Controlling cosine-Gaussian beams in linear media with quadratic external potential

Lifu Zhang; Lifu Zhang; Haozhe Li; Haozhe Li; Zhao Liu; Zhao Liu; Jin Zhang; Wangyang Cai; Yanxia Gao; Yanxia Gao; Dianyuan Fan

doi:10.1364/OE.418392

1. Introduction

The spreading of wave packets originated from dispersive or/and diffraction is a universal phenomenon, which is a limiting factor in many practical applications [1]. Therefore, it becomes important to develop methods to overcome or reduce such phenomenon. Non-diffracting waves, also known as localized waves are able to resist diffraction and have attracted significant and increasing research interest [2–5]. For instance, the different kinds of solitons are of the well-known example of non-diffracting waves in nonlinear regimes [6–8]. On the other hand, in the linear approximation, non-diffracting beams (or non-dispersive pulses) have also theoretically proposed and experimentally demonstrated [9–15]. Non-diffracting wave packets have also been expanded from optics to surface plasmonic, acoustics, and electronics [16].

Durnin et al. first found non-diffracting Bessel beams in 1987 [17,18], which stimulated much attention for finding another kinds of non-diffracting beams. Cosine beams (CBs), a type of non-diffracting beam, are exact solution with the form of a cosine function to the Helmholtz equation, whose intensity shape is independent of the propagation distance [19,20]. However, it is impossible to realize CBs physically because its total energy is infinite at any transverse plane. To overcome such problem, then finite-energy CBs were introduced by imposing a truncated factor on the CBs [21]. Cosine-Gaussian (CG) beam is a typical one of the truncated version [22], which can be considered as special cases of the Hermite-sinusoidal-Gaussian beams as the order of Hermite-polynomial function is zero [23,24]. Compared to the ideal CBs, CG beam still exhibits unique ability of diffraction-free and self-healing in a finite propagation distance and beyond that region they are inevitably dispersive [25]. Because of their self-healing and non-diffracting natures, there are much increasing attention towards the propagation dynamics of the CG beam in different system, including free space [26], turbulent atmosphere [27–30], uniaxial crystals [31], correlated beams [32–37], plasmonic [38–41] and even water waves in hydrodynamics [42,43], and so on.

As an effective method to control beams propagation, the external potentials, such as linear and parabolic potentials, have been introduced. Linear and dynamic linear potentials have been used to manipulate the beam trajectory during propagation [44–47]. As for parabolic potential, propagation dynamics of different kinds of beams, such as symmetric and asymmetric beams, as well as beams carried orbital angular momentum have been investigated [48–60]. There are some novel and interesting features during propagation process, including periodic inversion and focusing [48–54], and self-induced Fourier transformation [55]. Besides, the manipulation of the propagation dynamics of beams or pulses via its spectral phase modulation have also been disclosed [61,62].

The CG beams can be treated as a superposition of two Gaussian beams with different accelerating direction. The impact of linear potential on its propagation dynamic has been reported [44]. In this work, we investigated the propagation dynamics of on-axis and off-axis CG beams in linear media with external quadratic potential. We obtained the analytical expression of CG beams propagation in the presence of external quadratic potential by Fourier analysis, which is quite different from previous researches related the CG beams based on ABCD matrix. The analytical results were confirmed by directly numerical calculation of the Schrödinger equation with external quadratic potential. The propagation dynamics of dual CG beams as well as two-dimensional CG beams propagations are also discussed.

2. Theoretical model

In the paraxial approximation, beam propagation in a linear media with quadratic external potential can be described by the following dimensionless Schrödinger equation [48–54]

(1)$$i\frac{{\partial \phi }}{{\partial z}} + \frac{1}{2}\frac{{{\partial ^2}\phi }}{{\partial {x^2}}} - \frac{\textrm{1}}{\textrm{2}}{\alpha ^\textrm{2}}{x^\textrm{2}}\phi = 0,$$

where $\phi $ is the amplitude of the beam envelope; $x$and z represent the transverse coordinate and the propagation distance, respectively, and normalized by beam initial width ${x_0}$ and diffraction length $kx_0^2$; $\textrm{ }k = {{2{n_0}\pi } / \lambda }$ represents the wavenumber, $\lambda $ is the wavelength, ${n_\textrm{0}}$ is the linear refraction index. Parameter $\alpha $ stands the depth of the potential. When $\alpha = 0$, Eq. (1) describes the propagation of beams in free space.

