## Abstract

We report selective excitations of higher-order Hermite-Gaussian and Ince-Gaussian (IG) modes in a laser-diode-pumped microchip solid-state laser and controlled generation of corresponding higher-order and multiple optical vortex beams of different shapes using an astigmatic mode converter (AMC). Simply changing the pump-beam diameter, shape, and lateral off-axis position of the tight pump beam focus on the laser crystal within a microchip semispherical cavity can produce the desired optical vortex beams in a well controlled manner. Pattern changes featuring different IG and HG modes obtained by rotating the AMC are also demonstrated. Numerical simulation shows that the vortex structure is changed by controlled off-axis laser diode pumping, which could lead toward precise optical manipulation of small particles.

© 2008 Optical Society of America

## 1. Introduction

Two transverse lasing modes, Hermite-Gaussian modes (HGMs) and Laguerre-Gaussian modes (LGMs), have been widely investigated using analytical, numerical, and experimental techniques. These two modes separately form two complete families of exact and orthogonal solutions of the paraxial wave equation (PWE) in rectangular and cylindrical coordinates. Researchers have recently proposed a third complete family of transverse modes, namely Ince-Gaussian modes (IGMs), as a third complete family of PWE solutions in elliptic coordinates [1,2]. They have observed IGMs in a stable resonator in a laser-diode-pumped (LD-pumped) solid-state laser by breaking the symmetry of the cavity under tight pump beam focusing conditions [3,4]. These attempts include introducing off-axis pumping with a cross hair inside the cavity [3] and adjusting the azimuthal symmetry of the short laser resonator [4,5]. We have verified off-axis pump-beam focusing conditions for generating desired IGMs by numerical simulations of the model of LD-pumped microchip solid-state lasers [6].

On the other hand, researchers have widely used optical vortex beams [7], which are optical beams that have phase singularities mixed with wavefront curvature, in the study of optical tweezers [8–12], trapping and guiding of cold atoms [13–15], rotational frequency shift [16,17], and entanglement states of photons [18]. Optical vortex beams have been reported to be created by passing HGM beams through spiral phase-plates [19], computer-generated holographic converters [20], or astigmatic mode converters (AMC) [21].

This paper demonstrates selective excitations of higher-order HGMs and IGMs in a microchip NdGdVO_{4} laser with off-axis LD pumping and controlled generation of several corresponding higher-order optical vortices by an AMC, featuring a novel type of non-circular optical vortex originating from higher-order IGMs.

## 2. Experimental results

#### 2.1 Experimental setup

The experimental setup is shown in Fig. 1. The experiment was carried out using an LD-pumped 1-mm-thick 3 at.% Nd-doped a-plate Nd:GdVO_{4} crystal (refractive index n=2.0). A Nd:GdVO_{4} crystal was placed within a semispherical external cavity, where the sample was attached to a plane mirror M_{1} (99.8% reflective at 1064 nm and >95% transmissive at 808 nm), and a concave mirror M_{2} (99% reflective at 1064 nm, radius of curvature: 100 mm) was placed 10 mm away from the plane mirror. The two mirrors and the laser crystal were assembled into one body. An elliptical LD beam was transformed into a circular one and focused onto the Nd:GdVO_{4} crystal by using a microscope objective lens with a numerical aperture of 0.25 to obtain a tight focus giving a minimum spot size of approximately 75 µm at the crystal, while the lasing beam-waist spot size for the fundamental *HG _{00}* mode was 100 µm. The shape of the pump beam at the crystal was changed to non-circular by slightly changing the crystal along the lasing axis (z-axis) through an astigmatic aberration. The absorption coefficient for the LD wavelength of 808 nm was 74 cm

^{-1}. The resultant absorption length was as short as 135 µm. In all cases, linearly π-polarized emissions along the tetragonal c-axis were observed.

The lasing beam was delivered to an AMC through a mode-matching lens, as shown in Fig. 1, in which the separation of cylindrical lenses was precisely adjusted to *L*=2^{1/2}
*f*=70.71 mm (*f*=50 mm: focal length of cylindrical lenses). The optical design is summarized in Table 1. Both surfaces of each cylindrical lens were coated to be anti-reflective. The pair of cylindrical lenses was rotated by a step motor.

