Second harmonic generation in isotropic chiral medium with nonlocality of nonlinear optical response by heterogeneously polarized pulsed beams

K. S. Grigoriev; N. Yu. Kuznetsov; E. B. Cherepetskaya; V. A. Makarov

doi:10.1364/OE.25.006253

1. Introduction

Second harmonic generation (SHG) is one of the first nonlinear optical effects, discovered in the “laser era”. Being one of the most fundamental nonlinear effects, it is still a powerfull tool of nonlinear spectroscopy. Despite the great age of its discovery, SHG still holds the interest for theoretical research as well. While SHG is studied primarily for non-centrosymmetric media, the possibility of frequency doubling in the centrosymmetric ones is a very interesting class of problems in nonlinear optics for two reasons. They provide exclusive experimental techniques for nonlinear spectroscopy and they enlarge the understanding of physics behind optical processes. There are a few known ways to generate SH in the centrosymmetric medium. The first is the SHG at the surface, as the symettry of the medium is broken close to its surface [1,2]. The second is to use powerful pulses that generate plasma, the nonlinear response of which is far more complicated than that of the medium [3]. Very interesting and unusual ways to generate SH signal, such as degenerate six-photon processes [4] or using of twisted medium [5], are developed by theoretitians. Finally, one can consider even the isotropic medium with nonlocality of its quadratic optical response [6]. Recent research show that the second harmonic generation, previosly considered forbidden within plane wave approximation, is possible if one takes into account spatial finiteness and vectorial nature of optical beams. In this case, even the homogeneosly polarized fundamental beams generate the signal with strongly heterogeneous polarization distribution [7].

The developing of various techniques of fine structuring of light [8–10] can provide more ways of effective SHG generation because the optical response of the medium on multi-mode heterogeneously polarized light is more complex than on an ordinary evenly polarized Gaussian mode. Among the variety of complex light beams one can point out a very important kind of beams carrying non-zero orbital angular momentum, driven by singularities of their phase profile. These singularities are also called optical vortices and they are distinguished by their topological charge which is equal to total phase variation calculated along a small contour, enclosing the vortex. There is also a finer kind of optical singularities — polarization singularities (or C-points). These singularities appear in heterogeneously polarized fields and they are isolated points in the cross-sections of propagating light in which the polarization is purely circular. Polarization singularities can be both left-hand and right-hand polarized and are also distinguished by their topological charge, which is equal to total winding number of the polarization ellipse, calculated along a small contour, enclosing the C-point [11]. Previous research shows that qualitatively different polarization and intensity profiles of a signal beam can be generated by the fundamental beams, containing optical singularities, by selecting the values of their topological charges [12,13].

In this paper the possibility of SHG in the bulk of isotropic chiral medium is studied both analytically and numerically. The optical response is supposed to be nonlocal in space or in time. Special attention is paid to the transversal structure and polarization distribution of the fundamental light that can provide the effective generation of a signal pulse. In order to correctly describe the fine effects of nonlocality of the nonlinear response the fundamental light beam is considered beyond the plane wave approximation.

2. Structure of fundamental pulsed beam

We consider the propagation of a pulsed beam at the fundamental frequency ω within the parabolic approximation. The complex amplitudes of circularly polarized components of the pulsed beam $ℰ_{\pm}^{(ω)} = (ℰ_{x}^{(ω)} \pm i ℰ_{y}^{(ω)}) / \sqrt{2}$ satisfy the following parabolic equations:

[(\frac{\partial}{\partial z} + \frac{1}{u_{ω}} \frac{\partial}{\partial t}) - \frac{i}{2 k_{ω}} Δ_{⊥} + \frac{i k_{ω}^{^{″}}}{2} \frac{\partial^{2}}{\partial t^{2}}] ℰ_{\pm}^{(ω)} = 0,

where

Δ_{⊥} = \partial_{x}^{2} + \partial_{y}^{2}

is the transversal Laplacian, k_ω is a wave number at the fundamental frequency, u_ω and

k_{ω}^{″}

are group velocity and its dispersion coefficient at the same frequency. Let the pulse at the fundamental frequency have Gaussian time envelope and multi-mode transversal structure:

ℰ_{\pm}^{(ω)} (r, z, t) = f_{\pm} (r, z) G (r, z, t),

where r = (x, y) is the transversal radius-vector. The G function is the Gaussian envelope:

