Controllable conversion between Hermite Gaussian and Laguerre Gaussian modes due to cross phase

Guo Liang; Qing Wang

doi:10.1364/OE.27.010684

1. Introduction

The evolutions of beams of finite cross sections in the paraxial approximation can be described by the Fresnel diffraction theory [1, 2], and equivalently modeled by the well known paraxial wave equation [3]. Hermite-Gaussian (HG) beams, described by the product of Hermite polynomials and Gaussian functions, are the solutions to the paraxial wave equation in Cartesian coordinates, which form a complete and orthogonal set of functions and are called the “modes of propagation” with the Gaussian beam as their lowest-order member [4]. The number of zeros for the Hermite-Gaussian mode $H G_{m n}$ is equal to the corresponding mode number, that is, m zeros in the x direction and n zeros in the y direction, and the total number of its intensity peaks is given by the sum $m + n + 1$ . The HG beam contracts to a minimum diameter at the beam waist where the phase front is plane, and the area occupied by a mode increases with the mode number. Laguerre-Gaussian (LG) beams, described by the product of a generalized Laguerre polynomials and Gaussian functions, are another set of solutions to the paraxial wave equation in cylindrical coordinates. The LG beams $L G_{p l}$ with p and l being the radial and angular mode numbers, own $p + 1$ bright rings (intensity peaks) and l intertwined helical wavefronts (topological charge or vorticity) [5, 6]. The LG beams carry orbital angular momentum (OAM), and found numerous applications in singular optics [7–9], quantum informatics [10], optical micromanipulation [11], and free-space data transfer [12]. Different methods have been employed to generate the LG modes such as the computer-generated holograms [13], spiral phase plates [14], biaxial crystals [15], the spatial light modulators [16] and so on.

On the other hand, the vortex-free beam with nonzero OAM was also found [17], and is an elliptical Gaussian beam with the cross-phase [18]. The OAM can result in the rotations and the anisotropic diffractions for the fundamental elliptic Gaussian beam. To our knowledge, it has not been reported for the propagation properties of the higher-order Hermite Gaussian beams carrying such a cross phase. In this paper, we will show that the interconversion between the HG modes and the LG modes can be achieved by imposing the cross phase on them, and that the modes can be altered in a controllable manner by changing the cross phase coefficient. Due to the cross phase, a radially dependent rotation is predicted during the conversion process between HG and LG modes. Such a radially dependent rotation is different from that discussed recently in [19], where the rotations are achieved by operating on the phase structure of two superposed LG beams with opposite vorticities. Although the interconversion between the two modes has been realized with cylindrical lenses [20], the approach presented here is much different, which can be achieved in a simpler and controllable manner.

2. Theoretical model

The paraxial optical beam propagating in the bulk media is modeled by the paraxial wave equation [21]

2 i k \frac{\partial φ}{\partial Z} + (\frac{\partial^{2} φ}{\partial X^{2}} + \frac{\partial^{2} φ}{\partial Y^{2}}) = 0,

where φ is the slowly varying amplitude for the paraxial beam, k is the wave vector, Z is the longitudinal coordinate denoting the propagation direction of the beams, X and Y are the transverse coordinates normal to Z. If we we introduce a variable transformation of

x = X / w_{0}, y = Y / w_{0}, z = Z / (k w_{0}^{2})

with w₀ being the initial beam width of paraxial beam φ, then the paraxial wave equation (1) can be transformed into its dimensionless form

i \frac{\partial φ}{\partial z} + \frac{1}{2} (\frac{\partial^{2} φ}{\partial x^{2}} + \frac{\partial^{2} φ}{\partial y^{2}}) = 0 .

Miller has solved this paraxial wave equation (2), and obtained different eigenmodes in 17 coordinate systems [22]. Among them, the Hermite Gaussian modes in Cartesian coordinates and the Laguerre-Gaussian modes in cylindrical coordinates are most well known to us. These eigenmodes can evolve with their amplitude distributions invariable except for the beam-expanding due to their diffraction. Nonetheless, for the input of any profile, the output after a linear propagation can be expressed by the Fresnel diffraction integral [21]

φ (x, y, z) = - i \frac{exp (i z)}{2 π z} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} φ (x^{'}, y^{'}, 0) exp [i \frac{{(x - x^{'})}^{2} + {(y - y^{'})}^{2}}{2 z}] d x^{'} d y^{'},

where

φ (x, y, 0)

is the input beam at z = 0. Of course, if the input is not an eigenmode, it will experience a deformation during its evolutions, and can not keep its initial amplitude distribution. The discussed in the following is just in this case.

