM-polarized plane wave with a magnetic field in the direction of $\vec k^{\textrm {FW}}_{||}$ can not be produced, even though $\Delta \vec H_{||}$ has in general a nonzero component in this direction at every interface.
It is convenient to use the transfer-matrix formalism to calculate the second-harmonic field propagation through the multilayer structure. If we assume for definiteness, that the electric field is oriented along the $x$-axis and the magnetic field along the $y$-axis, we get the following equality for the fields at the second-harmonic frequency in the $m$-th and $m+1$-th layers:
Here the minus sign arises, since we get the field below the interface using the field above. The subscript "+" in Eq. (6) corresponds to the field just below the top interface of the respective layer. In contrast to the conventional transfer-matrix analysis, we get equations with nonzero right-hand side due to the presence of the surface nonlinear polarization at the interfaces. Writing down Eq. (6) for every single interface in the stack and combining them together, we end up with an equation of the form:
In particular, in the case of only one interface, i.e., without layers on the top of the substrate, the matrices in Eq. (9) reduce to
In general, the emitted field at the second-harmonic frequency contains both TE- and TM-polarized components. For this reason the boundary conditions Eq. (2) have to be divided into two terms, giving the jumps of the electric and the magnetic field for both polarizations. While for the general case of an arbitrary incident field at the fundamental frequency this could be a quite challenging task, for the plane-wave illumination this separation can be easily performed. Indeed, as shown above in Eq. (4), the jumps of the electric field $\Delta E_{||,m(m+1)}$ would have the same direction at every interface inside the stack, which coincides with the direction of $\vec k^{\textrm {FW}}_{||}$. At the same time, $\vec k^{\textrm {SH}}_{||} = 2 \vec k^{\textrm {FW}}_{||}$ for the field in every layer. If we put the $x$-axis along the direction of $\vec k^{\textrm {FW}}_{||}$, then the jumps $\Delta E_{||,m(m+1)}$ and $\vec k^{\textrm {SH}}_{||}$ are also directed along $x$-axis. Since the electric field in a plane wave is orthogonal to the wavevector, TE-polarized second-harmonic waves with electric field along the $x$-axis cannot be generated at any interface inside the stack. This means, that the jumps $\Delta E_{||,m(m+1)}$ would produce in every layer the TM-polarized second-harmonic waves with the electric field lying in the $xz$-plane and magnetic field along $y$-axis, therefore:
At the same time, the jumps of the electric and the magnetic field in Eq. (2) are independent of each other, so that $\Delta H_{||,m(m+1)}$ at every interface would in general possess both $x$- and $y$-components. The $x$-component would in turn produce the TE-polarized second-harmonic waves with magnetic field lying in the $xz$-plane and electric field along the $y$-axis. Since $\Delta E_{||,m(m+1)} \parallel \vec e_x$ for any $m$, the corresponding boundary conditions are:
The transfer matrices for TE- and TM-polarized fields are different, therefore both cases are to be considered separately. Below we derive the corresponding expressions for both TE- and TM-polarized second-harmonic waves. The treatment in the next subsections will proceed step by step as follows:
- In subsections 2.2 and 2.3 we present transfer matrices for TE- and TM-polarized fields at the second-harmonic frequency and derive the expressions relating the emitted fields at the second-harmonic frequency with the source terms and the transfer matrices.
- In subsection 2.4 we discuss, how the source terms are calculated from the fundamental field by solving the linear scattering problem at the fundamental frequency with the transfer matrix method.
- In subsection 2.5 we summarize all steps of our numerical algorithm and provide its general scheme.
- In subsections 2.6 and 2.7 we consider the special case of an ultrathin multilayer structure and derive the effective surface nonlinear tensor of the stack.
