Originating an integral formula and using the quantum Fourier transform to decompose the Hermite-Laguerre-Gaussian modes into elliptical orbital modes

Y. F. Chen; Y. H. Hsieh; K. F. Huang

doi:10.1364/OSAC.1.000744

1. Introduction

Optical vortex beams, carrying orbital angular momentum (OAM), have been used in a variety of applications including optical tweezers and microscopy [1,2], trapping and guiding of cold atoms [3–5], controlling the chirality of twisted metal nanostructures [6,7], quantum communication [8,9], and spiral interferometry [10]. Even though optical vortex beams can be obtained by directly generating the Laguerre-Gaussian (LG) modes with circular symmetry, it is generally difficult to make a laser oscillation in a pure LG mode. Alternatively, the optical vortex beams are generated by converting the high-order Hermite-Gaussian (HG) modes into LG modes via the use of an astigmatic mode converter (AMC) [11–16]. Abramochkin and Volostnikov originally used an AMC formed by a matched pair cylindrical lenses to generate the so-called Hermite-Laguerre-Gaussian (HLG) beams [12]. The HLG beams, a continuous transformation lying between HG and LG modes, can successively be realized by rotating the cylindrical lens about the optical axis by an angle. The HLG modes display a plentiful evolution of point dislocations and edge dislocations. It has been theoretically verified that the HLG modes can be expressed as the superposition of the degenerate HG modes by using the SU(2) algebra [17]. Nevertheless, the superposition based on the HG modes makes the representation of the HLG mode extremely complicated and leads to the hindrance of calculation, even only for computing the intermediate order mode.

On the other hand, since the paraxial wave equation for the spherical laser cavity is identical to the Schrödinger equation for the 2D harmonic oscillator [18], the laser transverse modes have been employed to explore the wave functions with quantum-classical correspondence [19,20]. The selective excitation was used to generate the elliptical orbital modes, which are laser transverse modes to be well localized on the elliptical orbits. It has been universally found that wave functions localized on periodic orbits are associated with striking quantum phenomena such as conductance fluctuations in mesoscopic semiconductor billiards [21,22], oscillations in photo-detachment cross sections [23,24], and shell effects in metallic clusters [25,26]. So far, the relationship between the HLG modes and elliptical orbital modes has not been explored and connected.

In this work, we first use the SU(2) representation of the HLG mode to manifest the successive transformation between HG and LG modes by varying the angle parameters. We then use the mathematical formula of Schrödinger coherent state to construct the time-dependent HLG-based coherent state and to verify it to be given by a closed form of Gaussian wave packet. By using the explicit closed form, the elliptical orbital mode related to the HLG modes is analytically derived as an integral of the Gaussian wave-packet state over the corresponding orbit. On the other hand, we also confirm that the elliptical orbital mode can be expressed as the superposition of the degenerate HLG modes. We further use the superposition formula and the quantum Fourier transform to verify that the HLG mode can be inversely expressed as the superposition of the elliptical orbital modes with different ellipticities and rotations. The novelty is that the whole expression for the HLG mode is only related to the integration of Gaussian wave-packet states without involving Hermite and Laguerre polynomials. It is believed that the derived formula can provide an important insight into quantum physics and laser transverse modes with OAM [27–33].

2. SU(2) representation for Hermite-Laguerre Gaussian modes

The transverse HG modes in the spherical cavity are identical to the eigenfunctions in the 2D isotropic harmonic oscillator. In terms of the creation operator, the HG modes are given by [34]

ψ_{n_{1}, n_{2}}^{(H G)} (\tilde{x}, \tilde{y}) = \frac{{(a_{1}^{†})}^{n_{1}}}{\sqrt{n_{1}!}} \frac{{(a_{2}^{†})}^{n_{2}}}{\sqrt{n_{2}!}} ψ_{0, 0}^{} (\tilde{x}, \tilde{y})

where

ψ_{0, 0}^{} (\tilde{x}, \tilde{y}) = \frac{1}{\sqrt{π}} e^{- {(\tilde{x} + \tilde{y})}^{2} / 2},

a_{1}^{†} = \frac{1}{\sqrt{2}} (\tilde{x} - \frac{\partial}{\partial \tilde{x}}),

a_{2}^{†} = \frac{1}{\sqrt{2}} (\tilde{y} - \frac{\partial}{\partial \tilde{y}}),

