## Abstract

A novel method is presented for the beam shaping of far field intensity distributions of coherently combined fiber arrays. The fibers are arranged uniformly on the perimeter of a circle, and the linearly polarized beams of equal shape are superimposed such that the far field pattern represents an effective radially polarized vector beam, or discrete cylindrical vector (DCV) beam. The DCV beam is produced by three or more beams that each individually have a varying polarization vector. The beams are appropriately distributed in the near field such that the far field intensity distribution has a central null. This result is in contrast to the situation of parallel linearly polarized beams, where the intensity peaks on axis.

©2009 Optical Society of America

## 1. Introduction

The propagation of electromagnetic fields from multiple fibers can result in complex far-field intensity profiles that depend intricately on the individual near field polarization and phase. Coherently phased arrays have been used in defense and communications systems for many years, where the antenna ensemble forms a diffraction pattern that can be altered by changing the antenna spacing, amplitude, and relative phase relationships. In the optical regime, similar systems have recently been formed by actively or passively locking the phases of two or more identical optical beams [1–4]. While coherently combined fiber lasers are increasingly gaining acceptance as sources for high power applications, most approaches involve phasing a rectangular or hexagonal grid of fibers with collimated beams that are linearly polarized along the same axis [5]. The resultant far field diffraction pattern therefore is peaked in the center. If the polarization is allowed to vary, as it does in radial vector beams, the diffraction pattern vanishes in the center [6], which can result in a myriad of device applications, discussed below.

Several types of polarization states have been investigated, including radial and azimuthal [6–18]. These beams are used in mitigating thermal effects in high power lasers [19–21], laser machining [22, 23], and particle acceleration interactions [24, 25]. They can even be used to generate longitudinal electric fields when tightly focused [26,27], and although they are usually formed in free space laser cavities using a conical lens, or axicon, they can also be formed and guided in fibers [28–30]. Typical methods for creating radially polarized beams fall into various categories: one scenario involves an intracavity axicon or another polarization-selective element in a laser resonator to generate the laser mode. It has been reported [31] that four Gaussian beams, rotated intra-cavity, can be effectively combined on the resonator exit mirror. Other approaches involve taking a single beam and rotating the polarization of portions of the beam to create an inhomogeneously polarized beam (typically radial or azimuthal). Such elements as a polarization-selective mirror [8,11,22], or an aperture used with strong-birefringence rods [7,19–21] are also widely used.

In this paper we investigate the generation of an effective radially-polarized beam by coherently combining Gaussian beams, each one with an appropriately oriented linear polarization. This new technique of generating a cylindrical vector beam has the advantage of simplicity, direct control over mode generation, and eliminates power propagation along the center axis for high power Gaussian beam applications. In each particular configuration of the fibers, experimental findings are in good agreement with predictions based on standard Fraunhofer diffraction theory.

## 2. Method and Experimental Setup

Our starting point in determining the propagation of polarized electromagnetic beams is the vector Helmholtz equation, ∇ × ∇ × **E**(*ρ*,*z*) - *k*
^{2}
_{0}
**E**(*ρ*,*z*) = 0. We use cylindrical coordinates, so that *ρ* is the usual transverse coordinate, *k*
_{0} = *ω*/*c*, and the usual sinusoidal time dependence has been factored out. Applying Lagrange’s formula permits the wave equation to be written in terms of the vector Laplacian: ∇^{2}
**E**(*ρ*,*z*) + *k*
^{2}
_{0}
**E**(*ρ*,*z*) = 0, which can then be reduced to the paraxial wave equation. The paraxial solutions are then inserted into the familiar Fraunhofer diffraction integral, which for propagation at sufficiently large *z*, yields the electric field,

where **E**
^{0}(**r**+**s**
_{j},0) is the incident electric field in the plane of the fiber array. It is clear that longitudinal polarization is not considered here, consistent with the paraxial approximation. The far field intensity will be a complex diffraction pattern that depends on the individual beam intensity profile in the near field as well its polarization state and optical phase. A diagram illustrating the geometry used is shown in Fig. 1, where an array of three circular holes of radius *a* are equally distributed on the circumference of a circle of radius *R*. The center of each hole is located at **s**
_{j} = *R*
**r**̂_{j}, where **r**̂_{j} = (**x**̂ cos *θ _{j}* +

