Recognition of multipolar second-order nonlinearities in thin-film samples

Kalle Koskinen; Robert Czaplicki; Tommi Kaplas; Martti Kauranen

doi:10.1364/OE.24.004972

1. Introduction

Second-order nonlinear optical processes provide the basis for frequency conversion and electro-optic modulation of light. A prime example here is second-harmonic generation (SHG), i.e., conversion of light at a fundamental frequency $ω$ to light at the doubled frequency $2 ω .$ Within the electric-dipole approximation of the light-matter interaction, second-order processes can occur only in non-centrosymmetric materials. This is a crucial limitation in the search for new second-order materials. On the other hand, the centrosymmetry of any material is broken at its surface, giving rise to an electric-dipole-allowed surface nonlinearity. This has found several applications in surface spectroscopy [1–3 ]. More recently, it has been shown that by elaborate stacking of several surface layers, one can even build up artificial metamaterials with appreciable second-order response [4,5 ].

The symmetry properties of magnetic-dipole and electric-quadrupole interactions are different from those of electric-dipole interactions. In consequence, such higher-multipolar interactions can allow second-order effects even in the centrosymmetric bulk [6–9 ]. In principle, higher-multipolar responses could lead to completely new types of second-order materials, but their design guidelines are poorly understood. Nanostructured materials, where the higher multipoles should be interpreted in terms of effective (Mie-type) response, could provide an avenue forward [10,11 ]. Multipole interactions have also other important uses, e.g., in directional optical antennas [12], sensing [13], and third-order nonlinear optics [14].

In spite of these opportunities, the separation between the dipolar surface and multipolar bulk nonlinearity has been a long standing problem [3,6,7 ]. A breakthrough was achieved by SHG using two non-collinear beams at the fundamental frequency. This technique provides relatively simple, yet distinct signatures for the dipolar surface and multipolar bulk responses [15,16 ] and was subsequently used to quantify the dipolar and multipolar responses of bulk glasses and gold films [10,17–19 ]. In all these works, the sample was such that the reflections between its front and back surfaces could be excluded from the analysis. New materials, however, are often convenient to characterize as thin films, and need to be analyzed using models that account for the multiple reflections between the various interfaces [20]. Although conceptually straight-forward, such models are tedious to implement.

In this paper, we show that thin films provide additional challenges in the recognition of higher-multipolar nonlinearities. More specifically, SHG from a thin film of silicon nitride (SiN), which has strong dipolar second-order nonlinearity, gives rise to signatures of apparent multipolar origin when analyzed using the model justified for bulk samples. We further show that these signatures arise from multiple reflections within the thin film and can be fully explained by considering only the dipolar nonlinearity. These results emphasize the importance of using the most complete models to describe nonlinear processes, which further complicate the recognition of potential materials with strong higher-multipolar responses.

2. Two-beam second-harmonic generation

The dipolar surface and multipolar bulk contributions to SHG from isotropic materials can be separated by two-beam SHG [15,16 ]. In this arrangement, two non-collinear beams at the fundamental frequency and in the same plane of incidence are applied on the sample and the SHG signal generated jointly by the two beams is detected [Fig. 1 ]. For symmetry reasons, the multipolar response can only be accessed by using the two non-collinear beams. In addition, the dipolar and multipolar SHG signals behave very differently when their dependence on the polarizations of the fundamental beams is considered. For sufficiently thick bulk samples, the two fundamental beams can be made to cross at the front interface in such a way that they are separated at the back surface and the back reflections miss the interaction volume [Fig. 1].

Fig. 1 Schematic of two-beam SHG measurement and the used notation.

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We choose z-axis perpendicular to the sample surface and x-axis (y-axis) parallel (perpendicular) to the plane of incidence. Lower (upper) -case letters denote quantities at $ω$ ( $2 ω$ ). The electric fields are $a$ and $b$ for the fundamental beam with larger $(θ_{a})$ and smaller $(θ_{b})$ angle of incidence, respectively. Such geometry is best analyzed in the $(\hat{p}, \hat{s})$ basis, where p (s) polarization is in (perpendicular to) the plane of incidence.

