Abstract
We exploit the SU(2) representation of the Hermite-Laguerre-Gaussian (HLG) mode to manifest the successive transformation between Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes. We theoretically confirmed that the time-dependent coherent state for the HLG modes can be simplified as a closed form of Gaussian wave packet. We further employ the explicit closed form to originate an integral of the Gaussian wave-packet state over the elliptical orbit to represent the elliptical orbital mode with fractional orbital angular momentum. On the other hand, we also derive the elliptical orbital mode as the superposition of the degenerate HLG modes. By using the derived formulae and the quantum Fourier transform, the HLG mode is inversely expressed as the superposition of the elliptical orbital modes. The derived representation unambiguously reveals the connection between HLG modes and bundles of elliptical orbits.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
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]
where , , , , 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]whereα 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 and , i.e., , correspond to the HG modes . On the other hand, the HLG modes with and , i.e., , correspond to the LG modes . Note that the azimuthal quantum number of LG mode is determined by and the radial quantum number is given by the smaller of n1 and n2. Figure 1 shows the calculated patterns of HLG modes with and 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 and to and . The OAM per photon for the HLG mode can be analytically found to be given by .3. Elliptical orbital modes
Schrödinger used the generating function of Hermite polynomials [36] to derive the Gaussian wave packet [37] as
In terms of , the central peak of the Gaussian wave packet in Eq. (7) can be found to mimic the classical motion , 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 aswhere the parameters u1 and u2 are given bythe variable θ represents 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 can be summed to give an expression in closed form,On the other hand, can be summed to give an expression in closed form,withThe explicit closed form in Eq. (11) is derived from substituting Eq. (5) into Eq. (8) to lead toThen, substituting Eq. (6) into Eq. (13) and after cumbersome algebra, can be organized as a separable form of and ,which can be combined with Eq. (7) to obtain Eq. (11).With the substitution of , the explicit closed form in Eq. (11) indicates that is a Gaussian wave packet with the central peak moving in the elliptical orbit of and . The stationary elliptical mode [38] derived from the wave packet can be given by
On the other hand, Eq. (10) reveals that the Gaussian wave packet is the superposition of all HLG modes . Substituting both expressions of Eqs. (10) and (11) into Eq. (15) and using the orthogonality relation,the stationary coherent state can be derived aswhere the HLG modes with have been regrouped by exploiting the new index K to express the superposed modes with and . For a given and , the spatial intensity of the mode is exactly concentrated on the elliptical orbit. The OAM per photon for the elliptical orbital mode can be analytically proven to be4. Discrete Fourier transform
The stationary coherent state can be seen to be a superposition of degenerate HLG modes with the phase factor in the weighting coefficient, where . The concept of the discrete Fourier transform can be used to divide the phase parameter ϕ into different values as with . The conventional discrete Fourier transform of a discrete function and its inverse are given by
Using the discrete Fourier transform upon the quantum state [39], the set can constitute a basis to represent the HG mode in an inverse way. Using the orthogonality identity of complex exponentials:the Fourier transform for Eq. (17) can be reduced toWithout loss of generality, we can use Eq. (22) for to obtainEquation (23) indicates that the HLG mode can be interpreted as a summation of the generalized elliptical modes 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 with and . Note that the mode with and is the HG mode . Figure 2
shows the calculated results for by using the integral formula in Eqs. (17) with , , and with . The calculated elliptical modes with can be seen to have different ellipticities and different directions for the major axes. Applying the calculated to Eq. (22), the resulting pattern for , as shown in the central part of Fig. 2, can be seen to be the HG mode .The second numerical demonstration is the modes with and . For this case, the mode is the LG mode . Figure 3
shows the calculated results for by using the integral formula in Eq. (17) with , and with . The calculated elliptical modes with can be seen to have different directions for the major axes but their ellipticities are the same. Applying the calculated to Eq. (22), the resulting pattern for , as shown in the central part of Fig. 3, can be seen to be the LG mode . Figure 4 shows the calculated results for the case of with , and with . The calculated elliptical modes with 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 into the elliptical orbital modes 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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