The analytical expression of general solution to Eq. (1) can be written as [50]

(2)$$\phi ({x,z} )= \int_{ - \infty }^{ + \infty } {\phi ({\xi ,0} )\sqrt {P({x,\xi ,z} )} } d\xi ,$$

where

(3)$$P({x,\xi ,z} )= \frac{{ - i\alpha \csc ({\alpha z} )}}{{2\pi }}exp \{{i\alpha [{{x^2} - 2x\sec ({\alpha z} )\xi + {\xi^2}} ]\cot ({\alpha z} )} \}.$$

After some algebra of the combination of Eqs. (2) and (3), one ends up with

(4)$$\phi ({x,z} )= f({x,z} )\int_{ - \infty }^{ + \infty } {[{\phi ({\xi ,0} )\exp ({ib{\xi^2}} )} ]\exp ({ - iK\xi } )} d\xi ,$$

where $b = {{\alpha \cot ({\alpha z} )} / 2}$, $K = \alpha x\csc ({\alpha z} )$, and $f({x,z} )= \sqrt {{K / {2\pi x}}} \exp [{i({b{x^2} - {\pi / 4}} )} ]$.

Notably seen in Eq. (4), the integral can be seen as the Fourier transform of $\phi ({\xi ,0} )\exp ({ib{\xi^2}} )$ with the spatial frequency K, that is if we chooses an exact input beam $\phi ({\xi ,0} )$, the analytical evolution solution will be obtained by computing the Fourier transform of $\phi ({\xi ,0} )\exp ({ib{\xi^2}} )$. In simple terms, we can describe the propagation of beams in a parabolic potential in terms of the periodic conversion from the initial beam to the Fourier transform with quadratic chirp: which process can be named self-Fourier transform [54].

3. Analytical and numerical results

3.1 Single CG beam

The initial beam was chosen as a general CG beam,

(5)$$\phi ({x,0} )\textrm{ = }\cos [{\omega ({x + {x_0}} )} ]\exp [{ - {\sigma^2}{{({x + {x_0}} )}^2}} ],$$

where $\omega $ is the modulation frequency, $\sigma $ constrains beam width, and ${x_0}$ is a transverse displacement. By use of the Euler's formula, Eq. (5) is equivalent to

(6)$$\phi ({x,0} )= \frac{1}{2}\exp [{ - {\sigma^2}{{({x + {x_0}} )}^2}} ]\{{\exp [{i\omega ({x + {x_0}} )} ]+ \exp [{ - i\omega ({x + {x_0}} )} ]} \}.$$

According to Eq. (4), the evolution of CG beam can be described by

(7)$$\begin{array}{l} \phi ({x,z} )\textrm{ = }\sqrt {\frac{{\alpha \csc ({\alpha z} )}}{{8\sigma \sqrt {{\sigma ^\textrm{2}}\textrm{ + }{{{b^\textrm{2}}} / {{\sigma ^\textrm{2}}}}} }}} \exp \left( {i\frac{{\alpha \cot ({\alpha z} ){x^2}\textrm{ + }{\varphi_0} - \pi }}{2}} \right)\\ \textrm{ } \times \left\{ {\exp \left[ { - \frac{{{{({{x_ - } - 2b\sigma {x_0}} )}^2}}}{{4({{\sigma^\textrm{2}}\textrm{ + }{{{b^\textrm{2}}} / {{\sigma^\textrm{2}}}}} )}}} \right]\exp ({i{\varphi_1}} )+ \exp \left[ { - \frac{{{{({{x_ + } - 2b\sigma {x_0}} )}^2}}}{{4({{\sigma^\textrm{2}}\textrm{ + }{{{b^\textrm{2}}} / {{\sigma^\textrm{2}}}}} )}}} \right]\exp ({i{\varphi_2}} )} \right\}, \end{array}$$

with ${x_ - } = \omega - \alpha \csc ({\alpha z} )x$, ${x_\textrm{ + }} = \omega + \alpha \csc ({\alpha z} )x$, $b = 0.5\alpha \cot ({\alpha z} )$, ${\varphi _0} = \textrm{artan}({{b / {{\sigma^2}}}} )$, $C = {x_0}{\sigma ^4}[{\alpha \csc ({\alpha z} )x\textrm{ + }b{x_0}} ]$, ${\varphi _1} ={-} {{({{{bx_ -^2} / 4} - {b^2}\omega {x_0} - C} )} / {({{\sigma^\textrm{4}}\textrm{ + }{b^\textrm{2}}} )}}$, and ${\varphi _2} ={-} {{({{{bx_\textrm{ + }^2} / 4} + {b^2}\omega {x_0} - C} )} / {({{\sigma^\textrm{4}}\textrm{ + }{b^\textrm{2}}} )}}$.

By using analytical expression of Eq. (7), we plot the propagation dynamics of CG beam as a function of propagation distance. Figure 1 shows the spatial evolution of the on-axis and off-axis CG beams with $\omega = 1$ and $\sigma = 0.1$ in a linear media with different potential depth $\alpha = 0.2$ and $\alpha = 0.4$. It can be clearly seen from Fig. 1 that the CG beam exhibits periodic evolution pattern with initial shape recurring at special propagation distance. The CG beam first splits into two branches and then they are focused into spots individually. With a further increase in propagation distance, such two sub-beams are combined again. The process of separation-focus-combination is reproduced periodically. The period of beam pattern and the distance between two focal spots are independent of the transverse displacement (${x_0}$). While they are very sensitive to the depth of potential ($\alpha $), which decreases with an increasing $\alpha $. This is apparently disclosed by comparing left column of Fig. 1 ($\alpha = 0.2$) with right one ($\alpha = 0.4$). By changing the values of transverse displacement (${x_0}$) from zero to nonzero, the CG beam is switched from on-axis state to off-axis states. As a result, the beam shape changes from symmetric to asymmetric with respect to $x = 0$. So, it is easy to conclude that the intensity period of symmetric beam evolution is double times that of asymmetric one in linear media with quadratic potential.

