Structure of polarization singularities of a light beam at triple frequency generated in isotropic medium by singularly polarized beam

K. S. Grigoriev; P. S. Ryzhikov; E. B. Cherepetskaya; V. A. Makarov

doi:10.1364/OE.25.025416

1. Introduction

The impact of polarization state of the fundamental radiation on the third harmonic generation efficiency in isotropic medium was studied theoretically and experimentally almost half a century ago [1]. It was demonstrated that in the plane wave approximation the signal at the triple frequency vanishes completely if the fundamental radiation is circularly polarized. The exceptional dependency of the nonlinear response of isotropic medium on the polarization state of the input radiation was previously analytically demonstrated in the processes of sum-frequency and second harmonic generation [2, 3]. In particular, it was shown that the usage of non-uniformly polarized light beams provides the increase of generation efficiency. At the present time there are various methods of controlling the polarization profile of laser radiation [4–7] which can be applied in nonlinear optics. Therefore, an interesting problem to study is the third harmonic generation efficiency in the case when the fundamental beam is circularly polarized only in separate points of its cross-section, which are called polarization singularities (or C-points). During the propagation of such beam, three-dimensional curves along which the light polarization remains purely circular (C-lines) are formed. Apart from two possible handedness of rotation of the electric field vector, C-lines and C-points are distinguished by the value of the topological charge, which is equal to the winding number of polarization ellipse rotation calculated along a small closed contour around the singularity. Usually, the topological charge has the least possible half-integer value ±1/2, while the singularities of a higher charge (still being a multiple of 1/2) are unstable and break into several C-points with lower topological charges when the light field is perturbed or propagates. Polarization singularities are analogs of phase singularities (optical vortices) in uniformly polarized light fields: the special points in which the radiation has zero intensity and undetermined phase [8]. Typically, nonlinear optical processes are realized in crystals, in which the interacting waves are linearly polarized to achieve the phase synchronism [9–11]. Thus, the majority of research, devoted to nonlinear transformation of light singularities, deal with scalar optical vortices and the changes of their topological charges. However, the polarization pattern of the fundamental light is not crucial for satisfying phase matching conditions in isotropic medium. This opens a possibility of exploiting the polarization state of the fundamental radiation for shaping the transversal distribution of intensity and polarization of the radiation at triple frequency.

The present paper is concerned with theoretical description of the third harmonic generation in isotropic medium in the case when the fundamental light beam carries a single C-point of an arbitrary type on its axis. The special attention is paid to the generation efficiency in the area close to the beam axis, in which the polarization state of the incident radiation is almost circular, as well as to the growth of C-points number in the signal beam, their topological characteristics and three-dimensional structure of formed C-lines. Such kind of nonlinear optical processes, involving the generation of new C-points and the changes of their topological charge and polarization handedness, are highly topical for modern optics, since they are directly related with the nonlinear interplay of both orbital and spin parts of angular momentum of the electromagnetic field.

2. Main equations

The slowly varying envelope of the electric field of the light beam at fundamental frequency ω propagating along z-axis is considered to be of the following form:

E_{ω} (x, y, z) = \frac{1}{β (z)} (E_{G} e + E_{L} \frac{p (x + i σ y) + q (x - i σ y)}{\sqrt{2} w β (z)} e *) \exp (- \frac{x^{2} + y^{2}}{w^{2} β (z)}),

where E_G and E_L are complex constants, k_ω is the wavenumber at the frequency ω, σ is either 1 or −1 and the function β(z) = 1 + i(z − z₀)/z_R describes the diffraction of the beam with waist size w and Rayleigh range z_R = k_ωw²/2. The unit polarization vector

e = (e_{x} + i σ e_{y}) / \sqrt{2}

in Eq. (1) determines the polarization handedness of the beam. When σ = 1 the beam is a superposition of the left-hand circularly polarized Gaussian mode and two right-hand circularly polarized Laguerre-Gaussian modes of the first order and when σ = −1 the polarization handedness of all three modes is changed to the opposite. The beam of the form Eq. (1) contains a single C-point on its axis, the topological features of which are determined by the parametes p, q. It was shown in [12], that, rotating the coordinate system around z-axis, one can always achieve the following representation of these parameters: p = cos(θ/2) and q = exp (iη) sin (θ/2), where 0 ≤ θ ≤ π and 0 ≤ η ≤ 2π. The topological charge of the C-point in the incident beam Eq. (1) is equal to 0.5 sgn cos θ regardless of the sign of σ.

When propagating in isotropic medium with cubic nonlinearity the fundamental beam generates the vector field of nonlinear polarization of the medium at frequency 3ω:

P^{NL} (x, y, z) = χ^{(3)} (E_{ω} \cdot E_{ω}) E_{ω},

where χ⁽³⁾ is the coefficient determining all the non-null components of a fourth-rank material tensor which is invariant under permutation of its three last indices. Within negligible pump deplition approximation we will use the following equation for the slowly varying envelope of the third-harmonic beam (see [13]):

(\frac{\partial}{\partial z} - \frac{i}{2 k_{3 ω}} Δ_{⊥}) E_{3 ω} = \frac{2 π i k_{3 ω}}{ε_{3 ω}} P^{NL} \exp [i (3 k_{ω} - k_{3 ω}) z] .