_{$\tilde{x} = \sqrt{2} x / w (z)$},

\tilde{y} = \sqrt{2} y / w (z)

,

w (z) = w_{o} \sqrt{1 + {(z / z_{R})}^{2}}

,

w_{o} = \sqrt{λ z_{R} / π}

,

z_{R}

is the Rayleigh range, λ is the photon wavelength, and n₁ and n₂ are the transverse orders in the x and y directions, respectively. Using the SU(2) algebra, the HLG modes, continuous transformation from HG to LG modes, can be generalized as [35]

Ψ_{n_{1}, n_{2}}^{(α, β)} (\tilde{x}, \tilde{y}) = \frac{{(b_{1}^{†})}^{n_{1}}}{\sqrt{n_{1}!}} \frac{{(b_{2}^{†})}^{n_{2}}}{\sqrt{n_{2}!}} ψ_{0, 0}^{} (\tilde{x}, \tilde{y}),

where

[\begin{array}{l} b_{1}^{†} \\ b_{2}^{†} \end{array}] = [\begin{matrix} e^{- i α / 2} \cos (β / 2) & e^{i α / 2} \sin (β / 2) \\ - e^{- i α / 2} \sin (β / 2) & e^{i α / 2} \cos (β / 2) \end{matrix}] [\begin{array}{l} a_{1}^{†} \\ a_{2}^{†} \end{array}],

α and β can be imaged as the azimuthal and polar angles for a point on the Poincaré sphere. It can be verified that the HLG modes with

α = 0

and

β = 0

, i.e.,

Ψ_{n_{1}, n_{2}}^{(0, 0)} (\tilde{x}, \tilde{y})

, correspond to the HG modes

ψ_{n_{1}, n_{2}}^{(H G)} (\tilde{x}, \tilde{y})

. On the other hand, the HLG modes with

α = π / 2

and

β = π / 2

, i.e.,

Ψ_{n_{1}, n_{2}}^{(π / 2, π / 2)} (\tilde{x}, \tilde{y})

, correspond to the LG modes

ψ_{n_{1}, n_{2}}^{(L G)} (\tilde{x}, \tilde{y})

. Note that the azimuthal quantum number of LG mode is determined by

l = n_{2} - n_{1}

and the radial quantum number is given by the smaller of n₁ and n₂. Figure 1

Fig. 1 Calculated patterns for $Ψ_{n_{1}, n_{2}}^{(α, β)} (\tilde{x}, \tilde{y})$ with $n_{1} = 4$ and $n_{2} = 3$ for several values of α and β.

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shows the calculated patterns of HLG modes

Ψ_{n_{1}, n_{2}}^{(α, β)} (\tilde{x}, \tilde{y})

with

n_{1} = 4

and

n_{2} = 3

for several values of α and β. The transition from the HG mode to LG mode can be clearly seen by varying the values from

α = 0

and

β = 0

to

α = π / 2

and

β = π / 2

. The OAM per photon for the HLG mode

Ψ_{n_{1}, n_{2}}^{(α, β)} (\tilde{x}, \tilde{y})

can be analytically found to be given by

< L_{z} > = ℏ (n_{1} - n_{2}) \sin α \sin β

.

3. Elliptical orbital modes

Schrödinger used the generating function of Hermite polynomials [36] to derive the Gaussian wave packet [37] as

g (x, u) = \sum_{n = 0}^{\infty} \frac{u^{n}}{\sqrt{n!}} e^{- | u |^{2} / 2} [\frac{1}{\sqrt{2^{n} n!}} H_{n} (\tilde{x}) e^{- {\tilde{x}}^{2} / 2}] = e^{- ({\tilde{x}}^{2} - 2 \sqrt{2} u \tilde{x} + u^{2} + | u |^{2}) / 2} .