3. Conversion from HG modes to LG modes

We firstly consider the propagations of the HG beam carrying such a special cross phase $\exp (i Θ x y)$ , with the input at z = 0 being

φ (x, y, 0) = H_{m} (x) H_{n} (y) exp (- \frac{x^{2} + y^{2}}{2}) exp (i Θ x y),

where the integers

m, n

are the modes orders, and Θ is the cross phase coefficient. The input beam (4) owns the orbital angular momentum (OAM) per optical power as

σ \equiv M / P = Im \int \int φ^{*} (x \frac{\partial i}{\partial y} - y \frac{\partial φ}{\partial x}) d x d y / \int \int φ^{*} φ d x d y = (m - n) Θ

. We have investigated the effects of the cross phase

\exp (i Θ x y)

on the evolutions of both an elliptic beam [23, 24]and a circularly symmetric beam [25], and shown that the cross phase can result in an equivalent anisotropic diffraction and an intensity rotations of the beams. The beams considered there are just the fundamental modes

H G_{00}

of Eq. (4). In fact, the mode of

H G_{00} \exp (i Θ x y)

can be rewritten as a complex-variable-function-Gaussian beam [23], which is also the solution to the paraxial wave equation [26] and keeps its shape invariant in the propagation as an eigen mode. However, it has not been reported for the propagation properties of higher-order HG mode carrying the cross phase, which can be investigated by the substitution of Eq. (4) into Eq. (3) in the following. Without loss of generality, we give the result of the integral (3) with m = 1 and n = 0 for an example

\begin{matrix} φ (x, y, z) = A (x - \frac{z Θ}{1 + i z} y) \exp (- \frac{x^{2} + y^{2}}{2 w_{1}^{2}} - \frac{x y}{2 w_{2}}) \\ \times \exp [i c_{1} (x^{2} + y^{2}) + i c_{2} x y], \end{matrix}

where

A = 2 (i - z) {[z^{2} (Θ^{2} + 1) - 2 i z - 1]}^{- 3 / 2}; w_{2} = - α / 4 Θ z,

w_{1} = \sqrt{α / [z^{2} (Θ^{2} + 1) + 1]}; c_{1} = z [z^{2} (Θ^{2} + 1)^{2} - Θ^{2} + 1] / 2 α,

c_{2} = Θ [z^{2} (Θ^{2} + 1) - 1] / α; α = z^{4} {(Θ^{2} + 1)}^{2} - 2 z^{2} (Θ^{2} - 1) + 1 .

For the beam (5), we can calculate its width ratio between $w_{x} = \sqrt{\int \int x^{2} | φ |^{2} d x d y / P}$ and $w_{y} = \sqrt{\int \int y^{2} | φ |^{2} d x d y / P}$ given by $ρ = w_{y} / w_{x} = \sqrt{1 + z^{2} (1 + 3 Θ^{2})} / \sqrt{3 + z^{2} (3 + Θ^{2})}$ , which approaches to the limit $ρ_{\infty} = \sqrt{(1 + 3 Θ^{2}) / (3 + Θ^{2})}$ when $z \to \infty$ . Obviously, the pattern after a linear propagation depends closely on the value of Θ. For the special case $Θ_{c} = 1$ , the width ratio $ρ_{z \to \infty} = 1$ indicating that the output exhibits a circularly symmetric profiles. From the dimensionless process between Eq. (1) and Eq. (2), one can obtain the critical value $Θ_{c} = 1 / w_{0}^{2}$ for the input (4) of the beam width w₀. Figure 1 shows the output patterns for different Θ. Three key points can be demonstrated from Fig. 1. First, the cross phase can enhance the diffraction of optical beams, which is consistent with the cases discussed in [23–25]. Second, the patterns are of Hermite Gaussian profiles for both low and large Θ, and the profiles for $Θ > Θ_{c}$ are the inverted ones of that for $Θ < Θ_{c}$ . Last but not least, for the critical case that $Θ = Θ_{c}$ , the HG beams can be converted to the LG modes as shown in the following. When $z \to \infty$ , the output given by Eq. (5) with $Θ = 1$ is deduced to $φ_{\infty} \propto (x - i y) \exp (- \frac{x^{2} + y^{2}}{4 z^{2}})$ , which is the LG beam with the radial and angular mode numbers being p = 0 and l = −1. This transformation process from the HG (1,0) mode to the LG (0,-1) mode is shown in Fig. 2 in details. During the process, two separated intensity peaks rotate, gradually achieve a constructive interference, and form an elliptic bright ring shown in Figs. 2 (a1)-2 (c1). When propagating further, the optical intensity will redistribute and approach to a circularly symmetric ring shown in Fig. 2(d1). The cross phase structure described by $\exp (i x y)$ evolve into the helical phase pattern described by $\exp [- i \arctan (y / x)]$ as shown in Figs. 2(a2)-2(d2).