G (r, z, t) = \frac{1}{β_{z} (z) β_{t} (z)} \exp [- \frac{r^{2}}{w^{2} β_{z} (z)} - \frac{{[t - (z - l_{0}) / u_{ω}]}^{2}}{τ_{0}^{2} β_{t}^{2} (z)}]

with characteristic waist size w and minimal duration τ₀. Propagation functions β_t,z are

β_{z} (z) = 1 + i (z - l_{0}) / l_{d},

β_{t} (z) = \sqrt{1 - i (z - l_{0}) / l_{d s},}

where l_d = k_ωw²/2 is the diffraction length and

l_{d s} = τ_{0}^{2} / (2 k_{ω}^{^{″}})

is the dispersion length of the pulsed beam, the waist of which has z-coordinate equal to l₀. Of course, the functions f_± are not arbitrary, because Eq. (2) must be the solutions of the propagation Eq. (1). The electric field vector distribution of the pulsed beam is

E = (ℰ_{+}^{(ω)} e_{+} + ℰ_{-}^{(ω)} e_{-}) \exp (- i ω t + i k_{ω} (z - l_{0})) .

Here $e_{\pm} = (e_{x} \mp i e_{y}) / \sqrt{2}$ are unit polarization vectors of right-hand (”+”) and left-hand (”−”) circularly polarized waves.

The fundamental pulse Eq. (6) is purely transversal vector field. In the optics of nonlocal medium it is crucial to take into account the longitudinal component of the beam electric field, which is usually considered small and omitted. This component inevitably arises due to the spatial and temporal finiteness of the pulsed beam and can be derived from Maxwell equation (∇·E) = 0:

ℰ_{∥}^{(ω)} (r, z, t) \approx \frac{i}{k_{ω}} [1 + g (z, t)] [(e_{+} \cdot L_{+} (r)) + (e_{-} \cdot L_{-} (r))] G (r, z, t) .

In this formula

L_{\pm} (r, z) = \nabla_{⊥} f_{\pm} - \frac{2 r f_{\pm}}{w^{2} β_{z}},

g (z, t) = \frac{2 i [t - (z - l_{0}) / u_{ω}]}{β_{t}^{2} k_{ω} u_{ω} τ_{0}^{2}},

where ∇_⊥ = (∂_x, ∂_y).

3. Spatially nonlocal nonlinear response

The general expression for quadratic polarization vector field of the medium takes into account both spatial and temporal nonlocality:

P_{i}^{(2)} (r, t) = \int_{- \infty}^{t} d t_{1} \int_{- \infty}^{t} d t_{2} \int d r_{1} \int d r_{2} χ_{i j k} (r_{1}, r_{2}, t_{1}, t_{2}) E_{j} (r - r_{1}, t - t_{1}) E_{k} (r - r_{2}, t - t_{2}),

where the kernel χ_ijk(r₁, r₂, t₁, t₂) is a tensor of the quadratic response. Let us first consider the case of spatially nonlocal nonlinear medium. The response function can be written in a following form:

χ_{i j k} (r_{1}, r_{2}, t_{1}, t_{2}) = χ_{i j k} (r_{1}, r_{2}) δ (t_{1}) δ (t_{2}),

where δ(t) is the Dirac delta function. As a rule, the nonlocality range of the medium is much shorter than the optical wavelength, so the dependences E_j(r − r₁) and E_k(r − r₂) can be decomposed into Taylor series. Leaving only the first derivatives, the general expression (10) can be simplified as following [7]:

P_{i}^{(2)} = γ_{i j k l} E_{j} \frac{\partial E_{k}}{\partial r_{l}}

Here all the derivatives and values of the electric field E are taken at the point (r, t) and γ_ijkl is the tensor of the nonlocal quadratic response. It is worth mentioning, that the “classical” term, proportional to the second power of the electric field, is absent in Eq. (12) because of the external and permutation symmetry of the corresponding tensor. The symmetry group of the isotropic chiral medium also causes the tensor γ_ijkl to have the following structure:

γ_{i j i j} = γ_{1}, γ_{i j j i} = γ_{2}, γ_{i i j j} = γ_{3}, γ_{i i i i} = γ_{1} + γ_{2} + γ_{3} .

