Nonlinear Cherenkov radiation in rotatory nonlinear optics

Zhongmian Zhang; Dazhi Lu; Haohai Yu; Huaijin Zhang; Yicheng Wu

doi:10.3788/COL202523.041901

1. Introduction

Cherenkov radiation is an important phenomenon where the medium will emit coherent electromagnetic radiation when the charged particles travel faster than the phase velocity of light^[1,2]. Up to now, Cherenkov radiation has had an extensive application in modern physics and photonics such as particle detection^[3], Cherenkov luminescence imaging^[4], and integrated light sources^[5]. Furthermore, Cherenkov radiation could also be demonstrated in optics, especially in nonlinear optics, since the nonlinear Cherenkov radiation (NCR) was first observed in the bulk crystals in 1970^[6–8]. In NCR processes, the incident light, dipoles, and the second harmonic correspond to charged particles, induced currents, and radiation fields of Cherenkov radiation, respectively. NCR maintains similar realization conditions to the Cherenkov radiation process, and the phase velocity of incident light should be faster than the second harmonic radiation’s phase velocity. The behavior of NCR can be artificially manipulated by phase matching in $χ^{2}$ in nonlinear photonic crystals^[9–12].

Rotation is an important parameter in modern physics and optics^[13–15], a degree of freedom that manipulates the spatial properties of motion. The polarization direction of light will rotate propagating in realistic materials with helical structures, which is known as circular birefringence and has great application in linear and nonlinear optics^[16–18]. In 2022, the optical phase matching condition of the second-harmonic generation (SHG) in the optically rotatory crystals is theoretically analyzed and associated with the phase of the coupled waves introduced by the rotational polarization direction, which provides a new way to realize phase matching conditions in nonlinear optical processes^[17]. Up to now, the nonlinear optical effects in rotatory nonlinear optics have been mainly concentrated in collinear phase matching conditions, and there is no detailed discussion of any non-collinear nonlinear optical process. With the introduction of the rotation parameter, the coupled waves will produce an additional rotation phase, thus changing the phase matching condition in the collinear nonlinear optical process. The phase velocity threshold for the Cherenkov radiation process will be manipulated by rotational parameters in the optics. In addition, rotating electrical polarization may expand the phase matching types in the nonlinear optical process and realize phase matching conditions with different polarization configurations simultaneously. The rotation parameter in optics may also function in other non-collinear phase matching processes such as nonlinear Bragg diffraction^[19], nonlinear Raman-Nath diffraction^[20,21], and the nonlinear Talbot effect^[22,23].

Here, we discuss the NCR process in optically rotatory and nonlinear optical crystals. The NCR process with rotational polarization direction produces different characteristics from the linearly polarized NCR process. With the rotational polarization direction, the phase velocity of coupled waves will be accelerated or decelerated by the rotational angular velocity, which realizes multiple types of NCR phase matching in quartz simultaneously. In addition, rotational electric polarization in NCR processes can induce a variation in the effective nonlinear coefficient of the quartz crystal and change the second harmonic electric field distribution of the NCR.

2. Results and Discussion

The NCR process in optically rotatory and nonlinear optical crystals is shown in Fig. 1(a). The incident wave propagating along the optical axis of a uniaxial crystal is linearly polarized light and drives the electric polarization vector. In the optically rotatory crystal, the electric field of incident waves will rotate dynamically and continuously change the direction of the induced electric polarization vector. The second harmonic is emitted by the dipoles as a spherical wave and is coherently superimposed along the Cherenkov angle $θ_{c}$ . Similarly, circular birefringence occurs in the SHG propagation direction, and the electric field vector of the SHG will rotate perpendicular to the coherent radiation direction. With the participation rotational angular velocity ${\vec{ρ}}_{i}$ , the rotational electric field introduces an additional rotation phase $Δ φ$ in the coupled waves, which will change the phase velocities of the coupled waves^[17]. Therefore, the wavefront of the second harmonic will be manipulated due to the additional rotation phase $Δ φ$ of the coupled wave. Figure 1(b) shows the schematic diagram of NCR angle variation with the participation of the rotational angular velocity ${\vec{ρ}}_{1}$ along the optical axis.