In terms of

u = \sqrt{N} e^{- i (ω t + ϕ)}

, the central peak of the Gaussian wave packet in Eq. (7) can be found to mimic the classical motion

\tilde{x} = \sqrt{2} Re (u) = \sqrt{2 N} \cos (ω t + ϕ)

, where the phase factor ϕ is related to the initial position. Using the representation of Schrödinger coherent state, the HLG-based coherent state can be expressed as

g^{(α, β)} (\tilde{x}, \tilde{y}, u_{1}, u_{2}) = \sum_{n_{1} = 0}^{\infty} \sum_{n_{2} = 0}^{\infty} \frac{u_{1}^{n_{1}}}{\sqrt{n_{1}!}} \frac{u_{2}^{n_{2}}}{\sqrt{n_{2}!}} e^{- \frac{| u_{1} |^{2} + | u_{2} |^{2}}{2}} Ψ_{n_{1}, n_{2}}^{(α, β)} (\tilde{x}, \tilde{y}),

where the parameters u₁ and u₂ are given by

[\begin{matrix} u_{1} \\ u_{2} \end{matrix}] = [\begin{matrix} \sqrt{N_{1}} e^{- i (θ + ϕ / 2)} \\ \sqrt{N_{2}} e^{- i (θ - ϕ / 2)} \end{matrix}],

the variable θ represents

ω t

to be in the range of 0 to 2π, and the parameters N₁, N₂, and ϕ are related to the elliptical orbit. By substituting Eq. (9) into Eq. (8), the wave packet state

g^{(α, β)} (\tilde{x}, \tilde{y}, u_{1}, u_{2})

can be summed to give an expression in closed form,

g^{(α, β)} (\tilde{x}, \tilde{y}, u_{1}, u_{2}) = \sum_{n_{1} = 0}^{\infty} \sum_{n_{2} = 0}^{\infty} \frac{N_{1}^{n_{1} / 2}}{\sqrt{n_{1}!}} \frac{N_{2}^{n_{2} / 2}}{\sqrt{n_{2}!}} e^{- \frac{N_{1} + N_{2}}{2}} Ψ_{n_{1}, n_{2}}^{(α, β)} (\tilde{x}, \tilde{y}) e^{- i (n_{1} + n_{2}) θ} e^{- i (n_{1} - n_{2}) ϕ / 2} .

On the other hand,

g^{(α, β)} (\tilde{x}, \tilde{y}, u_{1}, u_{2})

can be summed to give an expression in closed form,

g^{(α, β)} (\tilde{x}, \tilde{y}, u_{1}, u_{2}) = \frac{1}{\sqrt{π}} e^{- \frac{({\tilde{x}}^{2} - 2 \sqrt{2} v_{1} \tilde{x} + v_{1}^{2} + | v_{1} |^{2})}{2}} e^{- \frac{({\tilde{y}}^{2} - 2 \sqrt{2} v_{2} \tilde{y} + v_{2}^{2} + | v_{2} |^{2})}{2}}

with

[\begin{array}{l} v_{1} \\ v_{2} \end{array}] = [\begin{matrix} e^{- i α / 2} \cos (β / 2) & - e^{- i α / 2} \sin (β / 2) \\ e^{i α / 2} \sin (β / 2) & e^{i α / 2} \cos (β / 2) \end{matrix}] [\begin{array}{l} u_{1} \\ u_{2} \end{array}],

The explicit closed form in Eq. (11) is derived from substituting Eq. (5) into Eq. (8) to lead to

g^{(α, β)} (\tilde{x}, \tilde{y}, u_{1}, u_{2}) = \sum_{n_{1} = 0}^{\infty} \sum_{n_{2} = 0}^{\infty} \frac{u_{1}^{n_{1}}}{n_{1}!} \frac{u_{2}^{n_{2}}}{n_{2}!} e^{- (| u_{1} |^{2} + | u_{2} |^{2}) / 2} {(b_{1}^{†})}^{n_{1}} {(b_{2}^{†})}^{n_{2}} ψ_{0, 0} (\tilde{x}, \tilde{y}) .

Then, substituting Eq. (6) into Eq. (13) and after cumbersome algebra,

g^{(α, β)} (\tilde{x}, \tilde{y}, u_{1}, u_{2})

can be organized as a separable form of

a_{1}^{†}

and

a_{2}^{†}

,

g^{(α, β)} (\tilde{x}, \tilde{y}, u_{1}, u_{2}) = \sum_{m_{1} = 0}^{\infty} \sum_{m_{2} = 0}^{\infty} \frac{v_{1}^{m_{1}}}{m_{1}!} \frac{v_{2}^{m_{2}}}{m_{2}!} e^{- (| v_{1} |^{2} + | v_{2} |^{2}) / 2} {(a_{1}^{†})}^{m_{1}} {(a_{2}^{†})}^{m_{2}} ψ_{0, 0} (\tilde{x}, \tilde{y}),

which can be combined with Eq. (7) to obtain Eq. (11).