In the relations above the indices i, j = x, y, z, i ≠ j. The other components of the tensor are zero. In this case, the expression (12) for the quadratic polarization vector of the medium can be simplified in accordance with Eq. (13) in a following way:

P^{(2)} = γ_{1} E (\nabla \cdot E) + \frac{1}{2} γ_{2} \nabla (E \cdot E) + γ_{3} (E \cdot \nabla) E .

Within the negligible pump depletion approximation the fundamental field satisfies the Maxwell equation (∇ · E) = 0, so the first summand in the right-hand side of Eq. (14) is zero. The second summand is a conservative vector field (a gradient). Conservative nonlinear polarization vector field can only generate a SH signal at the border of the medium [14] and that is beyond the scope of the present article. Thus, we consider only the third summand in Eq. (14), which takes the following form after the substitution of Eqs. (2) and (6):

P_{s}^{(2)} = i γ_{3} [e_{z}, f_{+} \nabla_{⊥} f_{-} - f_{-} \nabla_{⊥} f_{+}] G^{2} \exp (- 2 i ω t + 2 i k_{ω} (z - l_{0})) .

It can be seen from the Eq. (15) that the fundamental pulsed beam must satisfy two conditions in order to generate a quadratic polarization field in the bulk of the medium. First, f₊ and f₋ can not be both constants, i.e. the fundamental pulse must consist of transversal modes higher than Gaussian. Second, f₊ can not be proportional to f₋ which means that the polarization state in the cross section of the fundamental beam must not be uniform. In this paper we consider the following transversal structure of the fundamental beam:

f_{\pm} (r, z) = E_{L} \frac{p (x + i y) + q (x - i y)}{\sqrt{2} w β_{z} (z)}, f_{\mp} (r, z) = E_{G}

where p, q, E_G and E_L are complex constants and |p|² + |q|² = 1. The beam consists of left-hand (right-hand) circularly polarized Gaussian mode and two right-hand (left-hand) circularly polarized Laguerre-Gaussian modes of the first order. The polarization handedness of the modes is determined by choosing the upper or lower signs in Eqs. (16). At the axis of the fundamental pulse (x = y = 0) only left-hand (right-hand) circularly polarized Gaussian mode is present, so the polarization singularity lies at the axis. Its topological charge is

\pm \frac{1}{2} sgn {| p |^{2} - | q |^{2}}

, where the sign is chosen to be the same as in the first expression in Eqs. (16). The finer properties of the C-point at the axis of the beams of such kind are thoroughly discussed in [15]. The substitution of Eq. (16) to Eq. (15) results in a simple expression for the quadratic medium polarization field, generated by a pulsed beam with a C-point:

P_{s}^{(2)} = \pm γ_{3} E_{G} E_{L} \frac{p e_{-} - q e_{+}}{w β_{z}} G^{2} \exp (- 2 i ω t + 2 i k_{ω} (z - l_{0})) .

Here the sign of the whole expression is the same as the sign chosen in Eqs. (16). The vector field Eq. (17) has homogeneous polarization state, expressed by the numerator of the fraction before G².

The generation and propagation of second harmonic pulsed beam is described by the following parabolic equation:

[(\frac{\partial}{\partial z} + \frac{1}{u_{2 ω}} \frac{\partial}{\partial t}) - \frac{i}{2 k_{2 ω}} Δ_{⊥} + \frac{i k_{2 ω}^{^{″}}}{2} \frac{\partial^{2}}{\partial t^{2}}] ℰ_{\pm}^{(2 ω)} = \frac{2 π i k_{2 ω}}{ε_{2 ω}} P_{s \pm}^{(2 ω)} \exp (i Δ k (z - l_{0})) .

In this equation the parameters k₂_ω, u₂_ω and $k_{2 ω}^{″}$ are analogous to k_ω, u_ω and $k_{ω}^{″}$ in Eq. (1) and ε₂_ω is the constant of dielectric permittivity at double frequency. The phase mismatch Δk = 2k_ω − k₂_ω and $P_{s \pm}^{(2 ω)}$ are complex amplitudes of circularly polarized components of the $P_{s}^{(2)}$ vector field:

P_{s}^{(2)} = (P_{s +}^{(2 ω)} e_{+} + P_{s -}^{(2 ω)} e_{-}) \exp (- 2 i ω t + 2 i k_{ω} (z - l_{0}))