The surface of an isotropic material has $C_{\infty v}$ symmetry, and thus the dipolar surface susceptibility tensor has three independent nonvanishing components $χ_{z z z}$ , $χ_{z x x} = χ_{z y y}$ , and $χ_{x x z} = χ_{x z x} = χ_{y y z} = χ_{y z y}$ [18]. The SHG source polarization for s-polarized SHG signal can then be shown to be of the form [17]

P_{s}^{d} = 2 χ_{x x z} (a_{s} b_{p} \sin θ_{b} + a_{p} b_{s} \sin θ_{a}),

where the fields are evaluated inside the nonlinear medium. The superscript d refers to dipolar origin of the source polarization and the subscripts s and p describe the beam polarizations.

The higher-multipole contributions are usually analyzed in terms of effective polarization. By accounting for the symmetry of the magnetic and quadrupole tensors, one finds that the effective polarization for isotropic media is of the form [8,9,16 ]

P^{MP} = β e (\nabla \cdot e) + γ \nabla (e \cdot e) + δ' (e \cdot \nabla) e,

where

β

,

γ

and

δ^{'}

are nonlinear material parameters,

e

denotes the total fundamental field and MP refers to higher multipoles. The first term in Eq. (2) makes no contribution and the second contributes to the effective surface response [6–9 ]. The distinguishable bulk contribution

(δ^{'})

gives rise to the following s-polarized source [17]

P_{s}^{δ'} = i k δ' \sin (θ_{a} - θ_{b}) (a_{s} b_{p} - a_{p} b_{s}),

where

k

is the wavenumber of the fundamental field. Note that in Eq. (1) the terms

a_{s} b_{p}

and

a_{p} b_{s}

are in-phase when the incident angles have the same sign contrary to Eq. (3), where these terms are out-of-phase. This difference is crucial for the separation of the dipolar and multipolar contributions as the polarizations of the fundamental beams are modulated.

In order to extend this formalism for thin films, we assume that the dipolar nonlinearity is still associated with $C_{\infty v}$ symmetry (with the tensor components given above), but could extend through the whole film, i.e., is not limited to surface only. This is compatible with the known structure of the SiN films, where such dipolar nonlinearity has “bulk” origin [21]. However, we assume that the possible higher-multipolar contributions can still be described by Eq. (3), based on isotropy in three dimensions. Such assumption is sufficient because any deviation of the measurements from the dipolar model would provide evidence of higher-multipole contributions.

The additional complications for thin-film samples arise from the fact that the width of the beams is generally much larger than the film thickness. As a result, parts of the beams reflected at both interfaces of the medium cannot be separated from another. Thus, the total field must be treated as an infinite series of upward and downward propagating partial waves [Fig. 2(a) ]. In addition, a SHG source sheet at any given location emits waves both in the downward ( $-$ ) and upward ( $+$ ) directions, which again both undergo multiple reflections at the top and bottom interfaces [Fig. 2(b)]. The total SHG signal is then obtained by integrating the source sheets through the thickness of the nonlinear film (D).

Fig. 2 Schematic of the two main consequences of reflections in thin samples. a) Multiple reflections contributing to the total field strength. b) SHG generated into two directions in the nonlinear medium.

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In order to account for all the reflections, each fundamental incident field (a or b) gives rise to upward ( $a_{+}$ and $b_{+}$ ) and downward ( $a_{-}$ and $b_{-}$ ) propagating fields inside the film. These fields are $e_{\pm} = e_{0} C_{\pm} e^{i w_{\pm} z}$ , where

C_{+} = \frac{t_{13} r_{32} e^{2 i | w | D}}{1 - r_{31} r_{32} e^{2 i | w | D}}, and C_{-} = \frac{t_{13}}{1 - r_{31} r_{32} e^{2 i | w | D}} .