2.2 TE-polarized second-harmonic wave
We first start with the case of a TE-polarized emitted second-harmonic wave. Upon assuming as above, that the electric field at the second-harmonic frequency in this case is directed along the $y$-axis, Eq. (8) attains the form:
The transfer matrix of a single $m-$th layer for TE-polarized waves is given as:
The transfer matrix of the whole stack is:
We note, that we use the standard approach by writing the transfer matrices for the tangential components of the electric and magnetic fields [31], rather than using the matrices for forward- and backward-propagating components of the electric field in each layer, as it is done in many other works on SHG in multilayer structures [32,33,35–39]. The reasons for that are twofold. On the one hand, to calculate the surface nonlinear polarization at each interface we need to use the total field, i.e., the sum of forward- and backward-propagating fields, but not the fields separately. On the other hand, boundary conditions in the presence of surface nonlinearity Eq. (2) give the jump of the total field again and it can be a tricky issue to divide Eq. (2) at each interface into two terms, giving the field jumps separately for forward- and backward-propagating fields. Therefore it seems to be much more convenient to write down the transfer matrices for the tangential components of the electric and magnetic fields without their separation into forward- and backward-propagating components.
Above (in the superstrate) and below the stack (in the substrate) we get two outgoing plane waves. Let us denote their amplitudes of electric field strength as $A^{+}$ and $A^{-}$ respectively. Given that we can use the simple relations between electric and magnetic field in the plane wave to obtain:
Substituting expressions Eqs. (13)–(17) into Eq. (12) one finally gets the following system of equations for the unknowns $A^{+}$ and $A^{-}$:
For the field in the superstrate $A^+$, we finally derive:
2.3 TM-polarized second-harmonic wave
For TM-polarized SHG we get the electric field lying in the $xz$-plane and the magnetic field directed along the $y$-axis, hence Eq. (8) attains the form:
The transfer matrix of the $m-$th layer for TM-polarized waves is given as:
The transfer matrix of the whole stack is found again as:
We denote the amplitudes of the $x$-component of the electric field strength in the outgoing plane waves above (in the superstrate) and below the stack (in the substrate) as $A^+$ and $A^-$ respectively. Then we apply the relations between electric and magnetic field in the plane wave to obtain:
Substituting these expressions Eqs. (21)–24 into Eq. (20) we arrive at a system of equations for the unknowns $A^{+}$ and $A^{-}$:
For the field in the superstrate $A^+$ we finally obtain:
2.4 Calculation of the source terms
When we derive Eqs. (19) and (26), we need to be able to calculate the source terms $\hat {S}^{\textrm {TE}}_{\Sigma }, \hat {S}^{\textrm {TM}}_{\Sigma }$. This implies that we express, for convenience, the surface nonlinear polarization at every interface in the stack through the parameters of the incident fundamental wave. It should also be noted that the undepleted pump approximation is used, meaning that we neglect the reverse action of the field at the second-harmonic frequency on the field at the fundamental frequency.
For the surface nonlinear polarization we use the expression as follows:
Let us start with a TE-polarized incident plane wave at the fundamental frequency. We denote the amplitude reflection coefficient of the whole stack for TE-polarized plane wave as $R_{\textrm {TE}}$ and the amplitude of the electric field in the incident wave as $E_{\textrm {I}}$. Following the assumptions from the previous sections we assume, that vectors $\vec k^{\textrm {FW}}_{||}$ and $\vec k^{\textrm {SH}}_{||}$ are directed along the $x$-axis, thus the electric field in the TE-polarized incident plane wave has to be directed along the $y$-axis and the magnetic field lies in the $xz$-plane. Then using the transfer-matrix approach and Eqs. (13)–(16) derived above and rewritten for the fundamental frequency, we find for the electric field at the fundamental frequency just below the $m$-th layer:
Here we explicitly add indices $"\textrm {FW}"$ to emphasize that matrices $\hat {M}^{\textrm {TE}}_{m, \textrm {FW}}$ together with all values $Z'_{m, \textrm {FW}}$ are calculated at the fundamental frequency, while in all other equations they are calculated for the second-harmonic frequency.