The propagation Eq. (18) are linear and heterogeneous, so the solution can be found as the convolution of its right-hand side and the Green’s function:

{\vec{ℰ}}^{(2 ω)} (r, t) = \pm γ_{3} E_{G} E_{L} \frac{2 π i k_{2 ω} l_{d}}{w ε_{2 ω}} (p e_{-} - q e_{+}) J_{nloc} (r, t)

The SH pulse has the same polarization pattern as the nonlinear polarization field Eq. (17) and its spatio-temporal intensity envelope is described by the integral J_nloc(r, t), which has the following form:

\begin{array}{l} J_{nlco} (r, z, t) = \int_{- l_{0} / l_{d}}^{ζ} \frac{{\tilde{β}}_{z}^{- 1} (ζ^{'}) d ζ^{'}}{B_{z} (ζ, ζ^{'}) B_{t} (ζ, ζ^{'})} \times \\ \times \exp (i Δ k l_{d} ζ^{'} - \frac{2 r^{2} {\tilde{β}}_{z} (ζ^{'})}{w^{2} B_{z} (ζ, ζ^{'})} - \frac{2 {[t - (z - l_{0}) / u_{2 ω} + ζ^{'} l_{d} / \overset{⌣}{u}]}^{2} {\tilde{β}}_{t}^{2} (ζ^{'})}{τ_{0}^{2} B_{t}^{2} (ζ, ζ^{'})}), \end{array}

B_{z} (ζ, ζ^{'}) = {\tilde{β}}_{z}^{2} (ζ^{'}) + i \frac{ζ - ζ^{'}}{k_{2 ω} / (2 k_{ω})} {\tilde{β}}_{z} (ζ^{'}),

B_{t}^{2} (ζ, ζ^{'}) = {\tilde{β}}_{t}^{4} (ζ^{'}) - i \frac{ζ - ζ^{'}}{k_{ω}^{^{″}} / (2 k_{2 ω}^{^{″}})} \frac{l_{d}}{l_{d s}} {\tilde{β}}_{t}^{2} (ζ^{'}),

{\tilde{β}}_{z} (ζ^{'}) = 1 + i ζ^{'}, {\tilde{β}}_{t} (ζ^{'}) = \sqrt{1 - i ζ^{'} l_{d} / l_{d s}} .

In the integral (21) ${\overset{⌣}{u}}^{- 1} = u_{2 ω}^{- 1} - u_{ω}^{- 1}$ is the group velocity mismatch coefficient and ζ = (z −l₀)/l_d. Similar functions, describing the intensity dynamics of the output monochromatic signal beam were obtained in [7], and it was shown that the maximum output is achieved when the phase mismatch coefficient is positive. In the present paper this effect, known as quasi-synchronism, is confirmed for the dynamics of both peak and integral power of the SH pulsed beam. Figure 1 illustrates the propagational dynamics of the peak and integral powers at positive, negative and zero phase mismatch coefficient; the curves are normalized on the maximum value of the power for the positive phase mismatch.

Fig. 1 The dependences of the (a) peak and (b) integral power of the SH pulsed beam on z-coodrinate. The curves are calculated at Δk = 0.5/l_d (thin solid line), Δk = 0 (thick line) and Δk = −0.5/l_d (dashed line). The dependences are normalized on the maximum values of power for the positive phase mismatch. The other parameters are ŭ⁻¹ = 0, l_ds/l_d = 5.

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Since the polarization state of the SH pulsed beam is homogeneous in both space and time, the main quantity that describes the polarization state is the third Stokes parameter $S_{3}^{(2 ω)}$ , which is equal to 1 for left-hand circularly polarized light and −1 for right-hand circularly polarized one. Using Eq. (20) one readily obtains:

S_{3}^{(2 ω)} = | p |^{2} - | q |^{2} .

The handedness of the polarization ellipse of the SH pulsed beam is the same as the handedness of the C-point in the fundamental pulsed beam if its topological charge is positive and the handedness of the ellipse is opposite in the case of negative topological charge of the input singularity. The sum orbital angular momentum of Laguerre-Gaussian modes of the fundamental beam is proportional to the same difference |p|² − |q|² and for this reason SHG in the bulk of a nonlocal isotropic medium is an example of nonlinear orbital-to-spin conversion. Figure 2 illustrates some typical polarization distributions of the signal and fundamental pulsed beams, integrated over time. The ellipses in these figures have the size proportional to the electric field amplitude in their centers, their geometrical features correspond to those of the polarization ellipses of the integrated radiation. Left-hand ellipses are colored in blue and right-hand ones are white.