Figure 1.(a) The NCR process in optically rotatory and nonlinear optical crystals. The red and green cylinders with arrowheads represent the propagation of the fundamental and second harmonic beams, respectively. M_i represents dipoles in the propagation pathway of fundamental light, and the blue spheres represent the second harmonic of the dipole radiation at different locations. The second harmonic coherently superimposes along the Cherenkov angle θ_c. (b) The schematic diagram of the NCR angle with participation of the rotational angular velocity ${\vec{ρ}}_{1}$ . The gray dashed line represents the equi-phase plane during coupled wave propagation. The black solid line represents the second harmonic wavefront of linearly polarized light in the nonlinear Cherenkov radiation process. The blue and green dashed lines represent the second harmonic wavefronts in nonlinear Cherenkov radiation with the participation of the rotational angular velocity ${\vec{ρ}}_{1}$ along the optical axis. With the additional rotation phase $Δ φ_{\pm {\vec{ρ}}_{1}}$ in the coupled waves, the angle of the NCR has changed accordingly.

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In an optical rotation NCR process, the rotational angular velocity ${\vec{ρ}}_{i} (θ) (i = 1, 2)$ will accelerate or decelerate the phase velocities of the fundamental wave or second harmonic simultaneously, thus making the variation of the Cherenkov angle more complicated. The accurate variation of the NCR angle can be calculated from the phase matching condition of NCR. For simplicity, the fundamental frequency wave and the polarization wave are considered as plane waves propagating along the Z-axis in the crystal. After considering circular birefringence during light propagation, the fundamental wave and polarization wave in the crystal can be expressed in exponential form as ${\vec{E}}^{ω} = {\frac{1}{2} \hat{x} A^{ω} e^{- i [ω t - ({\vec{k}}_{1} + {\vec{ρ}}_{1}) \cdot \vec{z}]} + \frac{1}{2} \hat{y} A^{ω} e^{- i [ω t - ({\vec{k}}_{1} + {\vec{ρ}}_{1}) \cdot \vec{z}]}} + {\frac{1}{2} \hat{x} A^{ω} e^{- i [ω t - ({\vec{k}}_{1} - {\vec{ρ}}_{1}) \cdot \vec{z}]} - \frac{1}{2} \hat{y} A^{ω} e^{- i [ω t - ({\vec{k}}_{1} - {\vec{ρ}}_{1}) \cdot \vec{z}]}},$ (1) ${\vec{P}}^{2 ω} = ε_{0} d : {\vec{E}}^{ω} {\vec{E}}^{ω} = {\vec{P}}_{1} e^{- i [2 ω t - 2 ({\vec{k}}_{1} + {\vec{ρ}}_{1}) \cdot \vec{z}]} + {\vec{P}}_{2} e^{- i [2 ω t - 2 ({\vec{k}}_{1} - {\vec{ρ}}_{1}) \cdot \vec{z}]} + {\vec{P}}_{3} e^{- i (2 ω t - 2 {\vec{k}}_{1} \cdot \vec{z})},$ (2)where $A^{ω}$ is the amplitude; $\hat{x}$ and $\hat{y}$ are the unit vectors along the X- and Y-axes, respectively; $ε_{0}$ is the permittivity of free space; $d$ is the matrix of the second-order nonlinear coefficients; ${\vec{k}}_{1}$ and ${\vec{ρ}}_{1}$ are the incident wave vectors and rotational angular velocity of the fundamental polarization wave along the direction of the optical axis; and ${\vec{P}}_{1}$ , ${\vec{P}}_{2}$ , and ${\vec{P}}_{3}$ represent the amplitudes of polarized wave components with different phase velocities. The phase velocities of different polarization wave components ${\vec{v}}^{p}$ could be expressed as $2 ω / 2 ({\vec{k}}_{1} + n {\vec{ρ}}_{1})$ . Here, $n = - 1, 0, 1$ . The second harmonic is the superposition of spherical waves radiated by dipoles in the propagation pathway. Both circular birefringence and linear birefringence occur in the second harmonic propagation, which means that the second harmonic wave is elliptically polarized light and can be decomposed as the superposition of circularly and linearly polarized lights. The wave function of the second harmonic radiated by any dipole can be written as ${\vec{E}}^{2 ω} = \frac{A_{1}^{2 ω}}{\vec{r}} e^{- i {2 ω t - [{\vec{k}}_{2} (θ_{c}) + {\vec{ρ}}_{2} (θ_{c})] \cdot \vec{r}}} + \frac{A_{2}^{2 ω}}{\vec{r}} e^{- i {2 ω t - [{\vec{k}}_{2} (θ_{c}) - {\vec{ρ}}_{2} (θ_{c})] \cdot \vec{r}}} + \frac{A_{3}^{2 ω}}{\vec{r}} e^{- i [2 ω t - {\vec{k}}_{2} (θ_{c}) \cdot \vec{r}]},$ (3)where $A_{i}^{2 ω}$ represents the amplitude of different second harmonic components and $\vec{r}$ is the position vector of the second harmonic away from the dipole; ${\vec{k}}_{2} (θ_{c})$ and ${\vec{ρ}}_{2} (θ_{c})$ are the second harmonic wave vector and rotational angular velocity of the SHG along the NCR angle. The phase velocities of second harmonic components ${\vec{v}}^{2 ω} (θ)$ could be expressed as $2 ω / [{\vec{k}}_{2} (θ) + m {\vec{ρ}}_{2} (θ)]$ . Here, $m = - 1, 0, 1$ . Different from the colinear phase matching process along the optical axis, an additional linearly polarized component exists in the second harmonic waves with a phase velocity of $2 ω / [{\vec{k}}_{2} (θ)]$ ^[17]. The increase in the second harmonic component means that there are more types of non-collinear phase matching than the collinear phase matching process. Considering the phase matching condition for NCR, the radiation angle is decided by an equation related to the SHG propagation direction angle in the crystal, which is expressed as $\cos θ_{c} = \frac{| {\vec{v}}^{2 ω} (θ_{c}) |}{| {\vec{v}}^{p} |} = \frac{2 | {\vec{k}}_{1} + n {\vec{ρ}}_{1} |}{| {\vec{k}}_{2} (θ_{c}) + m {\vec{ρ}}_{2} (θ_{c}) |} .$ (4)From Eq. (4), the optical rotation NCR process can be effectively demonstrated by the wave vector superposition of the second harmonic wave ${\vec{k}}_{2} (θ_{c})$ , polarization wave ${\vec{k}}_{1}$ , and rotational angular velocity ${\vec{ρ}}_{i} (θ_{i})$ . All possible phase matching types of optical rotation NCR are shown in Fig. 2. The introduction of rotational parameters in the Cherenkov radiation process can achieve nine types of phase matching simultaneously, which indicates that optical rotation is an effective means of expanding the NCR types.