#### 2.2 Selective excitation of higher-order HGM operations and donut-like vortex beam generation

Any Laguerre-Gaussian (LG) mode propagating along the z-axis can be decomposed into a set of Hermite-Gaussian (HG) modes as follows [22].

Here, *u* denotes the field function of eigenmodes, *N*=*l*+*p*, and *B _{k}* are real expansion coefficients that satisfy the normalization condition ∑

*B*

_{k}^{2}=1. In the case of

*p*=0, each photon carries an orbital angular momentum of

*lħ*[23]. The factor i

*k*, where i

^{2}=-1, corresponds to a π/2 relative phase difference between successive HG components.

On the other hand, any HG mode whose principal axes make an angle of 45° with the (x,y) axes (diagonal modes) can be decomposed into exactly the same constituents set with the same coefficients *B _{k}* but without the i

*k*factor. For example, an

*HG*mode, aligned at 45° to the (x,y) axes, can be expressed as two in-phase

_{1,0}*HG*and

_{1,0}*HG*modes aligned with the principal axes of the lenses. The decomposition of the diagonal

_{0,1}*HG*and

_{2,0}*LG*modes into u

_{2,0}_{2,0}, u

_{1,1}, u

_{0,2}is shown in Fig. 2.

Therefore, in principle, any HG mode with indices *m* and *n* (denoted HG_{m,n}), aligned at 45° to the principal axis of the lens, will then be converted into an LG mode with the same beam waist as a result of relative phase shifts introduced into the diagonal modes by an AMC [21]. For higher-order HGMs, the expansion and transformation is more complex, but the same principles apply. The separation of the cylindrical lenses *L* is made such that these orthogonal modes undergo different Gouy phase shifts. Any HG mode is transformed into a corresponding LG mode with *l*=*m*-*n* and *p*=min(*m*,*n*) after the lenses, where *l* is the azimuthal mode index and *p* describes the number of radial nodes in the field.

By shifting the position of tight beam focus laterally from the central axis of the laser cavity (i.e., off-axis pumping) and longitudinally to get a nearly ‘elliptic’ pump-beam focus due to aberration, we achieved a variety of higher-order *HG _{m,0}* mode oscillations. Mode number

*m*was increased with increasing shift

*d*, such that the pump-beam position was adjusted to the position of one of the brightest outermost spots, namely the ‘target spot’, depicted in Fig. 1.

When we increased *d* gradually, higher-order *HG _{m,0}* oscillations and their corresponding

*LG*modes (

_{l,p}*l*=

*m*,

*p*=0) were easily obtained successively at a fixed pump power. Example results are shown in Fig. 3, where θ is the rotation angle of the AMC depicted in

*Fig. 1*. The structural changes in beam patterns caused by the AMC rotation (i.e., azimuthal symmetry variation) for the

*HG*mode is shown in Fig. 4 and the movie. Note that the same pattern repeated at every 90° rotation of the AMC. These patterns manifested themselves in optical vortices possessing angular momentum of

_{4,0}*lħ*(

*l*: topological charge), whose diameter is proportional to

*l*

^{1/2}[23]. The mode conversion efficiency was measured to be >95% on average in the entire pump-power region up to 400 mW from a comparison of the input HGM power and the corresponding LGM power.

Similar structural pattern changes for HGMs have been demonstrated using a variable-phase-shift mode converter consisting of two π/2 converters at ±45° to the beam and an image rotator R(ϕ) comprising two Dove prisms [24]. When the two parts of the image rotator are rotated with respect to each other through an angle ϕ/2, the image between the two π/2 converters is rotated through an angle ϕ. In this case, unlike in our present experiment, the variable-phase-shift mode converter acts as a 2ϕ converter and the pattern repeats at every 180° rotation of ϕ [24].

