## Abstract

A rectangular-core (ribbon) fiber that guides and amplifies a single higher-order-mode (HOM) can potentially scale to much higher average powers than what is possible in traditional circular-core large-mode-area fibers. Such an amplifier would require mode-conversion at the input to enable interfacing with seed sources that typically output TEM_{00} mode radiation and at the output to generate diffraction-limited radiation for end-user applications. We present the first simulation and experimental results of a mode conversion technique that uses two diffractive-optic-elements in conjugate Fourier planes to convert a diffraction limited TEM_{00} mode to the HOM of a ribbon fiber. Mode-conversion-efficiency is approximately 84% and can theoretically approach 100%. We also demonstrate a mode-converter system that converts a single HOM of a ribbon fiber back to a diffraction-limited TEM_{00} mode. Conversion efficiency is a record 80.5%.

© 2012 OSA

## 1. Introduction

Scalable, high average power lasers are needed for materials processing and defense systems. Scientific applications such as laser-based guide stars for astronomy, gravitational wave detection, coherent remote wind sensing and laser based particle acceleration could also benefit from high average power lasers with diffraction-limited output radiation. Considerable attention has been focused on fiber-based lasers and amplifiers due to their potential for high average power combined with high beam quality and efficiency, compactness, and reliability [1].

Ytterbium doped fiber lasers and amplifiers at 1 μm have recently made tremendous progress and have been scaled to the multi-kW average power level with diffraction-limited beam quality. These systems are based on large-mode-area (LMA) step-index and photonic crystal (PC) based fiber amplifiers. The typical approach to power scaling in these fiber amplifiers is to increase the core size in each successive amplifier stage (while reducing the numerical aperture to maintain single-moded radiation), since the thresholds of nonlinearities and facet-damage increase with increasing mode-field-diameter (MFD).

We have theoretically analyzed the limits to power scaling of these fiber amplifiers by considering thermal, non-linear, damage and pump coupling limits as well as the fiber’s MFD limitations [2]. Our analysis shows that if the fiber’s MFD could be increased arbitrarily, 36 kW of power could be obtained with diffraction-limited quality from a fiber laser or amplifier. This power limit is determined by thermal and non-linear limits that combine to prevent further power scaling, irrespective of increases in mode size. However, based on practical considerations for the fiber amplifier’s bend diameter, we have also found that there is a practical limit to the achievable mode size - and that for this MFD there is an optimum fiber length that results in a laser whose maximum output power is 10-20 kW with good beam quality [2]. This is the physical limit to scaling the power of conventional fiber lasers.

Our models show that by moving from circularly-symmetric waveguides to ribbon-like rectangular-core fiber waveguides [3] as in Fig. 1, the single aperture power limit can be raised from 10 - 20 kW to > 100 kW.

The ribbon fiber waveguide has a rectangular core with a high width-to-height aspect ratio. This waveguide is single-moded in the thin dimension (y) and multi-moded in the wide dimension (x). The fiber is coiled only in the y-direction. Since higher-order-modes (HOM) are less susceptible to bend loss and mode mixing [4], we choose to propagate a particular HOM (in the x direction) in the ribbon fiber. By selective excitation of the desired HOM [5] and prevention of coiling in the x-direction, it is possible to minimize excitation of other HOMs in the ribbon fiber. Further, the area of the waveguide and the mode’s effective area can then be scaled by simply increasing the waveguide width. These ribbon fiber amplifiers can thus guide a higher order mode with a larger effective area and therefore generate much higher output power than is possible in circular-core fibers.

Since most seed lasers output TEM_{00} radiation, mode conversion is required to launch this radiation into the ribbon fiber’s HOM. Similarly, since many applications require diffraction-limited radiation, the ribbon fiber’s output HOM radiation needs to be converted back to the TEM_{00} mode. Mode conversion has recently been accomplished for circularly-symmetric laser outputs via interferometric elements [6] and mode coupling in dual-core PC fibers [7]. However, these tend to be quite complex or inefficient as the mode number is increased. Another conversion approach based on diffractive optical elements (DOEs) has been implemented for a number of years, for example, to transform the annular beam of CO_{2} lasers [8] to a uniform spatial amplitude profile, albeit with non-flat phase.