Fig. 1. Propagation dynamics of CG beam with $\omega = 1$ and $\sigma = 0.1$ in the linear media with a parabolic potential with $\alpha = 0.2$ (left column) and $\alpha = 0.4$ (right column). The displacement of CG beam ${x_0}$ is 20 (top row), 0 (middle row) and -20 (bottom row), respectively. The red and yellow dashed curves are trajectories of two Gaussian beams given by Eq. (7).

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In order to deep the understanding of periodic evolution presented in Fig. 1 and quantify the impact of $\alpha $ and ${x_0}$ on the cosine beam propagation, we further analyze Eq. (7). Within mind, CG beam described by Eq. (5) can be decomposed into two Gaussian beams with opposite linear chirp. Their trajectories can be deduced from Eq. (6)

(8)$${x_t} = \left\{ {\begin{array}{{c}} { - {x_0}\cos ({\alpha z} )\textrm{ + }{{\omega \sin ({\alpha z} )} / \alpha },\textrm{ red line}}\\ { - {x_0}\cos ({\alpha z} )- {{\omega \sin ({\alpha z} )} / \alpha }\textrm{, yellow line}} \end{array}} \right..$$

They are two periodic functions with the period

(9)$${T _{as}} = {{2\pi } / \alpha }.$$

Equation (9) reveals the beam with arbitrary shape exhibits periodic evolution whose period is only determined by the potential depth. Each trajectory consists of two parts. One part is cosine function with amplitude ${x_0}$ the other is sine shape with amplitude ${\omega / \alpha }$. When modulation frequency equals to zero ($\omega = 0$), there is only one trajectory as ${x_t} = {x_0}\cos ({\alpha z} )$. The two trajectories of CG beam evolution are shown in Fig. 1 and marked with red and yellow color. They are exactly the same as the beam evolutions obtained by numerical simulations.

To confirm analytical results shown in Fig. 1, we directly solve the Eq. (1) with input of CG beams by using split-step Fourier method. Figure 2 displays numerical results of spatial and spectral evolution of CG beams versus propagation distance. All parameters used in numerical simulations are same as Fig. 1. By comparing Figs. 2(a) and 2(c) to Figs. 1(c) and 1(a), they exhibit perfect agreement. In order to quantitatively compare numerical and analytical results, the spatial intensities at two special propagation distances obtained from numerical simulation and analytical expression are shown in Figs. 3(a) and 3(b). Here the on-axis CG beam was selected for comparison. When the propagation distance is the integer multiple of the intensity period ${T _s} = {{{T_{as}}} / 2}$, that is $z = m{T _s}$ with $m = 1,2,3 \cdots$, the corresponding beam's shape is same as the initial one $\phi ({x,z = m{T_s}} )= \phi ({x,0} )$; when the propagation distance is the odd integer multiple of the half intensity period, that is ${z_h} = {{({2m + 1} ){T _s}} / 2}$ with $m = 1,2,3 \cdots$, the beam's shape can be expressed as

(10)$$\phi ({x,{z_h}} )\textrm{ = }\sqrt {\frac{\alpha }{{8{\sigma ^2}}}} \exp \left( {i\frac{\pi }{4}} \right)\left\{ {\exp \left[ { - \frac{{{{({x\textrm{ + }{\omega / \alpha }} )}^2}}}{{{{4{\sigma^2}} / {{\alpha^2}}}}}} \right] + \exp \left[ { - \frac{{{{({x - {\omega / \alpha }} )}^2}}}{{{{4{\sigma^2}} / {{\alpha^2}}}}}} \right]} \right\}.$$

Equation (10) indicates the beam is coherent superposition of the two Gaussian beams with a separation ${{2\omega } / \alpha }$ determined by initial modulation frequency ($\omega $) and potential depth ($\alpha $). As seen in Figs. 3(a) and 3(b), the numerical results (black solid curves) are consistent with the analytical solutions (red dashed curves).

Fig. 2. Spatial (left column) and spectral (right column) evolution of cosine Gaussian beams with displacement ${x_0} = 0$ (top row) and ${x_0} = 20$ (bottom row). The other parameters are: $\omega = 1$, $\sigma = 0.1$ and $\alpha = 0.2$.