We perform now exactly the same derivation for a TM-polarized incident plane wave at the fundamental frequency. We introduce the amplitude reflection coefficient of the whole stack for a TM-polarized plane wave as $R_{\textrm {TM}}$ and the amplitude of the electric field in the incident wave as $E_{\textrm {I}}$. Following the assumptions from previous sections we assume, that vectors $\vec k^{\textrm {FW}}_{||}$ and $\vec k^{\textrm {SH}}_{||}$ are directed along the $x$-axis, thus the electric field in the TM-polarized incident plane wave lies in the $xz$-plane and the magnetic field is directed along the $y$-axis. Then using the transfer-matrix approach and Eqs. (21)–(23) derived above, we find for the electric field just below the $m$-th layer:
In order to find the normal component $E_{\textrm {TM},\perp }$ we use the following relation between the electric and the magnetic field of a plane wave in the $m+1$-th layer (i.e. just below $m$-th layer):
Here the component $H_{\textrm {TM},y}$ inside the stack is to be found directly from Eq. (29).
Finally, we want to write down the explicit expressions for the amplitude reflection coefficients at the fundamental frequency $R_{\textrm {TE}}$ and $R_{\textrm {TM}}$. If $E_{\textrm {I}, \textrm {TE}/\textrm {TM}}$ is the field incident on the stack at the fundamental frequency and $E_{\textrm {T}, \textrm {TE}/\textrm {TM}}$ is the transmitted field, one can write similar to Eqs. (18) and (28) for TE-polarization:
Again we add indices $"\textrm {FW}"$ for the matrix $\hat {M}^{\textrm {TE}}_{\Sigma , \textrm {FW}}$ and all values $Z'_{m, \textrm {FW}}$, when they are calculated at the fundamental frequency. From Eq. (31) one gets for the $R_{\textrm {TE}}$:
In a similar way for TM-polarization we get:
2.5 General scheme of proposed transfer matrix method for modeling of surface SHG in multilayer structures
In this section we aim to summarize the above findings both for the fields at fundamental and second-harmonic frequency to provide a general scheme for the calculation of the surface SHG in multilayer structures. In this section we do not impose any restrictions on the thicknesses of the layers in the stack in order to provide analytically strict results. Results for the case of ultrathin layers will be derived in the next section to calculate an effective surface nonlinearity for the whole stack.
Overall our transfer matrix method includes the following steps:
- 4. Construct matrices $\hat {M}_{\Sigma }, \hat {S}_{\Sigma }$ for the second-harmonic field using Eqs. (9). Depending on the polarization of the incident field at the fundamental frequency, either the TE-, or the TM-polarized wave will be produced at the second-harmonic frequency, or even both. Boundary conditions for each of them are to be found from Eqs. (10)–(11).
Also, if one is interested in finding the transmitted second-harmonic wave instead of the reflected one, this can be readily done directly from Eq. (18) or Eq. (25). This five-step procedure gives a general framework for finding the surface-driven SHG from an arbitrary multilayer structure. Our approach inherently takes into account multiple reflections inside the stack, both for the fundamental and for the second-harmonic field, and is valid for the three-dimensional case, i.e., for arbitrary illumination angles of the incident field. We also use the undepleted pump approximation, i.e., the reverse action of the second-harmonic field on the fundamental field is ignored, since it is usually expected to be negligibly small.
The procedure above, even though it assumed the plane-wave illumination, can be extended to the case of an arbitrary illuminating field at the fundamental frequency. Then steps (1)-(2) above have to be run for each Fourier component of the incident field at the fundamental frequency. Then after finding the surface nonlinear polarization Eq. (27) and the boundary conditions Eq. (2) at every interface, one has to expand the corresponding jumps of the electric and magnetic field into Fourier series at the second-harmonic frequency and to run steps (4)-(5) above again for each plane-wave component separately.
Besides that, our procedure allows extending to the case of arbitrary symmetry group of the surfaces. This means, that Eq. (27) may contain other terms arising from the surface anisotropy due to the presence of defects, surface crystallization, surface oxidation or other fabrication issues. All the above reasoning still holds in this case, except for another expression for the surface nonlinear polarization in step (3).