In this field, Siegman, early on showed that for any near-field electric field profile with a purely real wave front but with regions of positive and negative sign, a single DOE in the form of a binary phase plate cannot improve beam quality as measured by the M^{2} criterion [9]. The resulting far-field profiles have energy in the side-lobes and that results in no change to the M^{2}. Subsequently it has been shown that a single DOE (phase plate) [10] and a continuous phase plate [11]- both when combined with spatial filters can improve the beam quality of a laser operating in a single higher-order-mode. However, in these approaches, the conversion efficiency suffers, as energy in the side-lobes is rejected.

In this paper, we demonstrate a mode-converter system that efficiently converts TEM_{00} radiation to the ribbon-fiber’s HOM and vice versa. Our work builds on mode-converter approaches that use two DOEs for coherent beam combining of phase locked arrays of VCSELS [12] and sparse laser arrays [13]. The approach taken by the Leger group [13] is especially promising because it offers the potential for nearly 100% theoretical beam combination efficiency. We propose to apply this scheme (which utilizes two phase plates - placed in two conjugate Fourier planes) in the context of mode-conversion to generate the ribbon fiber’s HOM. In Section II, we discuss mode-conversion between a TEM_{00} mode input to a HOM of the ribbon fiber. In Section III, we describe mode-conversion between a HOM of the ribbon fiber and the TEM_{00} mode. In Section IV we summarize and describe potential applications for our work.

## 2. Conversion from TEM_{00} to HOM

#### 2.1. Description of approach

Any mode converter that converts a TEM_{00} mode to the multi-lobed higher-order-mode of a ribbon fiber needs to redistribute the energy from the gaussian profile to multiple-lobes and make the phase across the profile flip between 0 and *π*. Accomplishing these two tasks necessitates two diffractive-optic-elements (i.e. phase-only plates, one for each task) as illustrated in Fig. 2.

The first plate steers and reshapes the gaussian profile to generate multiple, appropriately-sized lobes of the HOM fiber in the plane of the second plate. The second phase plate impresses the required phase across the wavefront to achieve lobes whose phase alternates between 0 and *π*. By placing the first plate at the waist of the input TEM_{00} mode, we ensure that incident beam’s phase-front is flat for convinience. The second phase plate is placed in a conjugate Fourier plane with respect to the first phase plate. This plane also corresponds to the ribbon-fiber’s near-field. Since the fields in these two planes are related via the Fourier transform, it is possible to use the Gerchberg-Saxton (GS) algorithm [14] to retrieve the phase profiles in the two planes and in a straightforward manner calculate the phase-plate profiles.

#### 2.2. Modeling

For the purposes of simulation and experiments we use a sample (7* ^{th}*) HOM of a ribbon fiber (i.e. a rectangular 5 μm × 50 μm silica core with 0.1 NA and circular cladding) which has 7 lobes along the width of the core. The phase of each lobe is 0 or

*π*. Since the waveguide is single moded in the y direction, in this dimension the eigenmode can be transformed to the appropriate size using cylindrical lenses. Using the GS algorithm in 1-D, we calculate the phase profiles of the two DOEs that transform a diffraction-limited gaussian profile to the ribbon-fiber’s HOM.

Figure 3 shows the input and target mode profiles that are fed into the GS algorithm as well as the retrieved phases on DOE 1 and 2. DOE 1 has a phase-excursion that spans approximately 1.5 waves. It steers and reshapes the gaussian profile to achieve multiple-lobes in the plane of DOE 2. DOE 2 makes the field’s phase profile flip between 0 and *π*. The GS algorithm is iterative and for each iteration, it produces a convergence metric corresponding to the normalized overlap integral between the constraint and evolving amplitude. In our numerical implementation of the GS algorithm, this convergence metric peaks at nearly 97%.