Figure 2.Different nonlinear Cherenkov phase matching processes with the rotational polarization direction. The optical rotation NCR processes are demonstrated by the wave vector superposition of the second harmonic wave ${\vec{k}}_{2} (θ_{c})$ , polarization wave ${\vec{k}}_{1}$ , and rotational angular velocity ${\vec{ρ}}_{i} (θ_{i})$ . Here, the different Cherenkov phase matching conditions can be expressed as (a) $| {\vec{k}}_{2} (θ_{c}) - {\vec{ρ}}_{2} (θ_{c}) | \cos θ_{c} = | 2 ({\vec{k}}_{1} - {\vec{ρ}}_{1}) |$ , (b) $| {\vec{k}}_{2} (θ_{c}) - {\vec{ρ}}_{2} (θ_{c}) | \cos θ_{c} = | 2 {\vec{k}}_{1} |$ , (c) $| {\vec{k}}_{2} (θ_{c}) - {\vec{ρ}}_{2} (θ_{c}) | \cos θ_{c} = | 2 ({\vec{k}}_{1} + {\vec{ρ}}_{1}) |$ , (d) $| {\vec{k}}_{2} (θ_{c}) | \cos θ_{c} = | 2 ({\vec{k}}_{1} - {\vec{ρ}}_{1}) |$ , (e) $| {\vec{k}}_{2} (θ_{c}) | \cos θ_{c} = | 2 {\vec{k}}_{1} |$ , (f) $| {\vec{k}}_{2} (θ_{c}) | \cos θ_{c} = | 2 ({\vec{k}}_{1} + {\vec{ρ}}_{1}) |$ , (g) $| {\vec{k}}_{2} (θ_{c}) + {\vec{ρ}}_{2} (θ_{c}) | \cos θ_{c} = | 2 ({\vec{k}}_{1} - {\vec{ρ}}_{1}) |$ , (h) $| {\vec{k}}_{2} (θ_{c}) + {\vec{ρ}}_{2} (θ_{c}) | \cos θ_{c} = | 2 {\vec{k}}_{1} |$ , and (i) $| {\vec{k}}_{2} (θ_{c}) + {\vec{ρ}}_{2} (θ_{c}) | \cos θ_{c} = | 2 ({\vec{k}}_{1} + {\vec{ρ}}_{1}) |$ .