Fig. 2 Polarization distribution of fundamental pulse at z = 0 (a,c) and the corrseponding signal pulse (b,d), generated in the bulk of the medium with spatially nonlocal nonlinear response, at z = l_d, integrated over time. Left-hand polarization ellipses are filled and right-hand ones are empty. The C-point in the fundamendal beam is marked by filled (charge 1/2) and opened (charge −1/2) circle. The parameters of the fundamental beam are $p = \frac{\sqrt{3}}{2}$ , q = 0.5 exp(iπ/3) (a,b) and p = 0.5, $q = \frac{\sqrt{3}}{2} exp (2 i π / 3)$ (c,d). The other parameters are Δk = −0.5/l_d, ŭ⁻¹ = 0, l_ds/l_d = 10.

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Sum-frequency generation in the bulk of an isotropic chiral medium (aqueous solutions of D- and L-arabinose) was experimentally detected many years ago [16, 17]. The constant of local quadratic susceptibility, responsible for this process, was estimated in these papers as $χ^{(2)} \approx 6 \cdot 10^{- 11}$ CGSE. The efficiency of spatially nonlocal SHG can be estimated by the analysis of $| P_{s}^{(2)} |$ (Eq. (17)):

| P_{s}^{(2)} | \propto γ_{3} E_{G} E_{L} \frac{1}{w} = \frac{1}{k_{ω} w} (k_{ω} γ_{3}) E_{G} E_{L}

The first multiple in Eq. (26) is proportional to λ_ω/w, where λ_ω is the fundamental beam wavelength. For paraxial beams its values are 10⁻³ − 10⁻². However, in the collinear geometry of SHG the effective beam interaction length can increase up to its diffraction length l_d = k_ωw²/2. Together with the quasi-synchonism matching it allows one to compensate the decrease of efficiency caused by the first multiple in Eq. (26). The second multiple k_ωγ₃ can be roughly estimated [18] as χ⁽²⁾d/λ_ω ≈ 10⁻²χ⁽²⁾, where d is characteristic nonlocality range. The above estimations provide an optimistic value of SHG efficiency, namely, three to two orders of magnitude less than in arabinose solution. Besides, SHG in the spatially nonlocal medium has no threshold and this also allows one to observe it experimentally. The measured value of the SHG signal can also be raised by using short laser pulses with high peak intensities, which do not damage the nonlinear medium.

4. Temporally nonlocal nonlinear response

To consider the SHG in the bulk of the nonlinear medium with only temporal nonlocality of its quadratic response one must first derive the general expression for the quadratic polarization vector field, analogous to Eq. (12). This can be done by substituting Eq. (6) to Eq. (10), in which the response function has the reduced form

χ_{i j k} (r_{1}, r_{2}, t_{1}, t_{2}) = χ_{i j k} (t_{1}, t_{2}) δ (r_{1}) δ (r_{2}) .

The slowly varying envelope

{\vec{ℰ}}^{(ω)}

is decomposed into Taylor series in time t and after the integration one obtains

P_{i}^{(2 ω)} \approx {χ_{i j k} (ω, ω) ℰ_{j}^{(ω)} ℰ_{k}^{(ω)} + {i \frac{\partial χ_{i j k} (ω_{1}, ω_{2})}{\partial ω_{2}} |}_{ω, ω} {ℰ_{j}^{(ω)} \frac{\partial ℰ_{k}^{(ω)}}{\partial t} + i \frac{\partial χ_{i j k} (ω_{1}, ω_{2})}{\partial ω_{1}} |}_{ω, ω} \frac{\partial ℰ_{j}^{(ω)}}{\partial t} ℰ_{k}^{(ω)}} .

Here we limit ourselves to the first order of Taylor approximation. The function χ_ijk(ω₁, ω₂) is a Fourier transform:

χ_{i j k} (ω_{1}, ω_{2}) = \int_{0}^{\infty} d t_{1} \int_{0}^{\infty} d t_{2} χ_{i j k} (t_{1}, t_{2}) \exp (i ω_{1} t_{1} + i ω_{2} t_{2}) .