Fig. 1 Different intensity patterns for different Θ at the propagation $z = 30$ .

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Fig. 2 Transformation process from (1,0) Hermite Gaussian mode with the phase $\exp (i x y)$ to (0,-1) Laguerre Gaussian mode. (a1-d1) transformation of intensity profiles; while (a2-d2) transformation of phase profiles.

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To investigate the rotations in the transformation process from the HG modes to the LG modes, we now turn our attention to the output (5), which exhibits two different parts in its amplitude structure: the outer amplitude peak and the inner amplitude hollow. By setting $\partial_{x} | φ (x, y, z) |^{2} = \partial_{y} | φ (x, y, z) |^{2} = 0$ in Eq. (5), we can obtain the position of two peaks $x_{p} = \pm \sqrt{[[z^{2} (Θ^{2} + 1) + \sqrt{α} + 1] / 2}, y_{p} = \mp z Θ / x_{p}$ , from which the angular velocity of intensity peaks can be obtained as $ω_{o} = \partial_{z} \arctan (y_{p} / x_{p})$ . And we can also find the angular velocity of the inner intensity hollow $ω_{i} = Θ [z^{2} (Θ^{2} - 1) + 1] / [z^{4} (Θ^{2} - 1)^{2} + 2 z^{2} (Θ^{2} + 1) + 1]$ . From Fig. 3, it can be found that both the angular velocities and accelerations are different for the outer intensity peak and the inner intensity hollow, that is, radially dependent angular velocities and acceleration happen in the transformation process from the HG modes to the LG modes. At the first evolutional stage, the rotational directions of the outer peak and the inner hollow are opposite. After some distances of linear propagation, the inner and outer parts will tend to rotate in the same direction with different velocities, which implies the deformation of the intensity structure. It is quite different from the cases discussed in [19, 27], where the opposite rotation of the outer and the inner parts was achieved by the super positions of LG beams with opposite handedness. Moreover, the angular velocities and accelerations also relate to the value of Θ, and the rotation effects can be enhanced by the increasing of Θ.

Fig. 3 Angular velocities ω_o [green curves in (a)], accelerations Ω_o [green curves in (b)] of the outer peaks and ω_i [red curves in (a)], accelerations Ω_i [red curves in (b)] of the inner hollow during the same transformation process as Fig. 2. Dashed curves are plotted for $Θ = 0.5$ , while solid curves are plotted for $Θ = 1$ .

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For another higher-order $H G_{m, n}$ modes, they can evolve into the $L G_{p l}$ modes when $Θ = Θ_{c}$ , with the angular mode number $l = n - m$ and radial mode number $p = \min (m, n)$ , which are shown in Fig. 4 (different vorticities), Fig. 5 (different bright rings) and Fig. 6 (zero vorticity and nonzero bright rings). Specially, it is mentioned above that the beam describe by $H G_{00} \exp (i Θ x y)$ is an eigenmode to the paraxial wave equation (2), which can be confirmed by the evolution as shown in Figs. 6 (a1)-6(b1). The transformation process from the higher-order HG modes to the their corresponding LG modes is shown in Fig. 7, which is a little different from that of the lower-order modes (for example the conversion from $H G_{10}$ to $L G_{0, - 1}$ shown in Fig. 2. As can be seen from Fig. 7(e) that the radially dependent rotation of the beams is more obvious. The orientations of the inner hollow, the first and the second elliptic bright rings are all different.

Fig. 4 Transformation from (1,0) [(a1),(a2)] and (2,0) [(c1),(c2)] Hermite Gaussian mode with the phase $\exp (i x y)$ to (0,-1) [(b1),(b2)] and (0,-2) [(d1),(d2)] Laguerre Gaussian mode. (a1-d1) transformation of intensity profiles; while (a2-d2) transformation of phase profiles.

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Fig. 5 Transformation from (1,0) [(a1),(a2)] and (3,2) [(c1),(c2)] Hermite Gaussian mode with the phase $\exp (i x y)$ to (0,-1) [(b1),(b2)] and (2,-1) [(d1),(d2)] Laguerre Gaussian mode. (a1-d1) transformation of intensity profiles; while (a2-d2) transformation of phase profiles.