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Fig. 3. Intensity profiles of the CG beam when the propagation distance (a) $z = {T _s}$and (b) $z = {{{T _s}} / 2}$. Black solid and red dashed curves represent the numerical and analytical results, respectively. (c) Spatial intensity profile at $z = 0$ and spectral intensity profile $z = {{{T _{as}}} / 4}$; (d) Spatial intensity profile at $z = {{{T _{as}}} / 4}$ and spectral intensity profile at $z = 0$. Other parameters are: $\omega = 1$, $\sigma = 0.1$, ${x_0} = 0$ and $\alpha = 0.2$.

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A comparison of the spatial and spectral evolution plotted in Fig. 2 shows that the spectral pattern resembles the spatial one, there is only a phase shift of one quarter period (${\pi / {2\alpha }}$) and different scales. To understand such interesting feature of Fig. 2, by use of Fourier transformer, Eq. (1) can be written as in moment space

(11)$$i\frac{{\partial \hat{\phi }}}{{\partial z}} + \frac{1}{2}{\alpha ^2}\frac{{{\partial ^2}\hat{\phi }}}{{\partial {k^2}}} - \frac{1}{2}{k^2}\hat{\phi } = 0.$$

Equation (11) is similar to Eq. (1) and they become identical each other when k is replaced by $\alpha x$. Therefore, the scaling factor of intensity and transverse coordinates between in real space and in inverse space can be written as

(12)$${I_x} = \alpha {I_k},\textrm{ }x = {\alpha ^{ - 1}}k.$$

To verify these scaling relations, Figs. 3(c) and 3(d) present the input spatial and spectral shapes and output one at one quarter period for the case of on-axis CG beam. It can be clearly seen that the spatial (spectral) profile of incident CG beam is converted to the output spectral (spatial) profile. The scaling relationship shown in Figs. 3(c) and 3(d) proved the correction of Eq. (12). Especially, as $\alpha = 1$, Eq. (11) has the same form to the Eq. (1), therefore, they have the same solutions which expressed in real space and inverse space, respectively. Such beam solutions are also called self-Fourier beams. The physical reason behind such periodical evolution can be understood by noting by noting that phase shift generated by quadratic external potential. By neglecting diffraction effect, the solution of Eq. (1) is $\phi ({x,z} )= \phi ({x,0} )\exp ({ - 0.5i{\alpha^\textrm{2}}{x^\textrm{2}}z} )$. The effect of quadratic external potential is fully the same as those of graded-index medium [63] or strongly nonlocal nonlinear medium [64], which can performs a Fourier transform (FT) and fractional FT on the input beam. On the other hand, the FT of Gaussian function is still a Gaussian distribution. As $\alpha z = {\pi / 2}$, Eq. (7) denotes the beam at propagated distance $z = {{{T _{as}}} / 4}$ is deduced as the well-known FT of the incident beam.

Next, we carry out a series of numerical simulations to confirm the period ${T _s} = {{2\pi } / \alpha }$ and distance between two focus spots $\Delta {\kern 1pt} x = {{2\omega } / \alpha }$ obtained from analytical expression. Both of them depend on the potential depth. The period scales inversely with the potential depth. The separation between two focal spots varies linearly with modulation frequency and inversely with the potential depth. Figure 4 shows the dependence of ${T _s}$ and $\varDelta x$ with the potential depth or the modulation frequency. It can be clearly seen from Fig. 4 that the agreement between the analytical results and numerical simulations holds very well. Once again, the analytical predictions have been confirmed by numerical experiments.

Fig. 4. (a) The oscillating period versus the potential depth $\alpha $. The distance between the two peaks of Gaussian beam versus (b) the modulation frequency $\omega $ and (c) the harmonic potential depth $\alpha $, respectively. The black curve and red points correspond to the analytical solution and numerical results, respectively.

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3.2 Dual CG beams

Next, we investigate the two CG beams interaction governed by Eq. (1). We first consider the case of two beams with in-phase. So, the incident beam can be written as

(13)$$\begin{array}{c} \phi ({x,0} )= \cos [{\omega ({x + {x_0}} )} ]\exp [{ - {\sigma^2}{{({x + {x_0}} )}^{^2}}} ]\\ + \cos [{\omega ({x - {x_0}} )} ]\exp [{ - {\sigma^2}{{({x - {x_0}} )}^{^2}}} ]. \end{array}$$

By using the same analysis method described in section 3.1, the analytical expression of dual in-phase CG beams is obtained