#### 2.3. Experimental demonstration

Figure 4 shows an experimental layout of the mode-converter system which uses two diffractive optic elements implemented with phase-only spatial light modulators (SLM). Figure 3 shows the phases impressed upon the inputs to the two computer-controlled SLMs (Boulder Nonlinear Systems, Model: P512-1064). The test-laser’s output is collimated and magnified to a size that is appropriate for incidence on the first SLM. The diffracted output is picked-off by using a right-angle-prism and imaged onto the second SLM. The diffracted output of SLM-2 is again picked-off with another prism. The resulting far and near field intensities are measured using a standard CCD camera (Gentec E-O, Model: Beamage CCD12).

Figure 5 shows the experimental results which depict the far-field and near-field intensities after SLM-2 measured in the two Fourier planes. We first calculate the respective field amplitudes by taking the square root of the measured intensities. The far-field amplitude (like the intensity) is two lobed since it is the Fourier transform of near-field amplitude with lobes whose phase is nominally-manipulated by SLM-2 to alternate between 0 and *π*. To verify that the phases indeed alternate between 0 and *π* and calculate the mode-conversion efficiency, we use these amplitudes and the GS algorithm to retrieve the corresponding phases in the two Fourier planes.

The retrieved phases are shown in Fig. 6. The 7-lobed mode’s electric field profile is calculated using the amplitude (calculated from measured intensity) and retrieved phase. The overlap integral of this field with the theoretical electric-field profile for the ribbon fiber’s 7* ^{th}* eigenmode is 84%. The conversion efficiency is limited to a certain extent because of the inaccuracy in determining the exact location of the Fourier-planes and the convergence of the GS algorithm. Pixelization also results in high-frequency ripples in the near-field and a reduction in conversion efficiency with respect to the value of 97% predicted by our-model.

## 3. Conversion from HOM to TEM_{00} mode

#### 3.1. Description of approach and setup

In this section, we present results for a mode-converter system that converts the ribbon fiber’s HOM output back to the TEM_{00} mode. Figure 7 illustrates the experimental setup that again utilizes two phase plates - placed in two conjugate Fourier planes - at the output of the fiber.

For test purposes, we use a low-power, single-frequency laser for high-angle excitation (albeit inefficiently) of the 7* ^{th}* HOM of the ribbon fiber. This fiber has a rectangular 5 μm × 50 μm silica core with 0.1 NA and circular cladding. The phase of each of 7 lobes is 0 or

*π*. The fiber’s output is magnified before impinging upon the first SLM. The diffracted output is picked-off by using a right-angle-prism and imaged onto the second SLM. The first SLM redistributes the energy from the multiple-lobes to a gaussian profile at the plane of the second SLM. The second SLM makes the phase across the profile uniform. Its output is again picked-off with another prism. The resulting far field intensity is measured using a standard CCD camera. The power in the central-lobe is measured by placing a power-meter behind a appropriately sized slit.

#### 3.2. Modeling

To calculate the required phase profiles, we feed the GS algorithm with an image of the 7-lobed mode at the input to SLM-1 and that of the desired TEM_{00} mode at SLM-2. With the retrieved phase profiles in the two planes, we calculate the SLM-1 and SLM-2 profiles. The GS algorithm is iterative and for each iteration, it produces an error signal corresponding to the difference in intensities between the constraint and evolving amplitude. We were unable to drive the error signal to nearly zero in our numerical model calculations. As a result, the retrieved phases and calculated SLM profiles have some degree of inaccuracy in them. To compensate for these errors, we choose to keep the phase on SLM-1 constant and implement a genetic algorithm (GA) [15] that evolves the phase profile on SLM-2 until the power through the slit in the far-field of SLM-2 is maximized. Figure 8 shows the resulting phase profiles that are implemented on SLM-1 and SLM-2.