With the substitution of $θ = ω t$ , the explicit closed form in Eq. (11) indicates that $g^{(α, β)} (\tilde{x}, \tilde{y}, u_{1}, u_{2})$ is a Gaussian wave packet with the central peak moving in the elliptical orbit of $\tilde{x} = \sqrt{2} Re (v_{1})$ and $\tilde{y} = \sqrt{2} Re (v_{2})$ . The stationary elliptical mode [38] derived from the wave packet $g^{(α, β)} (\tilde{x}, \tilde{y}, u_{1}, u_{2})$ can be given by

Φ_{N_{1}, N_{2}}^{(α, β)} (\tilde{x}, \tilde{y}, ϕ) = \frac{1}{2 π} \int_{0}^{2 π} g^{(α, β)} (\tilde{x}, \tilde{y}, u_{1}, u_{2}) e^{i (N_{1} + N_{2}) θ} d θ .

On the other hand, Eq. (10) reveals that the Gaussian wave packet

g^{(α, β)} (\tilde{x}, \tilde{y}, u_{1}, u_{2})

is the superposition of all HLG modes

Ψ_{n_{1}, n_{2}}^{(α, β)} (\tilde{x}, \tilde{y})

. Substituting both expressions of Eqs. (10) and (11) into Eq. (15) and using the orthogonality relation,

\frac{1}{2 π} \int_{0}^{2 π} e^{i (n - n^{'}) θ} d θ = δ_{n, n^{'}},

the stationary coherent state

Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ)

can be derived as

\begin{array}{l} Φ_{N_{1}, N_{2}}^{(α, β)} (\tilde{x}, \tilde{y}, ϕ) = \frac{1}{2 π} \int_{0}^{2 π} \frac{1}{\sqrt{π}} e^{- \frac{({\tilde{x}}^{2} - 2 \sqrt{2} v_{1} \tilde{x} + v_{1}^{2} + | v_{1} |^{2})}{2}} e^{- \frac{({\tilde{y}}^{2} - 2 \sqrt{2} v_{2} \tilde{y} + v_{2}^{2} + | v_{2} |^{2})}{2}} e^{i (N_{1} + N_{2}) θ} d θ \\ = \sum_{K = - N_{2}}^{N_{1}} \frac{N_{1}^{(N_{1} - K) / 2}}{\sqrt{(N_{1} - K)!}} \frac{N_{2}^{(N_{2} + K) / 2}}{\sqrt{(N_{2} + K)!}} e^{- \frac{N_{1} + N_{2}}{2}} e^{- i (N_{1} - N_{2}) ϕ / 2} Ψ_{N_{1} - K, N_{2} + K}^{(α, β)} (\tilde{x}, \tilde{y}) e^{i K ϕ}, \end{array}

where the HLG modes

Ψ_{N_{1} - K, N_{2} + K}^{(α, β)} (\tilde{x}, \tilde{y})

with

n_{1} + n_{2} = N_{1} + N_{2}

have been regrouped by exploiting the new index K to express the superposed modes with

n_{1} = N_{1} - K

and

n_{2} = N_{2} + K

. For a given

(N_{1}, N_{2}, ϕ)

and

(α, β)

, the spatial intensity of the mode

Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ)

is exactly concentrated on the elliptical orbit. The OAM per photon for the elliptical orbital mode

Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ)

can be analytically proven to be

< L_{z} > = ℏ [(n_{1} - n_{2}) \sin α \sin β + 2 \sqrt{n_{1} n_{2}} (\cos α \sin ϕ + \cos β \sin α \cos ϕ)] .