2.6 Ultrathin multilayer structure
Our transfer-matrix method proposed in the previous sections can be easily implemented for numerical simulations of the surface SHG from any multilayer structures. At the same time, for the case of ultrathin layers in the stack all equations obtained above can be largely simplified to yield analytical expressions for the second-harmonic radiation. In this section we derive analytical expressions for the emitted second-harmonic waves for both polarizations of fundamental and second-harmonic wave. For this end we assume that all layers in the stack are ultrathin, namely the following condition holds for each $m$:
With the condition Eq. (34) one can also expect Eq. (1) to hold, when noncentrosymmetric materials are used among others in the considered multilayer structure.
Given the condition Eq. (34) is fulfilled for all layers, we can first expand the elements of the transfer matrix Eq. (13) for a TE-polarized second-harmonic wave up to the first-order terms of small parameters $k^{\textrm {SH}}_{z,m} h_m$:
In a similar way the source matrix $\hat {S}^{\textrm {TE}}_{\Sigma }$ for the TE-polarized second-harmonic wave can be expanded as:
Finally, expression Eq. (19) for the electric field amplitude in the superstrate can be readily expanded into a series over small parameters $k^{\textrm {SH}}_{z,m} h_m$. For this we introduce $A^+_0$ as the limiting value of $A^+$ when all $k^{\textrm {SH}}_{z,m} h_m \xrightarrow {} 0$. Then Eq. (19) yields:
In a similar way for a TM-polarized second-harmonic wave, provided that the condition Eq. (34) is fulfilled for all layers, we can again expand the entries of the transfer matrix Eq. (21) up to the first-order terms of small parameters $k^{\textrm {SH}}_{z,m} h_m$:
The source matrix $\hat {S}^{\textrm {TM}}_{\Sigma }$ can again be written as:
We expand once more the expression Eq. (26) for the field in the superstrate into a series over small parameters $k^{\textrm {SH}}_{z,m} h_m$, introducing $A^+_0$ as the limiting value of $A^+$ when all $k^{\textrm {SH}}_{z,m} h_m \xrightarrow {} 0$. Then Eq. (26) yields:
In a similar way, one has to expand also the components of the electric field inside the stack at the fundamental frequency. For a TE-polarized incident plane wave at the fundamental frequency using Eqs. (28) and (35) we get for the electric field just below the $m$-th layer:
For a TM-polarized incident plane wave at the fundamental frequency from Eqs. (29) and (38) the electric field just below the $m$-th layer can be found as:
The component $H_{\textrm {TM},y}$ inside the stack can be found from Eq. (29):
Finally, in order to put the expression Eqs. (41)–(43) to the final form, we expand the reflection coefficients for the stack $R_{\textrm {TE}}$ and $R_{\textrm {TM}}$ into a series of small parameters $k^{\textrm {SH}}_{z,m} h_m$. For $R_{\textrm {TE}}$ one gets from Eq. (32):
For $R_{\textrm {TM}}$ one obtains in a similar way from Eq. (33):
Obtained expressions for the components of the electric field allow calculation of the source terms and eventually let to express the amplitudes of the emitted second-harmonic waves in Eqs. (37) and (40) through the thicknesses and refractive indices of the layers and the amplitude reflection coefficients in the absence of the stack.