(14)$$\begin{array}{l} \phi ({x,z} )\textrm{ = }\sqrt {\frac{{\alpha \csc ({\alpha z} )}}{{8\sigma \sqrt \Delta }}} \exp \left( {i\frac{{\alpha \cot ({\alpha z} ){x^2}\textrm{ + }{\varphi_0} - \pi }}{2}} \right)\\ \textrm{ } \times \left\{ \begin{array}{l} \exp \left[ { - \frac{{{{({{x_ - } - 2b\sigma {x_0}} )}^2}}}{{4({{\sigma^\textrm{2}}\textrm{ + }{{{b^\textrm{2}}} / {{\sigma^\textrm{2}}}}} )}}} \right]\exp ({i{\varphi_1}} )+ \exp \left[ { - \frac{{{{({{x_ + } - 2b\sigma {x_0}} )}^2}}}{{4({{\sigma^\textrm{2}}\textrm{ + }{{{b^\textrm{2}}} / {{\sigma^\textrm{2}}}}} )}}} \right]\exp ({i{\varphi_2}} )\\ + \exp \left[ { - \frac{{{{({{x_ + } - 2b\sigma {x_0}} )}^2}}}{{4({{\sigma^\textrm{2}}\textrm{ + }{{{b^\textrm{2}}} / {{\sigma^\textrm{2}}}}} )}}} \right]\exp ({i{\varphi_3}} )+ \exp \left[ { - \frac{{{{({{x_ - } + 2b\sigma {x_0}} )}^2}}}{{4({{\sigma^\textrm{2}}\textrm{ + }{{{b^\textrm{2}}} / {{\sigma^\textrm{2}}}}} )}}} \right]\exp ({i{\varphi_4}} )\end{array} \right\}, \end{array}$$

with ${x_ - } = \omega - \alpha \csc ({\alpha z} )x$, ${x_\textrm{ + }} = \omega + \alpha \csc ({\alpha z} )x$, $b = {{\alpha \cot ({\alpha z} )} / 2}$, ${\varphi _0} = \textrm{artan}({{b / {{\sigma^2}}}} )$, ${\varphi _1} ={-} \frac{{{{bx_ - ^2} / 4} - {b^2}\omega {x_0} - C}}{{{\sigma ^\textrm{4}}\textrm{ + }{b^\textrm{2}}}}$, ${\varphi _2} ={-} \frac{{{{bx_\textrm{ + }^2} / 4} + {b^2}\omega {x_0} - C}}{{{\sigma ^\textrm{4}}\textrm{ + }{b^\textrm{2}}}}$, $C = {x_0}{\sigma ^4}[{\alpha \csc ({\alpha z} )x\textrm{ + }b{x_0}} ]$, ${\varphi _3} ={-} \frac{{{{bx_\textrm{ + }^2} / 4} - {b^2}\omega {x_0}\textrm{ + }D}}{{{\sigma ^\textrm{4}}\textrm{ + }{b^\textrm{2}}}}$, ${\varphi _4} ={-} \frac{{{{bx_ - ^2} / 4}\textrm{ + }{b^2}\omega {x_0}\textrm{ + }D}}{{{\sigma ^\textrm{4}}\textrm{ + }{b^\textrm{2}}}}$ and $D = {x_0}{\sigma ^4}[{\alpha \csc ({\alpha z} )x - b{x_0}} ]$.

As seen in Eq. (14), the propagation of dual CG beams is made of four Gaussian beams evolution, which accelerate along the four branches of trajectories. Correspondingly, their trajectories can be expressed as

(15)$${x_{tra}} = \left\{ {\begin{array}{{c}} {{x_0}\cos ({\alpha z} )+ {{\omega \sin ({\alpha z} )} / \alpha },\textrm{ dashed red}}\\ {\textrm{ }{x_0}\cos ({\alpha z} )- {{\omega \sin ({\alpha z} )} / \alpha },\textrm{ dashed yellow}}\\ { - {x_0}\cos ({\alpha z} )+ {{\omega \sin ({\alpha z} )} / \alpha },\textrm{ soild yellow}}\\ { - {x_0}\cos ({\alpha z} )- {{\omega \sin ({\alpha z} )} / \alpha },\textrm{ soild red}} \end{array}} \right..$$

According to Eqs. (14) and (15), the evolution of dual CG beams and the corresponding trajectories are shown in Fig. 5. Obviously, one can see that the dual CG beam is symmetric and the period ${T _s} = \pi {\alpha ^{ - 1}}$ still depends on the parameter $\alpha $. Its propagation dynamics is made up of the coherently superposition of two off-axis CG beams. Because of its symmetric profiles, their evolution is similar as that single on-axis CG. But the focusing region becomes more complex. It can be clearly seen from the trajectories shown in Fig. 5 that four sub-beams propagate at different speeds and approach each other, resulting in four focal regimes with multiple peaks. The envelope of these peaks exhibits Gaussian profile.

Fig. 5. Propagation of a dual CG beam in a parabolic potential, the other parameters are: $\omega = 1$, $\sigma = 0.1$, ${x_0} = 20$ and $\alpha = 0.2$. The solid and dashed curves in color of red and yellow represent the four branches of trajectories given by Eq. (17), respectively.