4. Discrete Fourier transform

The stationary coherent state $Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ)$ can be seen to be a superposition of $N + 1$ degenerate HLG modes $Ψ_{N_{1} - K, N_{2} + K}^{(α, β)} (\tilde{x}, \tilde{y})$ with the phase factor $e^{i K ϕ}$ in the weighting coefficient, where $N_{1} + N_{2} = N$ . The concept of the discrete Fourier transform can be used to divide the phase parameter ϕ into $N + 1$ different values as $ϕ_{n} = 2 π n / (N + 1)$ with $n = 0, 1, \dots, N$ . The conventional discrete Fourier transform ${F_{0}, F_{1}, \dots, F_{N}}$ of a discrete function ${f_{0}, f_{1}, \dots, f_{N}}$ and its inverse are given by

F_{k} = \frac{1}{N + 1} \sum_{n = 0}^{N} f_{n} e^{- i 2 π n k / (N + 1)},

f_{n} = \sum_{k = 0}^{N} F_{k} e^{i 2 π n k / (N + 1)} .

Using the discrete Fourier transform upon the quantum state [39], the set

{Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ)}

can constitute a basis to represent the HG mode

Ψ_{N_{1} - K, N_{2} + K}^{(α, β)} (\tilde{x}, \tilde{y})

in an inverse way. Using the orthogonality identity of complex exponentials:

\frac{1}{N + 1} \sum_{n = 0}^{N} \exp [i 2 π (K - S) n / (N + 1)] = δ_{K, S},

the Fourier transform for Eq. (17) can be reduced to

\begin{array}{l} \frac{1}{N + 1} \sum_{n = 0}^{N} Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n}) e^{i (N_{1} - N_{2}) ϕ_{n} / 2} e^{- i S ϕ_{n}} . \\ = \frac{N_{1}^{(N_{1} - S) / 2}}{\sqrt{(N_{1} - S)!}} \frac{N_{2}^{(N_{2} + S) / 2}}{\sqrt{(N_{2} + S)!}} e^{- \frac{N_{1} + N_{2}}{2}} Ψ_{N_{1} - S, N_{2} + S}^{(α, β)} (\tilde{x}, \tilde{y}) \end{array}

Without loss of generality, we can use Eq. (22) for

S = 0

to obtain

Ψ_{N_{1}, N_{2}}^{(α, β)} (\tilde{x}, \tilde{y}) = {[\frac{N_{1}^{N_{1} / 2}}{\sqrt{N_{1}!}} \frac{N_{2}^{N_{2} / 2}}{\sqrt{N_{2}!}} e^{- \frac{N}{2}}]}^{- 1} \frac{1}{N + 1} \sum_{n = 0}^{N} Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n}) e^{i (N_{1} - N_{2}) ϕ_{n} / 2} .

Equation (23) indicates that the HLG mode

Ψ_{N_{1}, N_{2}}^{(α, β)} (\tilde{x}, \tilde{y})

can be interpreted as a summation of the generalized elliptical modes

Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})

which are given by an integral of the Gaussian wave-packet state over the classical orbit, as shown in Eq. (17).

The first case for the numerical demonstration is the modes $Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})$ with $α = 0$ and $β = 0$ . Note that the mode $Ψ_{N_{1}, N_{2}}^{(α, β)} (\tilde{x}, \tilde{y})$ with $α = 0$ and $β = 0$ is the HG mode $ψ_{N_{1}, N_{2}}^{(H G)} (\tilde{x}, \tilde{y})$ . Figure 2

Fig. 2 Calculated results for $Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})$ by using the integral formula in Eq. (17) with $(N_{1}, N_{2}) = (8, 7)$ , $(α, β) = (0, 0)$ , and $ϕ_{n} = π n / 8$ with $n = 0, 1, \dots, 15$ . The central pattern for $ψ_{8, 7}^{(H G)} (\tilde{x}, \tilde{y})$ obtained by substituting the calculated $Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})$ into Eq. (22).

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shows the calculated results for

Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})

by using the integral formula in Eqs. (17) with

(N_{1}, N_{2}) = (8, 7)

,

(α, β) = (0, 0)

, and

ϕ_{n} = π n / 8

with

n = 0, 1, \dots, 15

. The calculated elliptical modes

Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})

with

(α, β) = (0, 0)

can be seen to have different ellipticities and different directions for the major axes. Applying the calculated

Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})

to Eq. (22), the resulting pattern for

Ψ_{N_{1}, N_{2}}^{(α, β)} (\tilde{x}, \tilde{y})

, as shown in the central part of Fig. 2, can be seen to be the HG mode

ψ_{N_{1}, N_{2}}^{(H G)} (\tilde{x}, \tilde{y})

.