2.7 Effective surface nonlinear tensor of the stack
The expansions of the fields inside the stack derived in the previous section for the ultrathin layers allow us to proceed with finding the effective surface nonlinear tensor of the stack. Indeed, for the stack of ultrathin layers one can intuitively expect the surface responses from all interfaces to sum up leading to the overall nonlinear response being provided by a surface nonlinear tensor of the form:
The ratio of the relative permittivities at the fundamental frequency arises, when the normal component of the electric field $E_z$ is used for the calculation of $P^{\textrm {NL}}$ and accounts for the jump of the normal component at every interface inside the stack, as compared to a single interface between the substrate and the superstrate. This means:
For example, for the case of isotropic surfaces in the stack the surface nonlinear tensor Eq. (48) attains the form:
Let us first assume a TE-polarized incident wave at the fundamental frequency. In this case in Eq. (27) $E^{(\omega )}_{\perp } = 0$ inside the stack, therefore only the second term with $\chi ^{(2)}_{\perp || \ ||}$ would contribute to SHG at each interface. The surface nonlinear polarization then possesses only a normal component $P^{\textrm {NL}}_{S, \perp }$, thus $\Delta \vec H_{||} = 0$ and $S^{\textrm {TM}}_{m(m+1), 2} = 0$ in Eq. (39). According to Eqs. (10)–(11) only the TM-polarized second-harmonic wave is generated in this case. The effective surface nonlinear tensor can be obtained in the form:
As the next step we assume a TM-polarized incident wave at the fundamental frequency. In this case all terms in Eq. (27) would lead to SHG at each interface. The surface nonlinear polarization possesses both normal and tangential components. From Eqs. (2), (27) we find that $\vec P^{\textrm {NL}}_{S, ||} || \vec E^{(\omega )}_{||}$, so that $\Delta \vec H_{||} \perp \vec E^{(\omega )}_{||}$ and $\Delta \vec H_{||} \perp \Delta \vec E_{||}$. According to Eqs. (10)–(11) one can see, that again only the TM-polarized second-harmonic wave is generated. Using Eqs. (40) and (47) and keeping only first-order small terms we get:
The obtained expressions Eqs. (51)–(52) give only first-order correction terms, which, upon neglecting the absorption in the ultrathin stack, contribute just to the imaginary part of the effective nonlinearity $\chi ^{(2)}_{\textrm {surf}, \Sigma }$. In order to take into account the changes of the real part of $\chi ^{(2)}_{\textrm {surf}, \Sigma }$ one needs to keep second-order terms in the expansions of Eqs. (35), (38), which, however, makes the equations much more cumbersome. Besides that, Eqs. (51)–(52) are derived for the two simplest cases of pure TE- or pure TM-polarized incident wave. For the plane-wave excitation with mixed polarization the fields for both polarizations are mixed through the terms $\chi ^{(2)}_{|| \perp ||, m(m+1)}$, leading to a more complicated output, particularly to the generation of both TE- and TM-polarized second-harmonic waves. The findings above can be generalized using Eqs. (37) and (40).
Finally, the derived expansions for effective surface nonlinearity of the stack $\chi ^{(2)}_{\textrm {surf}, \Sigma }$ allow us to estimate the applicability limits of the simplified Eqs. (48). The condition of smallness of the correction terms in Eqs. (51), (52) can be written as:
Equations (49)–(50) actually represent the generalization of the condition from Eq. (34) to the whole stack. Indeed, Eq. (34) provides the smallness of the thickness of a single layer in the stack and allows simplifying the transfer matrices for a single layer. However, for large number of layers in the stack one can expect the condition from Eq. (34) to be insufficient for accurate analytical treatment of the whole stack. The respective more general necessary conditions for this case were derived above and are given by Eqs. (49)–(50).
3. Surface SHG from an exemplary stack
To illustrate the above findings we perform calculations for an exemplary multilayer structure. The structure under consideration consists of 8 thin layers made of Si, SiO$_2$ and TiO$_2$ and located on the top of a glass substrate (BK7). The thicknesses and materials of each layer are listed in Table 1. Such ultrathin layers can be grown by existing deposition techniques, e.g., atomic layer deposition or molecular beam epitaxy [25–27]. The fundamental wavelength of the incident wave is assumed to be 1 $\mu$m, with a corresponding second-harmonic wavelength of $500$ nm. However, this particular wavelength choice does not restrict the generality of our results. As we do not rely on material resonances, similar results as found by us will appear for different wavelengths, albeit for correspondingly changed layer stack parameters. The relative permittivities of all materials at the fundamental and second-harmonic frequencies are taken from Refs. [48–50].
Table 1. The composition of the considered multilayer stack.
