Ex) Article Title, Author, Keywords
Current Optics
and Photonics
Ex) Article Title, Author, Keywords
Current Optics and Photonics 2020; 4(4): 267-272
Published online August 25, 2020 https://doi.org/10.3807/COPP.2020.4.4.267
Copyright © Optical Society of Korea.
Xiaoyuan Huang, Manna Chen, Geng Zhang, Ye Liu, and Hongcheng Wang^{*}
Corresponding author: wanghc@dgut.edu.cn
The propagation of a truncated Airy beam with spatial phase modulation (SPM) is investigated in Kerr nonlinearity with an optical lattice. Before the truncated Airy beam enters the optical lattice, a sinusoidal phase is introduced on the wave-front of the beam. The effect of the spatial phase modulation and optical lattice on propagation behavior is analyzed by direct numerical simulation. It is found that the propagation direction of a truncated Airy beam can be effectively controlled by adjusting the values of phase shift. The effects of optical amplitude, truncation factor, spatial modulation frequency, lattice period and lattice depth on the propagation are discussed in detail. By choosing a high modulation depth, the finite-energy Airy beam can be deflected with a large deflection angle in an optical lattice.
Keywords: Optical propagation, Truncated Airy beam, Phase modulation, Optical lattice
Recently, self-accelerating Airy beams have received a great deal of attention since they were first theoretically and experimentally demonstrated in 2007 [1, 2]. This concept and the finite-energy solution to the paraxial wave equation were transferred from the solution of the free-particle Schrödinger equation within the context of quantum mechanics [3]. Because the perfectly diffractionless Airy beams have infinite energy, they cannot be realized in experiment. To make them realizable in optics, an exponentially decaying factor was introduced to the ideal Airy beam. These truncated Airy beams can retain their unique properties of nondiffraction, self-acceleration, and self-healing over long distances. Now spatially truncated Airy beams have found applications in creating self-bending plasma channels [4], particle micromanipulation [5], and ultrafast self-accelerating pulse generation [6], etc. Self-accelerating Airy beams have also been widely investigated in nonlinear media, such as the nonlinear generation of Airy beams [7], spatial Airy solitons [8, 9], as well as spatiotemporal Airy light bullets [10]. Later, the evolutions of Airy-Gaussian beams with different intensity and phase profiles were also investigated in the nonlinear Kerr medium [11-13]. In the nonlinear regime, some interesting phenomena have been found in the interactions of truncated Airy beams. The interactions of two Airy beams and nonlinear accelerating beams in Kerr and saturable nonlinear media made it possible to form bound and unbound soliton pairs, as well as single solitons [14, 15]. In 2015 and 2016, Shen and his coworkers revealed a controllable manipulation of anomalous interactions between Airy beams in nonlocal nonlinear media, and nematic liquid crystals, [16, 17] respectively. The interaction of two Airy-Gaussian beams can lead to the formation of single breathers and breather pairs [18]. Recently, the mutual interaction of Airy beams were also investigated in photorefractive media [19] and in the fractional nonlinear Schrödinger equation [20]. In nonlocal nonlinear media, the interactions between a truncated Airy beam and a soliton beam can not only produce new optical beams but also generate and control breather solitons [21].
On the other hand, phase distribution has an important effect on the beam propagation and optical manipulation. For example, phase engineering of an axicon’s phase transmission function can result in an exponential growth of an on-axis intensity of a diffraction-free or Bessel beam [22]. Careful design of the initial phase can be used to create self-bending wave packets propagating along arbitrary prescribed convex trajectories [23]. In optical Kerr nonlinearity, an initial sinusoidal phase can be used to steer the propagation direction of a Gaussian beam [24]. The basic idea is to introduce a sinusoidal phase on the wave-front before the beam enters the nonlinear Kerr medium. This method was used to control the optical propagation of Gaussian beams or solitons in a photovoltaic crystal and optical lattices [25, 26]. Recently, the propagation of a finite-energy Airy beam with spatial phase modulation was investigated in an optical Kerr medium. It was found that optical deflection, optical splitting, and periodical oscillation can be realized by choosing proper modulation parameters [27]. In a photonic lattice, the presence of periodically varying refractive index in the transverse spatial dimension can also affect the propagation and localization of optical beams in nonlinear media. Then questions naturally arise: What will propagation dynamics be when a truncated Airy beam with SPM propagates in Kerr nonlinearity with a lattice potential? Can the spatial phase modulation steer the propagation effectively in an optical lattice? In this paper, we investigate the propagation of a finite-energy Airy beam in Kerr nonlinearity with a periodical lattice. The influences of optical field amplitude, modulation amplitude, spatial frequency and phase shift on the optical steering are discussed in detail.