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Quartz crystal is the first nonlinear optical crystal found to have an optical rotation effect and is a feasible nonlinear optical material for the study of optical rotation NCR due to its large rotational angular velocity. The optical rotation NCR process generated by quartz crystal is discussed in detail. The crystal quartz is a uniaxial crystal with a point group of 32, and its cyclotron tensor can be expressed as $\begin{matrix} (\begin{matrix} g_{11} & 0 & 0 \\ 0 & g_{11} & 0 \\ 0 & 0 & g_{33} \end{matrix}) . \end{matrix}$ (5)

Since the refractive index difference of circular birefringence is much smaller than that of natural birefringence, the refractive index $n^{'} / n^{''}$ of the crystal can be approximated as $n_{o} / n_{e}$ . And the rotational angular velocity of quartz crystal in the propagation direction can be expressed as $ρ = \frac{π}{λ \bar{n}}$ ( $g_{11} \sin^{2} θ + g_{33} \cos^{2} θ$ ). Here, $\bar{n} = \sqrt{n_{o} n_{e} (θ)}$ . The coefficients of the cyclotron tensor are based on the experiment in Ref. [24]. The refractive index of quartz is determined by the Sellmeier equation^[25]. The wavelength of fundamental frequency light in optical rotation NCR processes is 1030 nm. Since fundamental frequency light propagates along the optical axis, the phase matching types for NCR are “oo-o” and “oo-e.” Optical refraction exists at the crystal interface, which affects the propagation direction of the SHG. After considering the optical refraction at the crystal interface, the propagation direction $β_{o} / β_{e}$ of SHG is determined by the following equation: $\sin β_{o} = n_{2, o} \sin \cos^{- 1} | \frac{2 ({\vec{k}}_{1}^{o} + n {\vec{ρ}}_{1})}{{\vec{k}}_{2}^{o} (θ_{c}) + m {\vec{ρ}}_{2} (θ_{c})} |,$ (6) $\sin β_{e} = n_{2, e} \sin \cos^{- 1} | \frac{2 ({\vec{k}}_{1}^{o} + n {\vec{ρ}}_{1})}{{\vec{k}}_{2}^{e} (θ_{c}) + m {\vec{ρ}}_{2} (θ_{c})} | .$ (7)

The SHG propagation angles of different NCR processes are calculated and shown in Table 1. Here, the phase velocities of SHG and the polarization waves are recorded as ${\vec{v}}_{o}^{p / 2 ω}$ and ${\vec{v}}_{e}^{p}$ according to the refractive index of linear birefringence $n_{o} / n_{e}$ .

Table 1. Cherenkov Radiation Angles of Polarized Waves and the Second Harmonic Waves with the Optical Rotation at the Fundamental Wavelength of 1030 nm

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Table 1. Cherenkov Radiation Angles of Polarized Waves and the Second Harmonic Waves with the Optical Rotation at the Fundamental Wavelength of 1030 nm