Here

w

is the z-component of the wave vector (

w = ω n \cos θ / c,

where n is the refractive index at the fundamental wavelength and c is the speed of light), t (r) is transmission (reflection) Fresnel coefficient and indices 1 (air), 3 (film) and 2 (glass substrate) denote respective media. The SHG field exhibits similar behavior, but only gives rise to an overall scaling factor since we limit ourselves to only s-polarized signal. Note that all quantities need to be evaluated separately for beams a and b.

The local nonlinear polarization acts as a source for upward and downward propagating SHG fields. Phase-matching considerations are most conveniently accounted for by Green’s function formalism for nonlinear optics [22]. The total SHG field amplitude in the medium can then be written as

E_{s} \propto \sum_{α, β \in {+, -}} [P_{s}^{d} (a_{α}, b_{β}) + P_{s}^{δ'} (a_{α}, b_{β})] (G_{α, β} + R_{31}^{s} G'_{α, β}),

where

G_{α, β} = \frac{1 - exp (- i (w_{a, α} + w_{b, β} + W) D)}{i (w_{a, α} + w_{b, β} + W)}, and G'_{α, β} = \frac{1 - exp (- i (w_{a, α} + w_{b, β} - W) D)}{i (w_{a, α} + w_{b, β} - W)}

and subscripts

α, β \in {+, -}

denote the direction of propagation of the partial beams related to

a

and

b,

respectively,

W = 2 ω N \cos (Θ) / c

(N and

Θ

are the refractive index and the angle of incidence at the second-harmonic wavelength, respectively). Finally, by combining Eqs. (2), 4–6), the s-polarized SHG field can be written as

E_{s} \propto h_{s} a_{p} b_{s} + k_{s} a_{s} b_{p},

where

h_{s}

and

k_{s}

are linear combinations of

χ_{x x z}

[Eq. (1)] and

k δ'

[Eq. (3)], and can be written as

[\begin{matrix} h_{s} \\ k_{s} \end{matrix}] = [\begin{matrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{matrix}] [\begin{matrix} χ_{x x z} \\ k δ' \end{matrix}], where

\begin{array}{l} m_{11} = 2 \sin (θ_{a}) \sum_{α, β \in {+, -}} C_{α}^{a, p} C_{β}^{b, s} (G_{α, β} + R_{31}^{s} G'_{α, β}) \\ m_{12} = i \sum_{α, β \in {+, -}} C_{α}^{a, p} C_{β}^{b, s} (G_{α, β} + R_{31}^{s} G'_{α, β}) 1_{β} \sin (θ_{a} - 1_{α} 1_{β} θ_{b}), \\ m_{21} = 2 \sin (θ_{b}) \sum_{α, β \in {+, -}} C_{α}^{a, s} C_{β}^{b, p} (G_{α, β} + R_{31}^{s} G'_{α, β}) \\ m_{22} = - i \sum_{α, β \in {+, -}} C_{α}^{a, s} C_{β}^{b, p} (G_{α, β} + R_{31}^{s} G'_{α, β}) 1_{β} \sin (θ_{a} - 1_{α} 1_{β} θ_{b}) \end{array}

and

1_{\pm} = \pm 1.

The relationship between auxiliary coefficients

(h_{s}, k_{s})

and material parameters

(χ_{x x z}, k δ')

depends only on the geometry of the experiment, and the matrix

[m_{i j}]

describing it can be fully determined beforehand.

In the limit where all reflections are neglected, $h_{s}$ and $k_{s}$ are

\begin{array}{l} h_{s} = [2 χ_{x x z} \sin (θ_{a}) - i k δ' \sin (θ_{a} - θ_{b})] t_{13}^{a, p} t_{13}^{b, s}, \\ k_{s} = [2 χ_{x x z} \sin (θ_{b}) + i k δ' \sin (θ_{a} - θ_{b})] t_{13}^{a, s} t_{13}^{b, p} \end{array}

which can also be obtained directly from Eqs. (1) and (3) . Thus,

h_{s}

and

k_{s}

are perfectly in-phase when the multipolar parameter

k δ'

is zero. However, the complete expression of Eq. (8) can give rise to a phase difference between

h_{s}

and

k_{s}

even when

k δ'

vanishes.