Owing to the permutation symmetry χ_ijk(ω₁, ω₂) = χ_ikj (ω₂, ω₁) and, subsequently,

{\frac{\partial χ_{i j k} (ω_{1}, ω_{2})}{\partial ω_{1}} |}_{ω, ω} = {\frac{\partial χ_{i k j} (ω_{1}, ω_{2})}{\partial ω_{2}} |}_{ω, ω} .

The symmetry of the medium requires the χ_ijk (ω, ω) tensor to be proportional to Levi-Civita tensor: χ_ijk (ω, ω) = χ (ω, ω)e_ijk [19]. That allows one to simplify the equation (28):

{\vec{P}}^{(2 ω)} \approx i γ_{t} [\frac{\partial {\vec{ℰ}}^{ω}}{\partial t}, {\vec{ℰ}}^{ω}],

where γ_t = 2∂χ/∂ω₁, the derivative being taken at ω₁ = ω₂ = ω.

The vectors $\partial {\vec{ℰ}}^{(ω)} / \partial t$ and ${\vec{ℰ}}^{(ω)}$ in Eq. (31) are collinear as long as the fundamental pulse has identical polarization states at its leading and trailing edges. Let us then consider a composite fundamental pulse $ℰ_{Σ \pm}^{(ω)}$ , which consists of two pulses, separated in time by the delay 2δ and having the form Eq. (2):

ℰ_{Σ \pm}^{(ω)} (r, z, t) = ℰ_{1 \pm}^{(ω)} (r, z, t - δ) + ℰ_{2 \pm}^{(ω)} (r, z, t + δ) .

Each of the two pulses has different functions f_m_± (m = 1, 2) that describe its polarization state. After the substitution of Eq. (32) in Eq. (31) and the separation of the solenoidal part one obtains the following expression for the quadratic polarization vector field:

{\vec{P}}_{s}^{(2 ω)} = \frac{4 δ}{β_{t}^{2} k_{ω} τ_{0}^{2}} γ_{t} [e_{z}, (f_{2 +} \partial_{+} f_{1 +} - f_{1 +} \partial_{+} f_{2 +}) e_{+} + (f_{2 -} \partial_{-} f_{1 -} - f_{1 -} \partial_{-} f_{2 -}) e_{-} + \frac{1}{2} (f_{2 -} \nabla f_{1 +} - f_{1 -} \nabla f_{2 +} + f_{2 +} \nabla f_{1 -} - f_{1 +} \nabla f_{2 -})] G^{2} \exp (- \frac{2 δ^{2}}{τ_{0}^{2} β_{t}^{2}}) .

where the derivatives

\partial_{\pm} \equiv (\partial_{x} \mp i \partial_{y}) / \sqrt{2}

. Like in the case of spatially nonlocal nonlinear medium, the fundamental pulse must consist of transversal modes that are higher than Gaussian. Besides, if f_1± is proportional to f_2± respectively, then the signal at double frequency is zero, since there is no temporal variation of the polarization state of the fundamental pulse.

In this paper we focus on the particular case when the first pulsed beam $ℰ_{1 \pm}^{(ω)}$ has the form Eq. (16) and the second pulsed beam $ℰ_{2 \pm}^{(ω)}$ is the first one, transformed by a quarterwaveplate. Thus, f_2± = f_1∓ and the expression for the medium quadratic polarization vector is reduced to:

{\vec{P}}_{s}^{(2 ω)} = \pm γ_{t} E_{G} E_{L} \frac{4 i δ}{w β_{t}^{2} β_{z} k_{ω} τ_{0}^{2}} (p e_{+} + q e_{-}) G^{2} \exp (- \frac{2 δ^{2}}{τ_{0}^{2} β_{t}^{2}})

Here the sign of the whole expression is the same as the sign chosen in Eq. (16). The solution of the Eq. (18) in this case is found by the same convolution as in the case of spatially nonlocal medium:

{\vec{ℰ}}^{(2 ω)} = \mp γ_{t} E_{G} E_{L} \frac{8 π k_{2 ω} l_{d} δ}{w ε_{2 ω} k_{ω} τ_{0}^{2}} (p e_{+} + q e_{-}) J_{loc} (r, t),

in which the integral J_loc(r, t) slightly differs from J_nloc(r, t) in Eq. (21):

\begin{array}{l} J_{loc} (r, z, t) = \int_{- l_{0} / l_{d}}^{ζ} \frac{{\tilde{β}}_{z}^{- 1} (ζ^{'}) {\tilde{β}}_{t}^{- 2} (ζ^{'}) d ζ^{'}}{B_{z} (ζ, ζ^{'}) B_{t} (ζ, ζ^{'})} \times \\ \exp (i Δ k l_{d} ζ^{'} - \frac{2 r^{2} {\tilde{β}}_{z} (ζ^{'})}{w^{2} B_{z} (ζ, ζ^{'})} - \frac{2 {[t - (z - l_{0}) / u_{2 ω} + ζ^{'} l_{d} / \overset{⌣}{u}]}^{2} {\tilde{β}}_{t}^{2} (ζ^{'})}{τ_{0}^{2} B_{t}^{2} (ζ, ζ^{'})} - \frac{2 δ^{2}}{τ_{0}^{2} {\tilde{β}}_{t}^{2} (ζ^{'})}), \end{array}

where all the variables are defined in the same way as in Eq. (21). The signal pulsed beam is homogeneously polarized as in the case of spatially nonlocal medium. The integral (36) has the dynamics, similar to Eq. (21) and also reaches its maximum values when the phase mismatch is positive. The third Stokes parameter of the SH signal is also proportional to the orbital angular momentum of the fundamental light, but with the opposite sign:

S_{3}^{(2 ω)} = | q |^{2} - | p |^{2}

It is difficult to analyze this answer in terms of the topological charge of the input polarization singularity since it has opposite signs at the leading and trailing edges of the fundamental pulsed beam because of the quarterwaveplate. Figure 3 illustrates some typical examples of leading, central and trailing time layers of the fundamental pulse as well as polarization profile of the signal beam, integrated over time.

Fig. 3 Polarization distribution of fundamental pulse at z = 0 (a,b,c and e,f,g) in three different time layers (−τ₀ (a,e), 0 (b,f) and τ₀ (c,g)) and the corrseponding signal pulse (d,h), generated in the bulk of the medium with temporally nonlocal nonlinear response, at z = l_d, integrated over time. Left-hand polarization ellipses are filled and right-hand ones are empty. The C-point in the fundamendal beam is marked by filled (charge 1/2) and opened (charge −1/2) circle. The parameters of the fundamental beam are $p = \frac{\sqrt{3}}{2}$ , q = 0.5 exp(iπ/3) (a,b,c) and p = 0.5, $q = \frac{\sqrt{3}}{2} exp (2 i π / 3)$ (e,f,g). The other parameters are Δk = −0.5/l_d, ŭ⁻¹ = 0, l_ds/l_d = 10, δ/τ₀ = 0.45.

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Let us estimate the efficiency of this kind of SHG. The absolute value $| {\vec{P}}_{s}^{(2 ω)} |$ (Eq. (34)) can be estimated as following:

| {\vec{P}}_{s}^{(2 ω)} | \propto γ_{t} E_{G} E_{L} \frac{δ}{k_{ω} w τ_{0}^{2}} = \frac{1}{k_{ω} w} (δ / τ_{0}) (γ_{t} / τ_{0}) E_{G} E_{L}

The first multiple was shown above to have the order of 10⁻³ − 10⁻². Numerical calculations of the integral J_loc provide one with the optimal value of the second multiple (δ/τ₀ ≈ 0.5), when the maximum output is attained. To the best of our knowledge, the frequency dispersion of the quadratic nonlinearity of isotropic chiral optically active liquids was not measured directly. Despite this, the magnitude of the third multiple in Eq. (38) can be roughly estimated as χ⁽²⁾T/τ₀, where T is a characteristic time of the quadratic response of the medium. There is a reason to suggest that the values of T are much higher in optically active liquids compared to ordinary isotropic media. The suggestion is based on the analogy with abnormally high relaxation times of the cubic nonlinearity, which were measured during the self-focusing of laser pulses in isotropic phase of liquid crystals [20]. The relaxation time was reported to be strongly dependent on the temperature of the crystal and it could be equal or even exceed the duration of the laser pulse.