${\vec{v}}^{p}$
${\vec{v}}^{2 ω}$	$2 ω / 2 ({\vec{k}}_{1}^{o} - {\vec{ρ}}_{1})$	$ω / {\vec{k}}_{1}^{o}$	$2 ω / 2 ({\vec{k}}_{1}^{o} + {\vec{ρ}}_{1})$
$2 ω / [{\vec{k}}_{2}^{o} (θ_{c}) - {\vec{ρ}}_{2} (θ_{c})]$	11.648°	11.639°	11.631°
$2 ω / [{\vec{k}}_{2}^{o} (θ_{c})]$	11.665°	11.657°	11.648°
$2 ω / [{\vec{k}}_{2}^{o} (θ_{c}) + {\vec{ρ}}_{2} (θ_{c})]$	11.683°	11.674°	11.666°
$2 ω / [{\vec{k}}_{2}^{e} (θ_{c}) - {\vec{ρ}}_{2} (θ_{c})]$	11.718°	11.710°	11.701°
$2 ω / [{\vec{k}}_{2}^{e} (θ_{c})]$	11.736°	11.727°	11.719°
$2 ω / [{\vec{k}}_{2}^{e} (θ_{c}) + {\vec{ρ}}_{2} (θ_{c})]$	11.753°	11.745°	11.736°

According to the theoretical calculation, the SHG propagation angle of quartz crystal varies relatively little ( $10^{- 2} deg$ ) in the optical rotation NCR process, which is because the optically rotatory power $ρ_{i}$ ( $\sim 10^{1} - 10^{2} deg / mm$ ) is much smaller than phase mismatch ( $\sim 10^{4} deg / mm$ ). Ferroelectric nematic liquid crystals are the theoretically available material for the study of optical rotation NCR processes. Helieletric nematic liquid crystals have realized SHG with rotational polarization direction^[18] and can provide a reciprocal lattice vector with a maximum value of $10^{4} - 10^{5} deg / mm$ for phase matching. Moreover, the reciprocal lattice of the helieletric nematic liquid crystals could be tuned by the concentration of chiral dopant, thus artificially changing the Cherenkov radiation angle.

The rotational polarization direction in nonlinear optical processes not only alters the NCR angle but also influences the electric field distribution of SHG. The intensity $I_{2}$ of the second harmonic is proportional to the effective nonlinear coefficient ${(d_{eff})}^{2}$ in the nonlinear optical frequency conversion process. Circular birefringence occurs in the light propagation along the optical axis, thus affecting the component intensity of the electric field in the X- and Y-axes. The nonlinear optical coefficient of the quartz with a rotational polarization direction in the optical rotation NCR process is derived in detail. In an optical rotation NCR process, the SHG propagates off the optical axis in the form of elliptically polarized light, and its ellipticity is determined by $e = \frac{- g}{\frac{1}{2} (n^{'' 2} - n^{' 2}) \pm \frac{1}{2} {[(n^{2} - n^{'' 2}) + 4 g^{2}]}^{1 / 2}},$ (8)where $g = {(g_{11} \sin^{2} θ + g_{33} \cos^{2} θ)}^{1 / 2}$ . The ellipticity of SHG is less than $10^{- 2}$ , which means that the second harmonic radiation can be approximated as a linearly polarized wave. The second-order nonlinear optical coefficient tensor of the quartz crystal is $[\begin{matrix} d_{11} & - d_{11} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & - d_{11} \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] .$ (9)

Given Kleimann symmetry, the nonlinear optical coefficient of quartz can be expressed as $d_{x x x} = - d_{x y y} = - d_{y x y}$ . The left and right circular polarizations of the light are denoted by optical rotatory powers $- ρ_{i}$ and $ρ_{i}$ ( $i = 1$ , 2), respectively. Therefore, the effective nonlinear coefficient of the two nonlinear frequency conversion processes can be written as $d_{eff}^{o} = (\sin φ \cos 2 α - \cos φ \sin 2 α) d_{x x x},$ (10) $d_{eff}^{e} = - \cos θ (\cos φ \cos 2 α - \sin φ \sin 2 α) d_{x x x} .$ (11)Here, $φ$ is the azimuthal angle measured counterclockwise from the X-axis, $θ$ is the angle between the direction of Cerenkov radiation and the crystal Z-axis, $α = \pm ρ_{1} L$ is the rotational angle of polarization direction ( $α = 0$ for polarization along X-axis), and $L$ represents the propagation distance of the incident light along the optical axis in the quartz. Equations (10) and (11) indicate that the variation in the effective nonlinear coefficient of the crystal is due to the rotation of the fundamental frequency light since the SHG is approximated as linearly polarized light.