The separation of the dipolar and multipolar responses relies on simultaneous control of the relative phase and amplitude between factors $a_{p} b_{s}$ and $a_{s} b_{p} .$ This can be accomplished when one fundamental beam is linearly polarized along direction $\hat{a} = (\hat{p} - \hat{s}) / \sqrt{2}$ and the other is initially linearly polarized $\hat{b} = \hat{p},$ and then varied by a rotating quarter-wave plate [16]. If reflections are neglected, any phase difference between $h_{s}$ and $k_{s}$ would: 1) be seen as a difference between peak heights in the measured graph and 2) indicate a nonvanishing multipole response [Figs. 3(a) and 3(b) ]. However, if reflections are taken into account, similar phase difference can arise strictly from the multiple reflections in the film [Fig. 3(c)]. Thus, neglecting reflections for thin films can lead to incorrect values for nonlinear parameters or even false positives for the multipole response.

Fig. 3 Three simulations of a measurement: a) reflections neglected, no multipole contribution; b) reflections neglected, 10% multipole contribution; c) 5° phase difference between h _s and k_s, no multipole contribution.

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3. Experimental results

To verify our findings experimentally, we studied an 800 nm thick film of SiN previously verified to have $C_{\infty v}$ symmetry and dipolar bulk nonlinearity [21]. The film was fabricated by plasma enhanced chemical vapour deposition on a fused silica substrate. The SHG experiments were performed with a setup described by Rodriguez et al. in [17], which uses a pulsed Nd:YAG laser (wavelength 1064 nm, pulse length 60 ps, pulse energy 0.15 mJ and pulse repetition rate 1000 Hz). The initial powers and polarizations of the control (a) and probe (b) beams were set using linear polarizers and half-wave plates. The polarizations of the beams were initially set to $\hat{a} = (\hat{p} - \hat{s}) / \sqrt{2}$ and $\hat{b} = \hat{p},$ and the angles of incidence were $θ_{a} = 58.3 °$ and $θ_{b} = 32.0 °$ , i.e., same sign but different magnitude. During the measurement, the probe polarization was varied using a motorized quarter-wave plate while measuring the s-polarized SHG in transmission using an analyzer and a photomultiplier tube.

The experimental data was fitted using both models: the simplified one described by Eq. (9) [Figs. 4(a) and 4(b) ] and the detailed one described by Eq. (8) without multipolar contribution [Fig. 4(c)]. The traditional (simplified) model predicts a multipolar response due to difference between the peak heights and yields a multipolar contribution of approximately 7% with respect to $χ_{x x z}$ , with a mean squared error (MSE) of 3.6 in arbitrary units. However, the detailed model based on the dipolar response, multiple reflections, and propagation effects, produces a better fit (MSE 2.9) even though there are less free parameters.

Fig. 4 Experimental data fitted in three different ways: a) traditional model without multipole contribution; b) traditional model with multipole contribution and c) detailed model without multipole contribution.

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4. Conclusions

We have presented a careful theoretical and experimental analysis of two-beam second-harmonic generation in addressing magnetic and quadrupole contributions to the second-order nonlinearity of films with sub-wavelength thickness. Our results show that a simplified analysis of the results may lead to apparent multipolar responses of significant magnitude. However, the results can be fully explained by the dipolar response when multiple reflections and propagation effects within the nonlinear film are properly taken into account. The results underline the importance of using detailed theoretical models in analyzing the nonlinear responses of thin films. Reliable recognition of multipolar responses is likely to require a more extensive set of experiments to be performed and fully analyzed.

Acknowledgments

The authors would like to thank Outi Hyvärinen from the Optoelectronics Research Centre at Tampere University of Technology, for the sample preparation. We also acknowledge the Academy of Finland (265682, 287886 and 287651) for funding. K.K. acknowledges the Väisälä foundation for funding a personal fellowship.

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Recognition of multipolar second-order nonlinearities in thin-film samples

Abstract

1. Introduction

2. Two-beam second-harmonic generation

3. Experimental results

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (10)

Optics Express