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Let focus our attention on the focal regime. Compared to the Gaussian shape of focal spots for the case of single input CG beam, dual CG beams will form two multi-peaked structures at the propagation distance ${z_F} = {{({2m + 1} ){T_s}} / 2}$. While the locations of other two deviate from ${z_F}$. One location is $m{T_s} + {z_1}$, the other one is $m{T_s} + {z_2}$. From Eq. (16), ${z_{1,2}}$ can be obtained

(16)$${z_1} = {{\textrm{arctan}({{{\alpha {x_0}} / \omega }} )} / \alpha },\textrm{ }{z_2} = {{[{\pi - \textrm{arctan}({{{\alpha {x_0}} / \omega }} )} ]} / \alpha }.$$

The distance between ${z_{1,2}}$ and ${z_F}$ is

(17)$$\varDelta d = {{[{0.5\pi - \textrm{arctan}({{{\alpha {x_0}} / \omega }} )} ]} / \alpha }.$$

At the propagation distance $z = m{T_s}$ with m being a nonnegative integer, beam shapes is a recurrence of the initial beam, that is $\phi ({x,z = m{T_s}} )= \phi ({x,0} )$. When the propagation distance equals an odd integer multiple of the half period, that is $z = {{({2m + 1} ){T_s}} / 2}$ the beam is obtained

(18)$$\begin{array}{c} \phi (x )\textrm{ = }\sqrt {\alpha {{} / {({2{\sigma^2}} )}}} \cos ({\alpha {x_0}x} )exp ({i{\pi / 4}} )\\ \times \{{exp [{ - {{{\alpha^2}{{({x - {\omega / \alpha }} )}^2}} / {({4{\sigma^2}} )}}} ]+ exp [{ - {{{\alpha^2}{{({x + {\omega / \alpha }} )}^2}} / {({4{\sigma^2}} )}}} ]} \}. \end{array}$$

At propagation distance at ${z_1}$, it can be written as

(19)$$\begin{array}{l} \phi (x )\textrm{ = }\frac{{g(x )}}{\textrm{2}}exp [{{{ip({{\omega^2} + {J^2} - 4{\sigma^4}x_0^2} )} / {({{\sigma^2}B} )}}} ]\\ \textrm{ } \times \left\{ \begin{array}{l} exp [{ - {{{{({\omega + J} )}^2}} / {\rm B}}} ]exp [{i({\Gamma + {\gamma_2}} )} ]+ exp [{ - {{{{({\omega - J} )}^2}} / {\rm B}}} ]exp [{i({\Gamma - {\gamma_2}} )} ]\\ + 2\cos ({{\gamma_1}} )exp ({ - {{{J^2}} / {\rm B}}} )exp ({ - i\Gamma } )\end{array} \right\}, \end{array}$$

where $J = x\sqrt {{p^2} + 0.25{\alpha ^2}}$, $p = {\omega / {2{x_0}}}$, ${\varphi _ + } = \arctan ({{p / {{\sigma^2}}}} )$, $\Gamma = {{{p^2}\omega {\kern 1pt} {x_0}} / {({{\sigma^2}B} )}}$, $g(x )= {\left( {\frac{{{p^2} + {{{\alpha^2}} / 4}}}{{{\sigma^2}B}}} \right)^{0.25}}exp \left[ {i\left( {p{x^2} + {\pi / 4}\frac{{}}{{}} + {{{\varphi_ + }} / 2}} \right)} \right]$, ${\gamma _1} = {{({p\omega + \textrm{2}{\sigma^4}{x_0}} )J} / {({\textrm{2}{\sigma^2}B} )}}$, ${\gamma _2} = {{({p\omega - \textrm{2}{\sigma^4}{x_0}} )J} / {({\textrm{2}{\sigma^2}B} )}}.$ and ${\rm B} = {\sigma ^2} + {({{p / \sigma }} )^2}$.

With further increase propagate distance to ${z_2}$, the bean intensity profile is same as that at ${z_1}$, the expression changes only in phase, which can be written as

(20)$$\begin{array}{l} \phi (x )\textrm{ = }\frac{{h(x )}}{\textrm{2}}exp \left[ { - ip\frac{{{\omega^2} + {J^2} - 4{\sigma^4}x_0^2}}{{4{\sigma^2}B}}} \right]\\ \textrm{ } \times \left\{ \begin{array}{l} exp\left[ { - \frac{{{{({2\omega + J} )}^2}}}{{4{\rm B}}}} \right]exp [{ - i({\Gamma + {\gamma_2}} )} ]+ exp \left[ { - \frac{{{{({2\omega - J} )}^2}}}{{4{\rm B}}}} \right]exp [{ - i({\Gamma - {\gamma_2}} )} ]\\ + 2exp \left( { - \frac{{{J^2}}}{{4{\rm B}}}} \right)\cos ({{\gamma_1}} )exp ({i\Gamma } )\end{array} \right\}, \end{array}$$

with $h(x )= {[{{{({{p^2} + {{{\alpha^2}} / 4}} )} / {{\sigma^2}B}}} ]^{0.25}}exp [{ - i({p{x^2} - {\pi / 4} - {{\arctan ({ - {p / {{\sigma^2}}}} )} / 2}} )} ]$.