5. Conclusions

It was shown, that either of two mechanisms, namely, spatial and temporal nonlocality of quadratic response of an isotropic chiral medium, can provide the second harmonic generation in its bulk. For each of the mechanisms the general expression of the medium quadratic polarization vector field was derived. It was shown that the fundamental pulsed beam must consist of more than one transversal mode and should be heterogeneously polarized to generate a pulsed beam at double frequency. In the case of temporally nonlocal nonlinear response the fundamental pulsed beam must be additionally heterogeneously polarized in time. The pulsed beam with single polarization singularity generates homogeneously polarized SH signal and the third Stokes parameter of the SH radiation is proportional to the orbital angular momentum of the light at fundamental frequency. The functions, describing the spatio-temporal envelope of the signal pulsed beam were found in quadratures for both of the mechanisms: see Eqs. (20) and (35).

Funding

Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST «MISiS» (No K1-2015-025); Russian Foundation for Basic Research (Grant No 16-02-00154); grant of the President of the Russian Federation for state support of leading scientific schools (Grant No NSh-9695.2016.2).

References and links

1. J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987). [CrossRef]

2. Y. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989). [CrossRef]

3. M. Beresna, P. G. Kazansky, Y. Svirko, M. Barkauskas, and R. Danielius, “High average power second harmonic generation in air,” App. Phys. Lett. 95, 121502 (2009). [CrossRef]

4. M. M. Coles, M. D. Williams, and D. L. Andrews, “Second harmonic generation in isotropic media: six-wave mixing of optical vortices,” Opt. Express 21, 12783–12789 (2013). [CrossRef] [PubMed]

5. A. Bahabad and A. Arie, “Generation of optical vortex beams by nonlinear wave mixing,” Opt. Express 15, 17619–17624 (2007). [CrossRef] [PubMed]

6. P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical second-harmonic generation,” Phys. Rev. B 35, 4420–4426 (1987). [CrossRef]

7. S. N. Volkov, N. I. Koroteev, and V. A. Makarov, “Second-harmonic generation in the interior of an isotropic medium with quadratic nonlinearity by a focused inhomogeneously polarized pump beam,” J. Exp. Theor. Phys. 86, 687–695 (1998). [CrossRef]

8. S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express 19, 4085–4090 (2011). [CrossRef] [PubMed]

9. C. F. Phelan, K. E. Ballantine, P. R. Eastham, J. F. Donegan, and J. G. Lunney, “Conical diffraction of a gaussian beam with a two crystal cascade,” Opt. Express 20, 13201–13207 (2012). [CrossRef] [PubMed]

10. B. Yang and E. Brasselet, “Arbitrary vortex arrays realized from optical winding of frustrated chiral liquid crystals,” J. Opt 15, 044021 (2013). [CrossRef]

11. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 singular optics: Optical vortices and polarization singularities,” (Elsevier, 2009), pp. 293–363.

12. K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015). [CrossRef]

13. K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Formation of the lines of circular polarization in a second harmonic beam generated from the surface of an isotropic medium with nonlocal nonlinear response in the case of normal incidence,” J. Opt. 18, 014004 (2016). [CrossRef]

14. Y. R. Shen, The Principles of Nonlinear Optics, Wiley Series in Pure and Applied Optics (J. Wiley, 1984).

15. E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014). [CrossRef]

16. P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792–794 (1966). [CrossRef]

17. A. P. Shkurinov, A. V. Dubrovskii, and N. I. Koroteev, “Second harmonic generation in an optically active liquid: Experimental observation of a fourth-order optical nonlinearity due to molecular chirality,” Phys. Rev. Lett. 70, 1085–1088 (1993). [CrossRef] [PubMed]

18. V. Agranovič and V. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons, no. 18 in Interscience monographs and texts in physics and astronomy, v. 18 (Interscience Publishers, 1966).

19. J. A. Giordmaine, “Nonlinear optical properties of liquids,” Phys. Rev. 138, A1599–A1606 (1965). [CrossRef]

20. E. G. Hanson, Y. R. Shen, and G. K. L. Wong, “Experimental study of self-focusing in a liquid crystalline medium,” Appl. Phys. 14, 65–77 (1977). [CrossRef]

Second harmonic generation in isotropic chiral medium with nonlocality of nonlinear optical response by heterogeneously polarized pulsed beams

Abstract

1. Introduction

2. Structure of fundamental pulsed beam

3. Spatially nonlocal nonlinear response

4. Temporally nonlocal nonlinear response

5. Conclusions

Funding

References and links

Cited By

Figures (3)

Equations (38)

Optics Express