The second numerical demonstration is the modes $Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})$ with $α = π / 2$ and $β = π / 2$ . For this case, the mode $Ψ_{N_{1}, N_{2}}^{(α, β)} (\tilde{x}, \tilde{y})$ is the LG mode $ψ_{N_{1}, N_{2}}^{(L G)} (\tilde{x}, \tilde{y})$ . Figure 3

Fig. 3 Calculated results for $Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})$ by using the integral formula in Eq. (17) with $(N_{1}, N_{2}) = (2, 13)$ , $(α, β) = (π / 2, π / 2)$ , and $ϕ_{n} = π n / 8$ with $n = 0, 1, \dots, 15$ . The central pattern for $ψ_{2, 13}^{(L G)} (\tilde{x}, \tilde{y})$ obtained by substituting the calculated $Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})$ into Eq. (22).

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shows the calculated results for

Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})

by using the integral formula in Eq. (17) with

(N_{1}, N_{2}) = (2, 13)

,

(α, β) = (π / 2, π / 2)

and

ϕ_{n} = π n / 8

with

n = 0, 1, \dots, 15

. The calculated elliptical modes

Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})

with

(α, β) = (π / 2, π / 2)

can be seen to have different directions for the major axes but their ellipticities are the same. Applying the calculated

Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})

to Eq. (22), the resulting pattern for

Ψ_{N_{1}, N_{2}}^{(α, β)} (\tilde{x}, \tilde{y})

, as shown in the central part of Fig. 3, can be seen to be the LG mode

ψ_{N_{1}, N_{2}}^{(L G)} (\tilde{x}, \tilde{y})

. Figure 4

Fig. 4 Calculated results with $(N_{1}, N_{2}) = (4, 11)$ , $(α, β)$ $= (2 π / 5, 2 π / 5)$ , and $ϕ_{n} = π n / 8$ with $n = 0, 1, \dots, 15$ .The central pattern for $Ψ_{4, 11}^{(α, β)} (\tilde{x}, \tilde{y})$ obtained by substituting the calculated $Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})$ into Eq. (22).

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shows the calculated results for the case of

Ψ_{N_{1}, N_{2}}^{(α, β)} (\tilde{x}, \tilde{y})

with

(N_{1}, N_{2}) = (4, 11)

,

(α, β) = (2 π / 5, 2 π / 5)

and

ϕ_{n} = π n / 8

with

n = 0, 1, \dots, 15

. The calculated elliptical modes

Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})

with

(α, β)

= (2 π / 5, 2 π / 5)

can be seen to have different ellipticities and different directions for the major axes. From Figs. 2–4, it is found that the decomposition of the HLG mode

Ψ_{N_{1}, N_{2}}^{(α, β)} (\tilde{x}, \tilde{y})

into the elliptical orbital modes

Φ_{N_{1}, N_{2}}^{^{(α, β)}} (\tilde{x}, \tilde{y}, ϕ_{n})

can reveal the symmetrical structure clearly.

5. Conclusions

In summary, the SU(2) representation of the HLG mode has been used to manifest the successive transformation between HG and LG modes by varying the angle parameters. We have derived a closed form of Gaussian wave packet for the time-dependent HLG-based coherent state. We have further exploited the explicit closed form to derive the elliptical orbital mode as an integral of the Gaussian wave-packet state over the orbit. On the other hand, we have also dervied the elliptical orbital mode as the superposition of the degenerate HLG modes. We finally employ the derived formulae and the quantum Fourier transform to verify that the HLG mode is inversely expressed as the superposition of the elliptical orbital modes. The complete form for the HLG mode can clearly manifest the quantum-classical correspondence and provide an important insight into the laser transverse modes with OAM.

Funding

Ministry of Science and Technology of Taiwan (MOST107-2119-M-009-015).

Acknowledgments

This work was financially supported by the Research Team of Photonic Technologies and Intelligent Systems at NCTU within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan. This work is also supported by the Ministry of Science and Technology of Taiwan (Contract No. MOST107-2119-M-009-015).

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Originating an integral formula and using the quantum Fourier transform to decompose the Hermite-Laguerre-Gaussian modes into elliptical orbital modes

Abstract

1. Introduction

2. SU(2) representation for Hermite-Laguerre Gaussian modes

3. Elliptical orbital modes

4. Discrete Fourier transform

5. Conclusions

Funding

Acknowledgments

References

Cited By

Figures (4)

Equations (23)

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