#### 2.3 Forced IGM operations and non-circular vortex laser beam generation

The Ince-Gaussian (IG) modes propagating along the z-axis of an elliptic coordinate system of ** r**=(ξ, η, z) with mode numbers

*p*and

*m*and ellipticity

*ε*are given by [2]

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\times \mathrm{exp}i\left[\mathrm{kz}+\left\{\frac{{\mathrm{kr}}^{2}}{2R\left(z\right)}\right\}-\left(p+1\right){\psi}_{z}\left(z\right)\right],$$

$$\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.9em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\times \mathrm{exp}i\left[\mathrm{kz}+\left\{\frac{{\mathrm{kr}}^{2}}{2R\left(z\right)}\right\}-\left(p+1\right){\psi}_{z}\left(z\right)\right],$$

where the elliptic coordinate is defined in a transverse z plane as x=*f*(z) cos ξ cos *η*, y=*f*(z) sin ξ sin η, and ξ∊[0, ∞], *η* ∊[0, 2π]. Here, *f*(z) is the semifocal separation of IGMs defined as the Gaussian beamwidth, i.e., *f*(z)=*f*
_{0}
*w*(z)/*w*
_{0}, where *f*
_{0} and *w*
_{0} are the semifocal separation and beamwidth at the z=0 plane, respectively. Here, *w*(z)=*w*
_{0} (1+*z*
^{2}/*z*
_{R}
^{2})^{1/2} describes the beamwidth, *z _{R}*=

*kw*

_{0}

^{2}/2 is the Rayleigh length, and

*k*is the wave number of the beam.

*C*and

*S*are normalization constants, the subscripts

*e*and

*o*refer to even and odd IGMs, and

*C*(., ε) and S

_{p}^{m}_{p}

^{m}(., ε) are the even and odd Ince polynomials [2] of order

*p*, degree

*m*, and parameter ε, respectively. In Eqs. (1) and (2),

*r*is the radial distance from the central axis of the cavity,

*R*(z)=

*z*+

*z*

_{R}^{2}/z is the radius of curvature of the phase front, and

*Ψ*

_{z}(z)=arctan(

*z/z*). The parameters of ellipticity ε, waist

_{R}*w*

_{0}, and semifocal separation

*f*

_{0}are not independent, but related by ε=2

*f*

_{0}

^{2}/

*w*

_{0}

^{2}. The patterns of IGMs can be recognized by the rules that degree

*m*corresponds to the number of hyperbolic nodal lines and (

*p*-

*m*)/2 is the number of elliptic nodal lines. Example analytic patterns of

*IG*modes are shown in Fig. 5.

_{p,m}Let us examine the relationship between IGMs and HGMs. The transition from *IG ^{e,o}_{p,m}* to
${\mathrm{HG}}_{{n}_{x},{n}_{y}}$
occurs when ellipticity ε→∞, and their indices are related as follows [2]: For even IGMs,

*n*=

_{x}*m*and

*n*=

_{y}*p*-

*m*, whereas for odd IGMs,

*n*=

_{x}*m*-1 and

*n*=

_{y}*p*-

*m*-1.

Example results indicate that some *IG ^{o}_{p,1}* and

*IG*modes with ε=4 approach their corresponding

^{e}_{p,p}*HG*mode with ε=1000, as shown in Figs. 6(a)–6(c), respectively. Note that the shapes of the largest and most intense outside lobes of IG modes bent to become ‘half-elliptic’ or became more symmetric and circular than those of

_{m,0}*HG*modes for

_{p,0}*IG*and

^{o}_{p,1}*IG*modes, respectively [6], but their distances from the central axis are the same. Our idea for forcing single IG mode operations can be implemented by controlling the size and shape of the pump beam focus such that they match the outside lobes of the desired IG modes in a laser cavity, i.e., by performing controlled off-axis pumping [6].

^{e}_{p,p}For any plane *z*, the IG↔LG expansions are written as follows [2].

Here, σ={e, o} is the parity and ∑*D*
_{j}
^{2}=1. Once we know the IG↔LG relations, the IG↔HG formulas can be readily obtained by applying the already known LG↔HG expansions (1) in cascade with the IG↔LG expansions. Consequently, in principle, various mode patterns are expected to appear from the AMC with a change in the rotation angle, i.e., azimuthal symmetry. Below, we describe several examples.