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Fig. 6 Evolutions of (0,0) [(a1),(a2)] and (1,1) [(c1),(c2)] Hermite Gaussian mode with the phase $\exp (i x y)$ . (a1-d1) evolutions of intensity profiles; while (a2-d2) evolutions of phase profiles.

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Fig. 7 Transformation process from (2,1) Hermite Gaussian mode with the phase $\exp (i x y)$ to (1,-1) Laguerre Gaussian mode. (a1-d1) transformation of intensity profiles; while (a2-d2) transformation of phase profiles.

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4. Conversion from LG modes to HG modes

The discussions above show that the $H G_{m n}$ beams carry such a phase $\exp (i x y)$ can evolve into the $L G_{p l}$ modes with $l = n - m$ and $p = \min (m, n)$ . In the following, we will investigate the evolution of the LG beams when carrying the cross phase $\exp (i Θ x y)$ to see whether it is possible to transform to the corresponding HG modes. Without loss of generality, we take the following input as an example

φ (x, y, 0) = (x - i y) exp (- \frac{x^{2} + y^{2}}{2}) exp (i Θ x y),

which is the

L G_{0, - 1}

mode carrying a cross phase

\exp (i Θ x y)

. Substitution of the input (6) into the integral given by Eq. (3) gives that

\begin{matrix} φ (x, y, z) = B [x - \frac{1 - i z (Θ - 1)}{(Θ + 1) z - i} y] \exp (- \frac{x^{2} + y^{2}}{2 w_{1}^{2}} - \frac{x y}{2 w_{2}}) \\ \times \exp [i c_{1} (x^{2} + y^{2}) + i c_{2} x y], \end{matrix}

where

B = [(Θ + 1) z - i] A

, and the other parameters are the same as those in Eq. (5). When

z \to \infty

, Eq. (7) with

Θ = Θ_{c} = 1

is deduced to

φ_{\infty} \propto x \exp (- \frac{x^{2} + y^{2}}{4 z^{2}})

, which is indeed the

H G_{1, 0}

modes. Another example that the

L G_{0, - 2}

mode carrying the cross phase

\exp (i x y)

transforms to the

H G_{2, 0}

is shown in Fig. 8. It is found that all the

L G_{p l}

beams carrying the cross phase

\exp (i x y)

can evolve into their corresponding

H G_{m n}

modes with their modes numbers subjecting to

l = n - m

and

p = \min (m, n)

.

Fig. 8 Transformation process from (0,-2) Laguerre Gaussian mode with the phase $\exp (i x y)$ to (2,0) Hermite Gaussian mode. (a1-d1) transformation of intensity profiles; while (a2-d2) transformation of phase profiles.

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Before concluding, it should be noted that all the cases discussed in the paper are for positive Θ. For negative Θ, the rotation during the modes conversion exhibits the same patterns as that for positive Θ, and the only difference is the rotation direction for the former will be in an opposite direction.

5. Conclusion

In conclusion, we investigated an inter-conversion between the Hermite-Gaussian (HG) modes and the Laguerre-Gaussian (LG) modes, and found that the ${HG}_{m n}$ beams carrying a cross phase $\exp (i Θ x y)$ can evolve into the ${LG}_{p l}$ modes with $l = n - m$ and $p = \min (m, n)$ when $Θ = Θ_{c}$ , and vice versa. This conversion process is accompanied by the intensity rotations of optical beams, and their angular velocities and acceleration are both radially dependent. Initially, the outer intensity peak and the inner intensity hollow rotate in the opposite directions. After that they tend to rotate in the same direction with different velocities. The output patterns closely depend on the cross phase coefficient, different patterns can be generated for different cross phase coefficients. The theoretical results provide a controllable modes generation by engineering the phase structure.

This kind of modes conversion is achieved by the linear propagations after carrying the cross phase, and the scale of the converted modes will be large due to the diffraction effects. Therefore, the nonlinear evolution of both HG modes and LG modes carrying the cross phase will be further discussed.

Funding

National Natural Science Foundation of China (11604199); China Scholarship Council (201708410236).

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Controllable conversion between Hermite Gaussian and Laguerre Gaussian modes due to cross phase

Abstract

1. Introduction

2. Theoretical model

3. Conversion from HG modes to LG modes

4. Conversion from LG modes to HG modes

5. Conclusion

Funding

References

Cited By

Figures (8)

Equations (7)

Optics Express