**y**̂ sin

*θ*) is the unit radial vector, and

_{j}**r**is the coordinate relative to each center. The center of each fiber is separated by an angle

*θ*= 2

_{j}*π*(

*j*- 1)/

*N*, where

*N*is the number of beams. One can scale to any number of beams, constrained mainly by the radius

*R*. For identical Gaussian beams, the incident field is expressed as,

**E**

^{0}=

*E*

^{0}(

**r**)

**r**̂

_{j}, where

*E*

^{0}(

**r**) =

*E*

_{0}

*e*

^{-r2/w20}. This permits Eq. (1) to be separated,

where ℱ = *E*
_{0}
*k*
_{0}/*z*∫^{a}
_{0}
*rdr*
*J*
_{0}(*k*
_{0}
*rρ*/*z*)e^{-r2/w20}, and the prefactors that do not contribute to the intensity have been suppressed. It is clear from Eq. (2) that the far field transform preserves the radially symmetric polarization state of the system, as expected. Note that in the limit *w*
_{0}/*a* ≫ 1, and using the relation *xJ*
_{0}(*x*) = [*x*
*J*
_{1}(*x*)]′, we have ℱ(*ρ*,*z*) ≈ *E*
_{0}(*a*/*ρ*)*J*
_{1}(*k*
_{0}
*aρ*/*z*), which gives the expected Fraunhofer diffraction pattern for a circular aperture of radius *a*. In the opposite limit, where the Gaussian profile is narrow enough so that the aperture geometry has little effect, we have ℱ(*ρ*,*z*) ≈ *E*
_{0}
*k*
_{0}
*w*
^{2}
_{0}(2*z*) exp[-(*k*
_{0}
*ρ*
*w*
_{0}/(2*z*))^{2}]. For the multiple beam arrangements investigated, corresponding to *N* = 3,4,6, we then can calculate the intensity, *I _{N}* (

*ρ*,

*ϕ*), explicitly:

where *I _{N}* = ℓ

^{2}(

*ρ*,

*z*)퓚

_{N}(

*ϕ*;

*ρ*,

*z*),

*k*= (

_{x}*k*

_{0}/

*z*)

*ρ*cos

*ϕ*and

*k*= (

_{y}*k*

_{0}/

*z*)

*ρ*sin

*ϕ*. To rotate the array, one can perform a standard rotation

**r**′ = (

*ϕ*′)

**r**, so e.g., a

*π*/4 rotation would give, 𝓚

_{4}(

*ϕ*′) = 4(1 - cos(√2

*k*) cos(√2

_{x}R*k*)).

_{y}RWe employ two different methods to create the DCV beams previously described. In each method, the experimental configuration utilizes collimated Gaussian beams that are in phase. The arrays are cylindrically symmetric both with respect to intensity and polarization. We focus the beams with a transform lens in order to simulate the far field intensity at the lens focus. The focal spot is then imaged onto a camera by a microscope objective in order to fill the CCD array. The laser source is a 100 mW continuous wave (CW) Nd:YAG operating at *λ* = 1.064*μ*m.

The first method (for the case of *N* = 4) involves expanding a beam from the Nd:YAG by means of a telescope beam expander to a diameter of about 1cm. The diffractive optical element then creates an 8 × 8 array of beams. Four of the beams are reflected and made nearly parallel by a segmented mirror, and then linearly polarized upon passing through a set of four half-wave plates. The phases of each beam are made identical by having them pass through articulated glass slides.

The second method (for the case of *N* = 3 and 6), which is simpler for larger arrays, is shown in Fig. 2. In this method the Nd:YAG is coupled into a polarization maintaining fiber. This signal is then split into multiple copies by means of a lithium niobate waveguide 8-way splitter and 8-channel electro-optic modulator. Each path can then have a separate phase modulation to ensure proper phasing of the beams. These signals are then propagated through fiber and coupled out and collimated by a 1 in. lens. The lens is slightly overfilled so the Gaussian outputs of the fibers are truncated at 1.1 × (the *e*
^{-1} radius).