Here k₃_ω, ε₃_ω are the wavenumber and dielectric permittivity at the triple frequency. We naturally choose the zero initial conditions for the Eq. (3): E₃_ω(x, y, 0) = 0.

3. Structure of the signal field

The solution of Eq. (3) with right-hand part Eq. (2) can be found using the Green’s function:

\begin{array}{l} E_{3 ω} (x, y, z) = A {J_{1} (x, y, z) \frac{p (x + i σ y) + q (x - i σ y)}{\sqrt{2} w} E_{G} e + \\ + [p q J_{2} (x, y, z) + {(\frac{p (x + i σ y) + q (x - i σ y)}{\sqrt{2} w})}^{2} J_{3} (x, y, z)] E_{L} e *} . \end{array}

In this expression the constant A = 2iπk₃_ωz_Rχ⁽³⁾E_LE_G/ε₃_ω, and the functions J_1,2,3(x, y, z) are in general expressible only in quadratures:

J_{1} (x, y, z) = \int_{- \frac{z_{0}}{z_{R}}}^{ξ} \frac{\exp (i ν ξ^{'})}{{\tilde{β}}^{2} (ξ^{'}) B^{2} (ξ, ξ^{'})} \exp (- \frac{3 (x^{2} + y^{2})}{w^{2} B (ξ, ξ^{'})}) d ξ^{'},

J_{2} (x, y, z) = i \int_{- \frac{z_{0}}{z_{R}}}^{ξ} \frac{(ξ - ξ^{'})}{k_{3 ω} / k_{ω}} \frac{\exp (i ν ξ^{'})}{{\tilde{β}}^{3} (ξ^{'}) B^{2} (ξ, ξ^{'})} \exp (- \frac{3 (x^{2} + y^{2})}{w^{2} B (ξ, ξ^{'})}) d ξ^{'},

J_{3} (x, y, z) = \int_{- \frac{z_{0}}{z_{R}}}^{ξ} \frac{\exp (i ν ξ^{'})}{{\tilde{β}}^{2} (ξ^{'}) B^{3} (ξ, ξ^{'})} \exp (- \frac{3 (x^{2} + y^{2})}{w^{2} B (ξ, ξ^{'})}) d ξ^{'},

Here

ξ = (z - z_{0}) / z_{R}

,

\tilde{β} (ξ^{'}) = 1 + i ξ^{'}

,

B (ξ, ξ^{'}) = \tilde{β} ξ^{'} + 3 i k_{ω} (ξ - ξ^{'}) / k_{3 ω}

, the parameter ν = Δkz_R characterizes the dimensionless phase mismatch, where Δk = 3k_ω − k₃_ω.

The electric field Eq. (4) is a vector superposition of two fields with opposite circular polarization, given by unit vectors e and e*. Polarization singularities exist in these points of the cross-section of the beam at triple frequency where the intensity of one of those two components is zero. Let us note several impotant features of the obtained solution. The functions Eqs. (5)–(7) can not be zero in any point of space z > 0 and have Gaussian asymptotic on transversal coordinates (x, y). In particular, the absolute values of these function reach their maximum when x = y = 0. The dependencies |J_1,2,3| on z-coordinate are not monotone and describe the competition between the nonlinear up-conversion of the fundamental beam and the diffraction blurring of the signal one. The detailed analysis of the behaviour of the functions, similar to Eqs. (5)–(7) is given in the paper [13], in which the third harmonic generation in isotropic medium by uniformly polarized Gaussian beam was studied analytically. Also, similar functions were previously examined in [2, 14], where they governed the second-harmonic and sum-frequency generation efficiency in isotropic medium with nonlocality of nonlinear optical responce. Finally, it is worth noting that when either E_G or E_L is zero the common multiple A in Eq. (4) is also zero. This vanishing of nonlinear signal is consistent with the classical paper [1], in which the third-harmonic generation in isotropic medium by circularly polarized fundamental radiation is shown to be impossible.

The circularly polarized component of the beam at triple frequency given by Eq. (4) with the same handedness as of the C-point in the fundamental beam (polarization vector e) reaches zero only at the beam axis x = y = 0. The analysis of Eq. (4) shows, that a C-point is formed at the axis. Its polarization handedness is determined by the vector e^∗, i.e. it is opposite to the handedness of the singularity at the fundamental beam axis. At the same time one can notice that the phase profile of the first summand in Eq. (4) near the beam axis is in fact the same as the phase profile of the superposition of two Laguerre-Gaussian in the beam at the fundamental frequency. This leads up to an interesting consequence: not only the polarization handednesses of C-points at beams axes are opposite, but also their topological charges are. This effect is illustrated in the fig. 1, where the polarization distributions of the fundamental and third-harmonic beams are shown. The position of the C-points are marked by circles in this figure and the right-hand or left-hand extent of rotation of the electric field vector is depicted by opened and filled ellipses respectively. It is worth mentioning that the intensity at the signal beam axis can be sufficiently big, despite the fact that the fundamental beam is nearly circularly polarized at the region close to its axis.