#### 3.3. Experimental demonstration

Figure 9 shows the resulting intensity profiles at the input of SLM-2 and the far-field of SLM-2. We see that the intensity profile at SLM-2 has a number of ripples and does not represent an ideal TEM_{00} amplitude profile. In the laboratory, we notice that the amplitude of these ripples are time-varying as well. We attribute this to the following two reasons:

- The ribbon-fiber’s output consists potentially of a time-varying interference of the fiber’s eigenmodes. Our calculations suggest that the fraction of light in the 7
eigen-mode is around 85% or better. However, beating between this mode and other excited modes in the ribbon fiber could result in a time-varying phase at the input to SLM-1 and create time-varying ripples in the intensity profile measured at SLM-2.^{th} - Large phase jumps across pixels in SLM-1 due to phase wrapping could also cause ripples in the intensity at the input to SLM-2.

As a test, we implemented a GA on SLM-1 so as to optimize it better for achieving a smoother TEM_{00} amplitude profile at SLM-2. This effort was partially successful, in that we were able to reduce, but not eliminate, the magnitude of the intensity ripples at SLM-2. However, the GA-optimized SLM-1 phase had much higher magnitude-and-frequency phase transitions between pixels. Implementing this GA-optimized phase on SLM1, which has pixelation that causes the phase error to depend on the magnitude of the phase transitions between pixels, results in approximately 30% of the energy being thrown outside (i.e. outside the 4 times the 1/e^{2} half-width) of the Gaussian profile at SLM-2.

As a result of this experiment, we decided to retain the GS-derived phase on SLM-1 and implemented the GA-optimized phase on SLM-2. The phase errors in the TEM_{00}-like profile at SLM-2 are largely corrected and a smooth TEM_{00} mode is generated after passing through a slit in the far-field. By taking the ratio of the incident and transmitted powers through the slit, we estimate that the measured conversion efficiency is 80%. The beam-quality of the mode passing through the slits is nearly diffraction-limited and its
${\text{M}}_{x}^{2}=1.27$ and
${\text{M}}_{y}^{2}=1.07$.

In contrast, when we used the GS-derived phase on SLM-1 and SLM-2, we were only able to achieve about 65% conversion efficiency into the TEM_{00} mode in the far-field of SLM-2. Our calculations suggest that given the 15 μm pixels on SLM-2 and the beam-size at SLM-2, the conversion efficiency will be limited to approximately 82% due to lack of spatial resolution. The measured conversion-efficiency can be improved to the 90–95% level by expanding the beam-size at SLM-2 so that that residual phase error after SLM-2 is minimized. Further improvements towards the 100% level should be possible by generating a more-pure HOM in the ribbon-fiber and modifying the phase impressed by SLM-1 to generate a smoother amplitude profile at SLM-2.

## 4. Summary

In summary, we have discussed the design, simulation, and experimental results of a mode converter that takes the TEM_{00} mode output of a seed 1053 nm laser and converts it to the HOM of rectangular-core ribbon fibers. The DOE’s phase properties are derived by using the Gerchberg-Saxton algorithm. Phase retrieval and overlap efficiency calculations based on experimental measurement of the intensities in the fiber-facet’s near and far-fields show that the mode conversion efficiency is approximately 84%. We have also demonstrated mode-conversion between the HOM and TEM_{00} modes with a nearly-identical setup with 80% efficiency. Our analysis suggests that the conversion-efficiency can be improved to the 90-95% level. These results represent a key contribution in the technology of mode-converters based on phase-only modulation in the near and far-fields of the source for power scaling of ribbon fiber lasers and amplifiers.

As an extension, the mode-conversion technique described here can be applied to the output modes of a broad-class of slab, rod, and other gas lasers. Many side-pumped rod-based solid-state lasers have a significant amount of pump absorption along the edges of the crystal. Resonator designs that emphasize good beam quality output often leave a lot of the stored energy behind since the TEM_{00} mode doesn’t overlap as well with the pump absorption profile. Operating the resonator in a higher-order ”doughnut” mode might improve the overlap and extraction efficiency. This might increase the gain threshold for unwanted parasitic oscillations as well. In the end, it may turn out that mode-converters for some of these laser systems might benefit from some combination [16] of amplitude and phase modulation.

## Acknowledgments

This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. The authors would like to thank another colleague, Graham Scott Allen, for useful discussions during the course of this work.

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