The simulated Cherenkov second harmonic rings are based on quartz crystal with a length of about 5.5 mm. An incident wave propagates along the Z-axis in the quartz crystal, with a wavelength of 1030 nm and initial polarization parallel to the X-axis. Figure 3 shows the stimulated second harmonic rings of optical rotation NCR in quartz. Each NCR process is distinguished by the left circular polarization $- ρ_{i}$ and right circular polarization $ρ_{i}$ of the fundamental-frequency light and the second harmonic. In comparison, the NCR rings generated by linearly polarized fundamental frequency light and second harmonic are also simulated. The Cherenkov rings generated by “oo-o” and “oo-e” phase matching types are shown in Figs. 3(a) and 3(b), respectively. In the optical rotation NCR process, the distribution of the NCR ring changes accordingly with the rotational polarization direction of the incident wave. The rotation of the second harmonic will not affect the distribution of the NCR rings since the effective nonlinear coefficient of the crystal is only related to the rotation angle of the incident wave electric field. The final simulated SHG rings are shown in Fig. 3(c). The o- and e-polarized components of the second harmonic are superimposed into a ring because the Cherenkov radiation angles of different phase matched types are nearly identical. Theoretically, the o- and e-components of the second harmonic would separate with the increasing difference of NCR angles^[26]. Rotating the polarization direction in the NCR process will alter the effective nonlinear coefficient of the crystal, ultimately changing the field distribution of the second harmonic. The simulation results indicate that optical rotation symmetry has the potential to function as a new way to manipulate NCR process besides nonlinear photonic crystals. This property may have potential applications in vacuum ultraviolet precision detection and may inspire the development of modern photonics and physics.

Figure 3.Theoretically simulated Cherenkov second harmonic ring distributions under different phase matching types. The simulated optical rings are divided into two phase matching types: “oo-o” and “oo-e.” Each NCR process contains left circular polarization −ρ_i, right circular polarization ρ_i, and linearly polarization (ρ_i = 0) of the fundamental frequency wave and the second harmonic. (a) Theoretically simulated Cherenkov rings generated by the “oo-o” phase matching type. (b) Theoretically simulated Cherenkov rings generated by the “oo-e” phase matching type. (c) The final simulated SHG rings, containing the o- and e-polarized components of the second harmonic.

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The introduction of the rotation parameter in NCR is an effective way of manipulating optical behavior. Optical rotation exists not only in optical crystals but also in optical metasurfaces and liquid crystals^[27–29], which provides a wide range of potential materials for the application of optical rotation NCR. Ferroelectric nematic liquid crystal has large optical rotatory power comparable to the order of magnitude of phase mismatch and has been used to demonstrate the SHG^[19,29] and parametric down-conversion^[28], with an efficiency close to the nonlinear crystals, which offers a potential for highly efficient optical rotation NCR. In addition, the optical rotatory power of ferroelectric nematic liquid crystal can be artificially manipulated by applying a few volts or twisting the molecular orientation along the sample, which makes it possible to achieve angular tuning or wavelength tuning of optical rotation NCR.

3. Conclusion

In summary, the NCR process in optically rotatory and nonlinear optical crystals is theoretically analyzed in this work. Theoretical analyses show that the introduction of the rotation parameter in the NCR process will accelerate or decelerate the phase velocities of the coupled waves, expanding the phase matching types of NCR, which alter the NCR angle in quartz crystals. The effective nonlinear coefficient of quartz changes with the rotation of the incident wave electric field, contributing to the variation of the NCR second harmonic rings. This work explores the influence of rotational parameters in the non-collinear phase matching process, which may inspire broad influence on modern physics and photonics.

Category: Nonlinear Optics

Received: Aug. 12, 2024

Accepted: Oct. 12, 2024

Posted: Oct. 14, 2024

Published Online: Apr. 18, 2025

The Author Email: Dazhi Lu (dazhi.lu@sdu.edu.cn), Haohai Yu (haohaiyu@sdu.edu.cn)

DOI:10.3788/COL202523.041901

CSTR:32184.14.COL202523.041901