To illustrate the propagation dynamics, we consider a one-dimensional truncated Airy beam propagating in Kerr nonlinearity with a periodical lattice potential. In the paraxial approximation, the optical propagation can be described by the following equation:
where
The optical field of a transverse self-accelerating Airy beam with finite energy can written as
where
where
According to Eq. (4), the input beam is divided into several subbeams which propagate at different angles by the sinusoidal phase modulation. The initial amplitude of the
By using split step Fourier method, Eq. (1) can be simulated for the propagation dynamics of the phase-modulated Airy beam in an optical lattice. Under the combined effects of Kerr nonlinearity, self-deflection effect of the Airy beam, spatial phase modulation, and the localization of the optical lattice, the optical propagation of a truncated Airy beam can be effectively controlled by choosing different values of optical parameters.
In this section, we discuss how to steer the deflection of the finite-energy Airy beam in an optical lattice by changing the values of
In the case
To study the propagation dynamics at different spatial modulation frequency
To understand the optical steering in optical lattice better, we investigate the propagation dynamics of a finite-energy Airy beam with different values of truncation coefficient
Compared to that in the main lobe, the energy remaining in the side lobes becomes relatively low as the truncation coefficient increases. For example, most energy can be steered when
In this section, we discuss the effect of the lattice parameter on the optical steering. First, we discuss the effect of the modulation depth
Next, we discuss the effect of the lattice period on the steering. Figure 5 shows the optical steering by spatial phase modulation in optical lattice with several values of period
In conclusion, we have numerically investigated the propagation dynamics of a truncated Airy beam with sinusoidal phase in nonlinear Kerr media with optical lattice. The effects of optical field amplitude, modulation amplitude, spatial frequency and phase shift on the optical steering are discussed in detail. For arbitrary values of optical field amplitude, the phase modulation can make the truncated finite-energy Airy beam deflect as a soliton. Spatial modulation frequency, which affects the amount of the exciting subbeams, has great influence on the evolution. Though larger deflection angle can be reached at larger value
The effect of lattice depth on the steering has also been discussed in detail. The lattice can be seen as a barrier to restrain the deflection of the optical beam. In a lattice with high modulation depth, the restrain force becomes large. The steering of the Airy beam will become difficult in the case of extra high lattice depth. In addition, the control of optical beam becomes easier as the lattice period increases. The presented spatial phase modulation can provide an effective method to control the propagation of Airy beams, and the propagation properties may have important applications in optical switches, optical logic gates and optical waveguides.
Current Optics and Photonics 2020; 4(4): 267-272
Published online August 25, 2020 https://doi.org/10.3807/COPP.2020.4.4.267
Copyright © Optical Society of Korea.
Xiaoyuan Huang, Manna Chen, Geng Zhang, Ye Liu, and Hongcheng Wang^{*}
Correspondence to:wanghc@dgut.edu.cn
The propagation of a truncated Airy beam with spatial phase modulation (SPM) is investigated in Kerr nonlinearity with an optical lattice. Before the truncated Airy beam enters the optical lattice, a sinusoidal phase is introduced on the wave-front of the beam. The effect of the spatial phase modulation and optical lattice on propagation behavior is analyzed by direct numerical simulation. It is found that the propagation direction of a truncated Airy beam can be effectively controlled by adjusting the values of phase shift. The effects of optical amplitude, truncation factor, spatial modulation frequency, lattice period and lattice depth on the propagation are discussed in detail. By choosing a high modulation depth, the finite-energy Airy beam can be deflected with a large deflection angle in an optical lattice.
Keywords: Optical propagation, Truncated Airy beam, Phase modulation, Optical lattice
Recently, self-accelerating Airy beams have received a great deal of attention since they were first theoretically and experimentally demonstrated in 2007 [1, 2]. This concept and the finite-energy solution to the paraxial wave equation were transferred from the solution of the free-particle Schrödinger equation within the context of quantum mechanics [3]. Because the perfectly diffractionless Airy beams have infinite energy, they cannot be realized in experiment. To make them realizable in optics, an exponentially decaying factor was introduced to the ideal Airy beam. These truncated Airy beams can retain their unique properties of nondiffraction, self-acceleration, and self-healing over long distances. Now spatially truncated Airy beams have found applications in creating self-bending plasma channels [4], particle micromanipulation [5], and ultrafast self-accelerating pulse generation [6], etc. Self-accelerating Airy beams have also been widely investigated in nonlinear media, such as the nonlinear generation of Airy beams [7], spatial Airy solitons [8, 9], as well as spatiotemporal Airy light bullets [10]. Later, the evolutions of Airy-Gaussian beams with different intensity and phase profiles were also investigated in the nonlinear Kerr medium [11-13]. In the nonlinear regime, some interesting phenomena have been found in the interactions of truncated Airy beams. The interactions of two Airy beams and nonlinear accelerating beams in Kerr and saturable nonlinear media made it possible to form bound and unbound soliton pairs, as well as single solitons [14, 15]. In 2015 and 2016, Shen and his coworkers revealed a controllable manipulation of anomalous interactions between Airy beams in nonlocal nonlinear media, and nematic liquid crystals, [16, 17] respectively. The interaction of two Airy-Gaussian beams can lead to the formation of single breathers and breather pairs [18]. Recently, the mutual interaction of Airy beams were also investigated in photorefractive media [19] and in the fractional nonlinear Schrödinger equation [20]. In nonlocal nonlinear media, the interactions between a truncated Airy beam and a soliton beam can not only produce new optical beams but also generate and control breather solitons [21].