The electric field strength of the incident pump wave is taken $10^8$ V/m, what corresponds to an intensity of 1.33 GW/cm$^2$. For the top interface between silicon and air we use the values of the surface nonlinear tensor from [51] and use Miller’s rule to convert the tensor to other pump and second-harmonic frequencies [7]. Thus we take for the Si-Air interface the following values: $\chi ^{(2)}_{\perp \perp \perp } \approx 7 \cdot 10^{-18}$ m$^2$/V, $\chi ^{(2)}_{\perp || \ ||} \approx 3.7 \cdot 10^{-19}$ m$^2$/V, $\chi ^{(2)}_{|| \perp ||} \approx 3.7 \cdot 10^{-19}$ m$^2$/V. Since experimental data on surface nonlinearities for different interfaces is lacking and in general it does not seem to be possible to find experimentally established values of the surface nonlinear tensor for different types of adjacent media in our stack, we assume in a first approximation, that the strength of the surface nonlinearity is proportional to the ratio of the relative permittivities of adjacent materials. With this assumptions we use the above values for the Si-Air interface to find the estimated values of the surface nonlinear susceptibilities at all other interfaces. Although such approach can give just a rough estimate of the surface nonlinearities for some interfaces, it is irrelevant for testing the applicability of our TMM technique and should still provide correct order-of-magnitude values of parameters of the second-harmonic radiation. Besides that, special care should be taken of the direction of the surface nonlinear polarization at each interface. Namely, we assume the surface nonlinear polarization to be directed into the medium with smaller refractive index out of two adjacent media.
Figure 2 shows the amplitudes of the second-harmonic wave in reflection mode for both TE- and TM-polarized incident wave at the fundamental frequency obtained with our analytical approach. As a first step we are interested to compare the SHG from our multilayer structure with the surface SHG from a single interface with the effective nonlinear susceptibility tensor $\chi ^{(2)}_{\textrm {surf}, \Sigma }$ derived above. For this reason we have also plotted in Fig. 2 for comparison the numerically calculated field from a single interface between air and substrate with the effective nonlinear susceptibility $\chi ^{(2)}_{\textrm {surf}, \Sigma }$. Figure 2 yields relatively good agreement to the $\chi ^{(2)}_{\textrm {surf}, \Sigma }$, even though the total thickness of the stack at second-harmonic frequency is already non-negligible, since, for example for $\theta = 0^\circ$ we get
which is not much smaller than 1 anymore. According to the expansions for $\chi ^{(2)}_{\textrm {surf}, \Sigma }$, derived in the previous section, we are interested to estimate the discrepancies $\xi _{\textrm {TE}/\textrm {TM}}, \zeta _{\textrm {TE}/\textrm {TM}}$ from Eqs. (49)–(50) and check whether the conditions Eqs. (49)–(50) can be fulfilled. Figure 3 illustrates the dependence of these parameters on the angle of incidence both for fundamental and second-harmonic frequency. One can see that in general the requirement of both $\xi$ and $\zeta$ being much smaller than 1 can not be reliably achieved for the considered stack from Table 1. The values of $\zeta _{\textrm {TE}/\textrm {TM}}$ could be reduced by using a substrate with larger dielectric permittivity, which in turn increases the values of $\xi _{\textrm {TE}/\textrm {TM}}$. Therefore in order to satisfy the conditions Eqs. (49)–(50) and to justify the Eq. (48) one needs to take a stack of smaller thickness and/or smaller relative permittivities of constitutive materials. We have directly checked, that, when decreasing the thicknesses of the layers in our stack, the blue curves in Fig. 2 corresponding to the SHG from the stack are getting closer to the green curves, which show the surface SHG from a single interface with the effective surface nonlinearity $\chi ^{(2)}_{\textrm {surf}, \Sigma }$. On the other hand, it is worth noting, that fabrication of significantly thinner layers as compared to ones in Table 1 could be quite challenging.Fig. 2. Amplitude of the upwards outgoing second-harmonic wave (SHW) from the considered stack for: (a) TE-polarized; (b) TM-polarized incident wave at the fundamental frequency, obtained using proposed TMM from the stack (blue solid line) and using effective surface nonlinear tensor Eq. (48) for a single interface between the substrate and the air (green dashed line).