When we introduced the appropriate asymmetry into the pump-beam focus profile by slightly tilting the pump-beam direction (oblique pumping) [6], we could achieve *IG ^{e}_{p,0}* or

*IG*mode oscillations. Delivering these IGMs into the AMC yielded the corresponding optical vortices. An example result is shown in Fig. 7, where the structural change of outgoing beam patterns from the AMC is demonstrated in the videos. Note that the asymmetric shapes of the vortices differ from the donut-like patterns formed from

^{o}_{p,1}*HG*modes shown in Fig. 4. By controlling the pump focus to be tighter and more circular than that for selective excitation of

_{m,0}*HG*, we successfully obtained a variety of

_{m,0}*IG*mode oscillations by systematic off-axis pumping. A typical example demonstrating structural changes is shown in Fig. 8. In both cases, the intensity patterns are not circular, but have rectangular-like or squarish shapes and the intensity profiles of the vortices changed critically around θ=45°, as shown in Figs. 7 and 8. This point will be discussed again theoretically later. Mode conversions between

_{p,p}*IG*and

^{o}_{3,1}*HG*as well as

_{3,0}*IG*and

^{e}_{3,3}*IG*modes took place when the AMC angle approached 0° and 90°. The pattern repeated at every 90° rotation of the AMC, similar to Fig. 4.

^{o}_{3,1}As for *HG _{m,n}* (

*n*≠0) and

*IG*modes with large

*p*values, complicated beam pattern changes were observed by rotating the AMC, as shown in Figs. 9–10.

## 3. Theoretical results

#### 3.1 Simulation model

To obtain selective excitations of lasing modes, we simulated the initial stimulated field with a partially coherent field [25] in the space-frequency domain to avoid dependence between the initial field selection and the conversion field in a stable laser cavity using Endo’s method [26]. The stimulated initial field propagating back and forth in the resonator is mimicked by the Fresnel-Kirchhoff integration [27]; and the optical fields changed by laser mirrors and gain medium are introduced by modifying the optical field at each position under the off-axis pumping condition [6,28]. The loaded gain at each station is assumed to be homogeneously broadened. The gain medium was simulated as several gain sheets and the saturated gain at each gain sheet *i* is expressed as

where *g _{0}*(

*x,y*) is the loaded gain,

*g*(

_{i0}*x,y*) is the small signal gain, and

*I*(

_{s}*x,y*) is the saturation intensity. In this paper, we assume that I

_{s}≈1 kW/cm

^{2}, which is reasonable for general solid-state lasers [27]. The symbols

*Ĩ*

^{+}

_{i}and

*Ĩ*

^{-}

_{i}are the average right-going and left-going optical intensities defined as

Here, *I*
^{+}
_{i} (*q*) and *I*
^{+}
_{i} (*q*) denote the intensities of the *q*th iteration step and α is a summation over a period of the cavity’s photon decay time. For example, in this simulation, the reflectivities of the two mirrors were set to r_{1}=100.0% and r_{2}=99%. Thus, the parameter α is given by r_{1}r_{2}=0.99.

The amplification of the optical field *E*(*x,y*) passing through gain sheet *i* with thickness *d* is expressed as

where the 1/2 is necessary because the gain is defined by the amplification of the optical intensity. With this simulation method, after a certain number of iterations, according to the boundary condition, the cavity will find the lasing mode distribution *E*(*x, y*) that satisfies

To model the performance of the mode converter we used a wave propagation algorithm, i.e., Fast Fourier transform method [29,30]. Outgoing fields from the AMC were numerically simulated.

#### 3.2 Numerical results

Numerical simulations were carried out using relevant parameters for the laser cavity, mode-matching lens, and AMC that we used in the experiment. They are summarized in Table 1. Typical examples are shown in Figs. 11, 12, and 13, respectively. The numerical results well reproduced the experimental observations shown in Figs. 4, 7, and 8.

On the other hand, intensity profiles for IGM-originating vortices (Figs. 12 and 13) are not circular donut-like shapes unlike those for HGM-originating vortices (Fig. 11), but rectangular- or square-like shapes and they change critically around θ=45°, similar to the experimental results.

Let us examine the structural change of such IGM-originating optical vortices with the rotation of the AMC around θ=45°. The computed interference fringes corresponding to intensity profiles at θ=35°, 45°, and 55° in Figs. 12 and 13 are shown in Fig. 14, where interference fringes were calculated by causing the vortex beam to interfere with a tilted plane wave, where the tilt angle relative to the propagation direction was 1°.