## 3. Discussion

Experimental results for the radial vector beam generation found good agreement with theoretical calculations, as illustrated in Fig. 3. Clearly, the net interference patterns and symmetry correlate well with the corresponding calculations. For *N* = 4 (middle row), the central null is surrounded by a checkerboard peak structure. This pattern clearly differs from the linear polarization by a rotation of *π*/4, based on a simple phase argument. The *N* = 6 case is shown in the bottom row, where the intensity vanishes at the origin, followed by the formation of bright hexagonal rings. The 3 beam configuration is shown in the top row, where again there is satisfactory agreement between the calculated and measured results. Any observed discrepancies may be reduced with an appropriately incorporated feedback system.

A substantial fraction of the energy in the plots of Fig. 3 is contained within the first regions of high intensity peaks. To define an effective measure of this, we integrate the intensity over a circular region of radius *ρ*, and normalize it to the intensity integrated over the entire image plane. This fraction, 𝒰_{N}, can be determined for a given number of fibers, *N*. If one considers a circular region that extends up to the primary peaked intensity patterns, it is often possible to accurately calculate this analytically. This is shown in particular for the *N* = 4 case, 𝒰_{4}, where,

$$+3{J}_{3}\left(2R{k}_{0}\rho /z\right)\left(\genfrac{}{}{0.1ex}{}{{k}_{0}^{2}{\rho}^{2}{w}_{0}^{2}}{\mathrm{Rz}}-2\genfrac{}{}{0.1ex}{}{z{w}_{0}^{2}}{{R}^{3}}-4\genfrac{}{}{0.1ex}{}{z}{R}\right)]\left]-\genfrac{}{}{0.1ex}{}{24z}{R}{J}_{1}\left(2R{k}_{0}\rho /z\right)\right\}.$$

Inserting the corresponding parameters, *w*
_{0} = 42*μ*m and *R* = 2.7*w*
_{0} into Eq. (6), we find 𝒰_{4} ≈ 0.57, which effectively characterizes the dominant field distribution contained within the given circle. Performing an analogous calculation at the same radius for *N* = 6,8 and 10 yields 𝒰_{6} ≈ 0.69, 𝒰_{8} ≈ 0.81, and 𝒰_{10} ≈ 0.84, respectively. Thus, as expected, increasing the number of combined beams generally enhances the fraction, 𝒰_{N}, and ultimately the desired radially polarized DCV beam emerges.

We have shown by employing two different configurations, and by carefully controlling the polarization, a central null can be created in the far field. In previous work, the central portion of a phased array of lasers typically has been a peaked function. Now for the first time, a central null has been experimentally demonstrated by coherently combining Gaussian beams, each one with an appropriately oriented linear polarization. Our technique allows direct control over which mode will be generated from the same system in contrast to single aperture lasers using an axicon to generate an annular mode inside a laser cavity.

Although the polarization symmetry is expected to be preserved in the far field, the observed central null in the far field intensity distributions indirectly confirms that the beams have a radial polarization distribution. Our experimental technique required each beam’s near field polarization to be individually aligned and each beam was directly verified to have a linear polarization aligned radially from the center of the ensemble. Further direct evidence of the radial polarization could be obtained after passing the beams through a linear polarizer in different orientations, or alternatively through a *λ*/2 retardation plate in different orientations in conjunction with a polarizing beam splitter.

As for scaling, there is no fundamental limitation on the number of beams used. Several methods are available to incorporate a greater number of beams, including concentric beam placement (following the same beam-combining prescription described in this paper) that could scale in roughly the same pattern as a Bessel beam. It is also possible to use fiber lasers with active phasing, thus potentially scaling to many hundreds of beams [5]. Our system also has a practical advantage over typical high power Gaussian beam applications which have the drawback of having the beam concentrated near the center, where the reflecting beam director (such as a telescope obscuration) resides. Our device on the other hand, has no power propagating along the center axis of the beam, and thus for high power applications requiring a center obscuration telescope beam director, our proposed system offers new advances.

## Acknowledgments

This project was supported in part by a grant of supercomputer resources provided by the DoD High Performance Computing Modernization Program (HPCMP) and NAVAIRs ILIR program sponsored by ONR.

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