Fig. 3. Values of the discrepancies $\xi _{\textrm {TE}/\textrm {TM}}$ and $\zeta _{\textrm {TE}/\textrm {TM}}$ from Eqs. (49)–(50) both for fundamental (FW) and second-harmonic (SH) frequencies.
With the proposed transfer-matrix method we can readily analyse, for instance, the dependence of the second-harmonic field on the thicknesses of the layers. To illustrate it we take the same multilayer stack from Table 1, but additionally assume that the thicknesses of all layers are multiplied by an extra factor $\alpha$, which is constant for all layers. Figure 4 shows the amplitude of the upwards outgoing SHW from the considered stack vs. the angle of incidence $\theta$ and the scaling factor $\alpha$ of the layer thicknesses in the stack. The curves in Fig. 2 then correspond to the horizontal lines in Fig. 4 for $\alpha = 1$. One can see from Fig. 4 that, while the angle of the maximal amplitude for both polarizations changes just slightly with $\alpha$, the maximal amplitude exhibits nonmonotonic dependence on $\alpha$ with pronounced maxima and minima.
Fig. 4. Amplitude of the upwards outgoing SHW from the considered stack [in $kV/m$] for: (a) TE-polarized; (b) TM-polarized incident wave at fundamental frequency vs. the angle of incidence $\theta$ and the scaling factor $\alpha$ of the thicknesses of the layers in the stack.
Next, it is also of interest to examine how the second-harmonic radiation scales with the number of layers. In order to do this, we take a simple periodic stack, where each period consists of layers 1 and 2 from Table 1, i.e. $h_1 = 3$ nm (Si), $h_2 = 5$ nm (SiO$_2$). The whole stack contains $N$ such periods and is located on the same substrate, as in Table 1. The results for this case are plotted in Fig. 5(a). The differences between the dependences for TM- and TE-polarized incident waves at the fundamental frequency on the layer pair number is due to the fact that for TE-polarization only one term of the nonlinear susceptibility, $\chi ^{(2)}_{\perp || \ ||}$, is contributing to SSHG at each interface (see Eq. (27)). On the other hand, for TM-polarizations, all three terms of the nonlinear susceptibility in Eq. (27) are contributing, leading to a more complex interference. One could intuitively expect close to linear dependence for ultrathin layers in the stack. However, it is seen that strong interference of multiple reflected waves both at the fundamental and at the second-harmonic frequency leads to a more complex behaviour already for $N \lesssim 10$. This fact is attributed to the thickness of the whole structure, which becomes comparable to the second-harmonic wavelength inside the media for these values of $N$. To check this we performed similar calculations, when the thicknesses of both layers in each period were reduced to 1 nm. The respective curves are depicted in Fig. 5(b) and turn out to be in satisfactory agreement with the expected linear functions within the considered range of $N$.
Fig. 5. Maximal amplitude of the upwards outgoing SHW (among all values of the angle of incidence) from the considered stack [in $kV/m$] vs. the number of layer pairs: (a) $h_1 = 3$ nm (Si), $h_2 = 5$ nm (SiO$_2$); (b) $h_1 = 1$ nm (Si), $h_2 = 1$ nm (SiO$_2$).