Figures 6(b)–6(c) display the analytical and numerical results of beam's intensity distribution at three propagation distances ${z_1}$, ${z_2}$ and ${{{T_s}} / 2}$. For comparison, the case of incident beam profile was shown in Fig. 6(a). The numerical results show an excellent agreement with analytical predictions. As the beams propagate along the media, sub-beams are created after splitting process, and then come close to each other. At the propagation distance ${z_1}$ shown in Fig. 6(b), the shape is constructed by an oscillatory structure covering the center range and other two smooth peaks located on its wings. The peaks on both sides are Gaussian profiles, but the innermost two sub-beams merge together to form a multiple peaks structure, resulting from its coherent superposition. Because their relative phase difference is equal to that at input, it still a CG beam with the same modulation frequency and different width compared with that of input beam. This feature can be directly observed by comparing the numbers of multiple peaks. As the propagation distance is increased to halve period $z = {{{T_s}} / 2}$, the inner and outer beams with farthest separation fully overlap each other, leading to form two new CG beams with small modulation frequency compared to that of incident Gaussian beam. Because the accumulated phase shifts of inner beams are different from that of outer beams. As a result, the modulation frequency changes. But the new CG beams are symmetric with respect to $x = 0$. At the propagation distance ${z_2}$, the beam profile is mirror symmetric to that at ${z_1}$. Although, the both shapes are the same, the left and right parts are exchanged.

Fig. 6. Spatial intensity profiles of the dual CG beam in the parabolic potential at (a) $z = 0$, (b) $z = {{{T_s}} / 2} + \varDelta d$ and (c) $z = {{{T_s}} / 2}$.The other parameters are: $\omega = 1$, $\sigma = 0.1$, ${x_0} = 20$ and $\alpha = 0.2$. Black solid and red dashed curves represent the numerical results and analytical solution, respectively.

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3.3 Two-dimensional CG beam

In the former sections, we have showed the harmonic oscillation and self-Fourier transform of the one-dimensional CG beam propagating in linear media with parabolic potential. The question is, will the same phenomena still appear when the beam extends to two dimensions? Next, we analyze the two-dimensional CG beam propagation, which was governed by

(21)$$i\frac{{\partial \psi }}{{\partial z}} + \frac{1}{2}\left( {\frac{{{\partial^2}}}{{\partial {x^2}}} + \frac{{{\partial^2}}}{{\partial {y^2}}}} \right)\psi - \frac{\textrm{1}}{\textrm{2}}{\alpha ^\textrm{2}}({{x^\textrm{2}} + {y^2}} )\psi = 0.$$

Equation (21) describes the linear two-dimensional harmonic oscillator, similar to the one-dimensional case, it has many well-known solutions and we choose one of them in the form

(22)$$\psi ({x,y,z} )={-} i\frac{{f({x,y,z} )}}{{2\pi }}\int {\int_{ - \infty }^{ + \infty } {\psi ({\xi ,\eta } )exp [{ib({{\xi^2} + {\eta^2}} )- i({{K_x}\xi + {K_y}\eta } )} ]d\xi d\eta } } ,$$

where $f({x,y,z} )= \alpha \csc ({\alpha z} )exp [{ib({{x^2} + {y^2}} )} ]$, ${K_x} = \alpha x\csc ({\alpha z} )$ and ${K_y} = \alpha y\csc ({\alpha z} )$, we consider an input with the form of two-dimensional CG beam

(23)$$\psi ({x,y} )= exp [{ - {\sigma^2}({{x^2} + {y^2}} )} ]\cos ({\omega {\kern 1pt} x} )\cos ({\omega {\kern 1pt} y} ).$$

After some algebra one ends up with the analytical solution

(24)$$\psi ({x,y,z} )= \frac{{f({x,y,z} )exp \left( { - i\frac{\pi }{2}} \right)}}{{8({ib - {\sigma^2}} )}}\left\{ \begin{array}{l} exp \left[ {\frac{{x_ -^2 + y_ -^2}}{{4({ib - {\sigma^2}} )}}} \right] + exp \left[ {\frac{{x_ -^2 + y_ +^2}}{{4({ib - {\sigma^2}} )}}} \right]\\ + exp \left[ {\frac{{x_ +^2 + y_ -^2}}{{4({ib - {\sigma^2}} )}}} \right] + exp \left[ {\frac{{x_ +^2 + y_ +^2}}{{4({ib - {\sigma^2}} )}}} \right] \end{array} \right\},$$

with ${x_ - } = \omega - {K_x}$, ${x_ + } = \omega + {K_x}$, ${y_ - } = \omega - {K_y}$, and ${y_ + } = \omega + {K_y}$.