Fig. 1 Polarization distribution in the waist cross-section (z = 0) of the fundamental beam (a) and the maximum intensity cross-section (z = 1.5z_R) of the signal beam (b). C-points with topological charge 1/2 (−1/2) are marked by filled (open) circles. Right-hand polarized ellipses are opened and left-hand ones are filled. The fundamentsl beam parameters are E_G/E_L = 3, θ = π/4, η = 0, σ = −1, the phase mismatch Δk = 0.

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The other circularly polarized component of the signal beam which has the same polarization handedness as of the Laguerre-Gaussian modes of the fundamental beam (polarization vector e^∗) reaches zero it two diametrically opposite points of the cross-section, thus forming two additional C-points in the cross-section of the signal beam. Due to the complex form of J₂ and J₃ functions the positions of these singularities can only be found by numerical methods. The topological charges and the polarization handedness of these C-points are the same as of the singularity in the incident beam. As the radiation propagates, these C-points move away from the beam axis and the intensity of light is decreasing in them.

In the paper [14] the emergence of the C-lines in the signal beam at the sum-frequency was explained by the fact that the circularly polarized components of the nonlinear polarization field of the medium carry phase singularities, which were called G-lines by the authors. Herewith, the shape of the C-lines in the signal beam can be remarkably different compared to the shape of the G-lines especially when the phase mismatch is high. Let us compare the shapes of C-lines and G-lines in the problem of the third harmonic generation. To find the G-lines structure we will use the explicit expression of P^NL, which can be easily obtained by the substitution of Eq. (1) to Eq. (2):

\begin{array}{l} P^{NL} = \frac{χ^{(3)} E_{G} E_{L}}{β^{3} (z)} {\frac{p (x + i σ y) + q (x - i σ y)}{\sqrt{2} w β (z)} E_{G} e + \\ + {(\frac{p (x + i σ y) + q (x - i σ y)}{\sqrt{2} w β (z)})}^{2} E_{L} e *} \exp (- 3 \frac{x^{2} + y^{2}}{w^{2} β (z)}) . \end{array}

It is clear that both right-hand and left-hand circularly polarized components of the field P^NL attain zero value only on the beam axis at any z. Without loss of generality we will consider below an incident beam carrying left-hand polarization singularity (e = e₋). Thus, the left-hand circularly polarized component of the nonlinear polarization at triple frequency has a phase singularity on its axis, which is identical to the phase singulaity of the right-hand circularly polarized component of the fundamental beam. The C-line, corresponding to this G-line, is this very line of polarization singularity with reversed polarization handedness and topological charge, that was described above. The right-hand circularly polarized component of the nonlinear medium polarization carries a phase singularity with doubled topological charge, compared to that of the fundamental beam. However, instead of generating a single C-line with doubled charge, it produces two C-lines, each of which having the same charge, as the C-line in the fundamental beam. The shape of these two lines significantly changes as the phase mismatch Δk increases (see. fig. 2). One can notice the formation of helical structure of the C-lines, analogous to the effect previosly discussed in [14]. The figure. 2 illustrates two examples of C-lines in the signal beam at different signs of the phase mismatch.

Fig. 2 Examples of the C-lines in the third-harmonic beam at various signs of the phase mismatch (a) Δk = 5/z_R, (b) Δk = −5/z_R. C-lines with topological charge 1/2 (−1/2) are marked by filled (open) circles. Two C-lines with positive topological charge are coloured in different tones to be better distinguished from each other. The parameters of the fundamental beam are E_G/E_L = 3, θ = π/4, η = 0, σ = −1.

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4. Conclusions

We have for the first time obtained the analytic expression for the electric field of the light beam at triple frequency, generated in the bulk of isotropic medium within the negligible pump depletion approximation. It is shown that the signal beam at triple frequency contains three polarization singularities. One of them lies on its axis and its polarization state and topological charge are opposite to those features of the singularity in the fundamental beam. Two other singularities lie symmetrically with respect to the beam axis and has the same features as the C-point in the incident beam. The presence of phase mismatch between wave vectors of the fundamental and triple frequency causes these two lines to attain helical structure. The helicity of a spiral depends on the sign of the phase mismatch and the sign of the topological charge of the singularity.

Funding

President of the Russian Federation for Support of Leading Scientific Schools (Grant NSh-9695.2016.2); Russian Foundation for Basic Research (Grant No 16-02-00154); Ministry of Education and Science of the Russian Federation in the framework of the Increase Competitiveness Program of the NUST MISiS (No. K2-2017-003).

Acknowledgments

We would like to thank Dr. Igor Perezhogin and BS Nikita Kuznetsov for fruitful discussions.

References and links

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Structure of polarization singularities of a light beam at triple frequency generated in isotropic medium by singularly polarized beam

Abstract

1. Introduction

2. Main equations

3. Structure of the signal field

4. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (2)

Equations (8)

Optics Express