Finally, we aim to compare the amplitude of the second-harmonic radiation from the exemplary stack in Table 1 to the second-harmonic field from a bulk layer of similar thickness made of noncentrosymmetric material with strong bulk nonlinearity in order to find out, whether our surface-driven nonlinearity could yield a conversion efficiency comparable to the bulk nonlinearity. To do this, we took a layer of GaAs located on the same substrate. The second-order bulk nonlinear polarization in GaAs is given as:
where indices $i,j,k,$ correspond to crystal symmetry axes. GaAs is a III-V semiconductor with zinc-blende crystalline structure, which belongs to the $\overline {4}3m$ symmetry group. This symmetry group imposes the following symmetry properties for the bulk nonlinear tensor: with all non-zero entries sharing the same value [1,52]. We fix in calculations this value to $\chi ^{(2)}_{ijk} = 100$ pm/V [53] and suppose, that the top plane of the GaAs layer represents the (100) face. Figure 6 demonstrates the SHG from such a layer calculated both for TM-polarized incident wave polarized in the crystal $xz$-plane and for TE-polarized incident wave with electric field vector making an angle $45^\circ$ with both $x$- and $y$- crystal axis. The thickness of the layer was increased to $70$ nm with respect to the thickness of considered stack (30 nm) in order to increase the amplitude of upwards propagating SHW for TM-polarized incident field. Keeping in mind some uncertainty in the values of the surface nonlinearities used in Fig. 2, we can still conclude from comparison of Fig. 2 and Fig. 6, that the conversion efficiency of the surface-driven SHG from an ultrathin multilayer structure can be expected to significantly exceed the conversion efficiency of the bulk-driven SHG from the layers of media with strong bulk nonlinearity of comparable thickness. The obtained results indicate that the considered structure can be promising for applications in the frequency conversion at the nanoscale. Besides that, we can also expect the nano-patterning of such stacks to lead to further increase of the conversion efficiency [54,55] similar to what has been obtained in all-dielectric metasurfaces [16,56–58].Fig. 6. Amplitude of the upwards outgoing SHW from a GaAs layer of thickness $70$ nm, located on the glass substrate (BK7). The top plane coincides with (100)-plane of the crystal. TM-polarized incident wave is polarized in $xz$-plane; in the case of TE-polarized incident wave the electric field vector forms angles $45^\circ$ both with $x$- and $y$-axis.
4. Conclusion
We have studied second-harmonic generation from a multilayer stacked structure solely driven by the surface nonlinearity arising at the interfaces between adjacent layers in the stack. In such structure composed of ultrathin dielectric layers (of the order of several nm) the surface nonlinear response is expected to be dominating, both for centrosymmetric and for noncentrosymmetric materials. Due to the large number of interfaces the overall surface-driven SHG from the stack can be largely boosted in comparison to a single flat or even nano-patterned layer.
We have developed a theoretical framework for calculation of the second-harmonic field from such a structure based on the transfer matrix method. Our approach inherently takes into account multiple reflections within the stack for both the field at fundamental frequency and the second-harmonic field. With the proposed technique we have derived the effective nonlinear susceptibility tensor describing the nonlinear optical properties of such a multilayer structure as long as the layers used in the stack are ultrathin. The effective surface nonlinearity of such stacked structure can be adjusted through the total number of layers and the material parameters of each layer. It was shown, that for deeply subwavelength thicknesses of the layers the contributions of single interfaces are expected to sum up leading to enhanced surface-driven SHG.
With simulations for an exemplary multilayer structure we have demonstrated that a multilayer stacked structure could serve as an efficient source of second-harmonic radiation compared to bulk layers of nonlinear materials of similar thickness. The proposed structure is therefore expected to be of interest as a new concept for boosting the nonlinear conversion efficiency in nanophotonic applications. In particular, it can be interesting to consider the nano-patterning of the stack into a dielectric metasurface in order to further gain performance through the excitation of Mie-type resonances.
Appendix: first-order correction terms for the effective surface nonlinear tensor $\chi ^{(2)}_{\Sigma , ijk}$
For TE-polarized incident wave at fundamental frequency combining Eqs. (40) and (45) and keeping only first-order small terms we find:
Here we assume for simplicity, that all parameters without $"\textrm {FW}"$ subscript are taken at second-harmonic frequency.
For TM-polarized incident wave at fundamental frequency combining Eqs. (40) and (47) and keeping only first-order small terms we obtain:
Funding
Bundesministerium für Bildung und Forschung (13N14877, 05M20ZBM); Deutsche Forschungsgemeinschaft (3988 16777 - SFB 1375, SE 2749/1-1); H2020 Marie Skłodowska-Curie Actions (No 675745 (MSCA-ITN-EID NOLOSS)).
Disclosures
The authors declare no conflicts of interest.
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