Note that ‘in-line’ type vortices were formed along the longer elliptic axis and rotated by 90° across θ=45°, as indicated by red dots for both cases. The linear array of vortices (with high beam purity and power) may find applications for optical tweezers or for guiding condensates. Similar in-line vortices have been created in the form of helical IG modes with an argon laser operating at 514.5-nm wavelength by using complex amplitude and phase masks encoded onto a liquid-crystal display, in which an intensity pattern consists of an ‘elliptic’ ring [31].

The structure of optical vortices is expected to depend on the ellipticity ε even if IGMs possess the same mode numbers (*p, m*) and parity. The computed intensity profiles, phase portraits, and interference fringes for *IG ^{0}_{3,1}* mode are shown in Fig. 15 together with those for

*HG*, where the LD pump-focus position was changed in the simulation. The intensity profile changed from a rectangular-like to circular donut-like shape as ε increased, and spatially separated singular points merged into the common central point accordingly. The numerical result suggests that one can control the property of optical vortices for optical manipulation of small particles by changing the ellipticity of IGMs with controlled off-axis LD pumping [6]. A systematic experimental study of this issue is in progress.

_{3,0}## 4. Summary and discussion

We have produced circular donut-like, rectangular-like, and square-like vortices possessing multiple topological changes from a well-designed astigmatic mode converter (AMC) by using a microchip solid-state laser operating in higher-order Hermite and Ince-Gaussian modes with controlled laser-diode off-axis end pumping. Structural beam pattern changes with continuous rotation of the AMC, which feature conversions among three orthogonal modes (i.e., HGM, IGM, and LGM) leading to successful vortex formations, have been demonstrated experimentally and well reproduced by numerical simulation. It has been also verified by numerical simulation that the intensity profiles and spatial distribution of phase singularities of multi-vortex laser beams can be changed by the controlled laser-diode off-axis pumping and AMC rotation. The present control of optical vortices by adjusting the pump-focus lateral position and shape at the laser crystal with off-axis laser-diode pumping suggests the potential of IGMs for advanced manipulation of small particles using microchip solid-state lasers.

Finally, let us briefly address advantages and disadvantages of AMC approach for generating multiple vortex laser beams in comparison with the spatial light modulator (SLM) approach, which employs a holographic element [32] or an appropriate amplitude and phase mask encoded onto a liquid-crystal display [31].

In AMC approach, vortex laser beams are generated by a mode conversion of higher-order transverse lasing modes that are directly emitted from the microchip laser. Such beams are the superposition of eigenmodes of the paraxial wave equation: thus, they are also the solution of PWE. Therefore, the shape invariance of the resulting vortex beam is ensured against propagation [5] and the beam quality is not degraded in the far-field. While, in SLM approach, the performance of vortex generations (e.g., conversion efficiency and mode purity) from the fundamental transverse mode is limited by the precision of techniques for fabricating a sophisticated vortex lens. Other disadvantages of SLM include high loss, small acceptance angle and high cost [32]. In our AMC approach, high conversion efficiencies (i.e., more than 95 %) from HGMs or IGMs to various structures of vortices were achieved with the simplest, solid device configuration with low cost. The advantages of SLM approach might be the dynamic control of phase plates, i.e., flexibility [32]. In the liquid-crystal display type of SLM [31], for example, the electrical control of a phase plate can be utilized for dynamic control of vortices [31]. In AMC approach, on the other hand, such a dynamic control of phase shift is not attained in general. However, controlling spatial arrangements of phase singularities and shapes of vortex beams has been demonstrated to be possible based on controlled IGMs by simply controlling the pump position, shape and the “rotation” of AMC [Figs. 7–8 and Figs. 14–15] toward applications for optical tweezers or guiding condensates. A variety of mode conversions among three orthogonal HG, IG and LG orthogonal lasing eigenmodes and the potential of producing various laser-beam patterns by controlling the AMC angle have also been shown [Figs. 7–10 and Figs. 12–13]. In the conventional AMC approach, the AMC angle is fixed to 45 degree and such rotation-dependent non-trivial modal structural changes by the precise rotation control of AMC was not reported so far. It might require sophisticated control of phase plates or electrics in SLM approach for producing various laser-beam patterns and vortices.

## Acknowledgment

This work was supported in part by a grant from the National Science Council of Taiwan, R.O.C., under contract no. NSC 96-2112-M-006 -019 -MY3.

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