On the other hand, phase distribution has an important effect on the beam propagation and optical manipulation. For example, phase engineering of an axicon’s phase transmission function can result in an exponential growth of an on-axis intensity of a diffraction-free or Bessel beam [22]. Careful design of the initial phase can be used to create self-bending wave packets propagating along arbitrary prescribed convex trajectories [23]. In optical Kerr nonlinearity, an initial sinusoidal phase can be used to steer the propagation direction of a Gaussian beam [24]. The basic idea is to introduce a sinusoidal phase on the wave-front before the beam enters the nonlinear Kerr medium. This method was used to control the optical propagation of Gaussian beams or solitons in a photovoltaic crystal and optical lattices [25, 26]. Recently, the propagation of a finite-energy Airy beam with spatial phase modulation was investigated in an optical Kerr medium. It was found that optical deflection, optical splitting, and periodical oscillation can be realized by choosing proper modulation parameters [27]. In a photonic lattice, the presence of periodically varying refractive index in the transverse spatial dimension can also affect the propagation and localization of optical beams in nonlinear media. Then questions naturally arise: What will propagation dynamics be when a truncated Airy beam with SPM propagates in Kerr nonlinearity with a lattice potential? Can the spatial phase modulation steer the propagation effectively in an optical lattice? In this paper, we investigate the propagation of a finite-energy Airy beam in Kerr nonlinearity with a periodical lattice. The influences of optical field amplitude, modulation amplitude, spatial frequency and phase shift on the optical steering are discussed in detail.
To illustrate the propagation dynamics, we consider a one-dimensional truncated Airy beam propagating in Kerr nonlinearity with a periodical lattice potential. In the paraxial approximation, the optical propagation can be described by the following equation:
where
The optical field of a transverse self-accelerating Airy beam with finite energy can written as
where
where
According to Eq. (4), the input beam is divided into several subbeams which propagate at different angles by the sinusoidal phase modulation. The initial amplitude of the
By using split step Fourier method, Eq. (1) can be simulated for the propagation dynamics of the phase-modulated Airy beam in an optical lattice. Under the combined effects of Kerr nonlinearity, self-deflection effect of the Airy beam, spatial phase modulation, and the localization of the optical lattice, the optical propagation of a truncated Airy beam can be effectively controlled by choosing different values of optical parameters.
In this section, we discuss how to steer the deflection of the finite-energy Airy beam in an optical lattice by changing the values of
In the case
To study the propagation dynamics at different spatial modulation frequency
To understand the optical steering in optical lattice better, we investigate the propagation dynamics of a finite-energy Airy beam with different values of truncation coefficient
Compared to that in the main lobe, the energy remaining in the side lobes becomes relatively low as the truncation coefficient increases. For example, most energy can be steered when
In this section, we discuss the effect of the lattice parameter on the optical steering. First, we discuss the effect of the modulation depth
Next, we discuss the effect of the lattice period on the steering. Figure 5 shows the optical steering by spatial phase modulation in optical lattice with several values of period
In conclusion, we have numerically investigated the propagation dynamics of a truncated Airy beam with sinusoidal phase in nonlinear Kerr media with optical lattice. The effects of optical field amplitude, modulation amplitude, spatial frequency and phase shift on the optical steering are discussed in detail. For arbitrary values of optical field amplitude, the phase modulation can make the truncated finite-energy Airy beam deflect as a soliton. Spatial modulation frequency, which affects the amount of the exciting subbeams, has great influence on the evolution. Though larger deflection angle can be reached at larger value
The effect of lattice depth on the steering has also been discussed in detail. The lattice can be seen as a barrier to restrain the deflection of the optical beam. In a lattice with high modulation depth, the restrain force becomes large. The steering of the Airy beam will become difficult in the case of extra high lattice depth. In addition, the control of optical beam becomes easier as the lattice period increases. The presented spatial phase modulation can provide an effective method to control the propagation of Airy beams, and the propagation properties may have important applications in optical switches, optical logic gates and optical waveguides.