As can be seen from Fig. 7, we can clearly see recurrence of incident beam at the integer multiple of the period ${T_s}$, and when the propagation distance is an odd integer multiple of the half period ${z_{odd}} = {{({2m + 1} ){T_s}} / 2}$, the beam will be divided into four spots, whose position can be computed from the analytical solution expression:

(25)$$\psi ({x,y} )= \frac{\alpha }{{8{\sigma ^2}}}exp \left( {i\frac{\pi }{2}} \right)\left\{ \begin{array}{l} exp \left( {\frac{{x_ -^2 + y_ -^2}}{{{{4{\sigma^2}} / {{\alpha^2}}}}}} \right) + exp \left( {\frac{{x_ -^2 + y_ +^2}}{{{{4{\sigma^2}} / {{\alpha^2}}}}}} \right)\\ + exp \left( {\frac{{x_ +^2 + y_ -^2}}{{{{4{\sigma^2}} / {{\alpha^2}}}}}} \right) + exp \left( {\frac{{x_ +^2 + y_ +^2}}{{{{4{\sigma^2}} / {{\alpha^2}}}}}} \right) \end{array} \right\},$$

with ${x_ - } ={-} x + {\omega / \alpha }$, ${x_ + } = x + {\omega / \alpha }$, ${y_ - } ={-} y + {\omega / \alpha }$, and ${y_ + } = y + {\omega / \alpha }$.

Fig. 7. (a) Iso-surface plot of the propagation of a two-dimensional CG beam in a parabolic potential. Intensity distribution of the two-dimensional CG beams at the propagation distance (b) $z = 0$, (c) $z = {{{T_s}} / 2}$and (d) $z = {T _s}$. Other parameters are: $\omega = 1$, $\sigma = 0.01$ and $\alpha = 0.2$.

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According to the properties of Gaussian beam, one can see that the beam divided into four spots, as shown in Fig. 7(c), whose intensity profiles are of Gaussian distributions. Their center coordinates are $({ - {\omega / \alpha }, - {\omega / \alpha }} )$, $({ - {\omega / \alpha },{\omega / \alpha }} )$, $({{\omega / \alpha }, - {\omega / \alpha }} )$ and $({{\omega / \alpha },{\omega / \alpha }} )$, which can be directly seen from the Eq. (26). These four focal spots located at two diagonals of square, not at center perpendicular lines. The reason is that x and y dimensions are indeed found to be correlated during beam propagation. As seen in Fig. 7, the modulation frequencies in two dimensions are the same, the focal spots are going to appear on the diagonal. It is also easy to understand by use of the relation between sum and difference of angles in trigonometry functions as $2\cos ({{\omega_1}x} )\cos ({{\omega_2}y} )= \cos ({{\omega_1}x + {\omega_2}y} )+ \cos ({{\omega_1}x - {\omega_2}y} )$. Therefore, the focal spots are satisfied relationship ${\omega _2}x \pm {\omega _1}y = 0$, and the corresponding coordinates are $({{{ \pm {\omega_1}} / \alpha },\textrm{ }{{ \pm {\omega_2}} / \alpha }} )$. Moreover, the positions of focal spots can be controlled by varying the modulation frequency of CG beam. This relationship also indicates the focal spots do not appear at center perpendicular lines in two dimensional cases. It is the prerogative of a one-dimensional CG beam.

4. Conclusion

In summary, we have analytically and numerically investigated the propagation dynamics of on-axis and off-axis CG beams in a medium with an external quadratic potential. The CG beams are the interference pattern originated from two Gaussian beams with different accelerating directions. The analytical expression of the CG beams propagation in a quadratic potential showed their periodic evolution described by the Fourier transform of the initial beam with a quadratic chirp. Due to the presence of external quadratic potential, the CG beams experience a periodic oscillation process, which can be characterized as splitting, focus and coalescence. The oscillated period and the distance between two focal spots are inversely proportional to the potential depth ($\alpha $). They remain the same for both on-axis and off-axis CG beams. While the trajectory of sub beams made a transition from sine curves to cosine-like curves as the incident on-axis CG beams were substituted by the off-axis one. Compared to the on-axis CG beams, the off-axis CG beams are asymmetric with respect to $x = 0$, Consequently, The period of intensity evolution of on-axis CG beam is half of that of on-axis one.

The dual CG beams, can be treated as a combination of two CG beams with transverse displacements, will perform complex interference and form four multi-peaks structure near the phase transition point in a period. Two-dimensional CG beams have the similar phenomenon to the one-dimensional case. But at the focal point, there are four focal spots located at four corner, their locations and interval between focal points are determined by transverse modulation frequencies and the depth of quadratic potential. The present investigation will be helpful to deepen understanding of CG beams with quadratic external potential, and also point out the potential applications of beam shaping technology.

Funding

National Natural Science Foundation of China (61975130).

Disclosures

The authors declare no conflicts of interest.

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Controlling cosine-Gaussian beams in linear media with quadratic external potential

Abstract

1. Introduction

2. Theoretical model

3. Analytical and numerical results

3.1 Single CG beam

3.2 Dual CG beams

3.3 Two-dimensional CG beam

4. Conclusion

Funding

Disclosures

References

Cited By

Figures (7)

Equations (25)

Optics Express