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Abstract
A relatively simple technique for coupling lasers in an array is presented. It is based on the insertion of an intracavity optical element in the far-field plane of a degenerate cavity laser that is used to form an array of lasers. We show that it is possible to control the selection of the lasers to couple regardless of the array geometry. An intracavity spherical lens in the far-field plane is numerically and experimentally investigated and the results compared with those from the more complicated Talbot diffraction for coupling lasers. With an intracavity cylindrical lens in a two dimensional square array geometry, it is possible to obtain controlled one-dimensional coupling, and with an intracavity binary phase element it is possible to obtain versatile couplings.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Coupling of lasers plays a significant role in many basic and applied investigations such as finding the ground state solution of complicated potentials landscape [1,2], studying dissipative topological defects [3], mapping the classical XY model [2,4] and solving hard computational problem [1,5]. As a result, many different techniques for coupling lasers have been developed. They involve evanescent waves [6], anti-guided waves [7], manipulations of mirrors [8] and Fourier and fractional Talbot diffractions [9–12]. Unfortunately, coupling with these techniques cannot be readily controlled, because they often suffer from spurious and undesired coupling, and usually depend on the specific geometry of the lasers in an array. This is particularly true for an array of lasers formed in a degenerate cavity laser (DCL) where the coupling between individual lasers is done with Talbot diffraction and/or Fourier diffraction. With Talbot diffraction, coupling is generally between nearest neighbor lasers, whereas with Fourier diffraction coupling is between all lasers and achieved with an amplitude filter (typically an aperture) [10,13]. Fractional Talbot diffraction was exploited to efficiently couple up to $1500$ lasers formed in a DCL, where it was found that the coupling between the lasers depends on the geometry of the array [2,3,10,14].
In this article, we present a new technique where the coupling of lasers in an array can be controlled regardless of the geometry of the array. Such control is achieved with intracavity elements that are inserted in the Fourier plane (far-field). Unlike amplitude filters in Fourier diffraction coupling, which only control the amplitude distribution [13], our intracavity elements control both the phase and the amplitude distributions. With an intracavity spherical lens, we experimentally and numerically show that the results are similar to those obtained with the Talbot diffraction, but with a more robust and easier to control coupling. When a cylindrical lens replaces the spherical lens, we show that is possible to obtain one-dimensional coupling at any desired orientation even for two-dimensional arrays of lasers. Finally, we show that with an intracavity binary phase element (BPE), it is possible to obtain versatile control of the coupling.
2. Degenerate cavity laser for forming and coupling lasers in an array
In our investigations, we used a degenerate cavity laser (DCL) [14] to form an array of lasers [2,3,10]. Three DCL arrangements are presented in Fig. 1. The most basic arrangement, shown in Fig. 1 (a), is comprised of a Nd:YAG gain medium (to obtain lasing at $\lambda =1064nm$), a flat back mirror with high $99.5\%$ reflectivity, a flat front mirror with $80\%$ reflectivity (output coupler), two Fourier lenses in a $4f$ telescope configuration and a mask of holes adjacent to the output coupler to form an array of lasers. To reduce the effects of thermal lensing, optical pumping was provided by quasi-CW $100\mu s$ pulsed flash lamps operating at $1Hz$. The $4f$ telescope configuration images the back mirror plane onto the output coupler plane ensuring that any hole in the mask is precisely imaged onto itself after a single round trip. Accordingly, when lasing, each hole in the mask leads to the formation of an independent laser. In our experiments, the focal length of the telescope lenses was $f=20cm$, their diameter $5.08cm$ and the diameter of the gain medium $0.95cm$. Accordingly, since the cross section of the gain medium is very large compared to the diffraction limited spot size of the telescope, a large number of lasers and a large number of transverse modes were obtained with the DCL [15].
Fig. 1. Degenerate cavity laser (DCL) arrangements. (a) Basic arrangement for forming an array of lasers. (b) Coupling of the lasers by means of fractional Talbot diffraction. (c) Coupling of the lasers by means of a lens with focal distance $f'$ in the far-field plane (Fourier plane at focal distance $f$ from Lens $1$ and Lens $2$). The lasers in (a) are uncoupled, while the lasers in (b) and (c) are coupled. O.C: Output coupler.
The configuration geometry of the mask of holes, adjacent to the output coupler and denoted as the near-field, establishes the geometry of the array of the lasers. In our investigations, the mask was a square array of holes each with a diameter of $200\mu m$ and a period between them $a=300\mu m$. The field at the Fourier plane between the two lenses of the telescope is denoted as the far-field and is the Fourier transform of the near-field [16].
Figure 1 (b) shows the DCL arrangement that includes fractional Talbot diffraction commonly used to couple lasers [2,3,10]. Here, the output coupler is shifted from its original position by a quarter of the Talbot distance $Z_{T/4}$. Since the light travels back and forth, the light from the mask returns back to the mask by half of the Talbot distance $Z_{T/2}$. In general, the image of a periodic array that is illuminated with an incident plane wave laser field is repeated at specific fractions of the Talbot distance [17,18], where the Talbot distance $Z_{T}$ is defined as $Z_{T}=\frac {2a^{2}}{\lambda }$ from the periodic array, with $a$ the period of the array and $\lambda$ the wavelength of the illumination [18]. The self-image of the array is phase-shifted by half of the array period $a/2$ at half of the Talbot distance. Thus the light from the periodic array of lasers interferes with light of the phase shifted image, inducing coupling between nearest neighbor lasers. Next nearest neighbor coupling was also observed, but weaker than nearest neighbors coupling [2]. Typically, the geometry of the array influences the coupling between the lasers. The fill factor of the array also influences the coupling. Specifically, a smaller fill factor increases the coupling as well as the loss [19]. Note, since the Talbot distance has quadratic dependence on the period of the array, it is difficult to couple lasers in an array whose period is above $1mm$, whereby $Z_{T}\approx 2m$ at $1064nm$.
Figure 1 (c) shows a new and relatively simple DCL arrangement with an intracavity element inserted in the far-field plane (Fourier plane). When the optical element is a spherical lens (e.g. Lens $3$ with focal length $f'$), a spherical phase is added, that could have the same effect of free space propagation with Talbot diffraction over a distance $Z_{T/4}$ from the near-field when the focal length $f'$ (derivation in Appendix) is :
where $f$ is the focal length of the telescope lenses (Lens $1$ and Lens $2$). In our experiments $Z_{T/4}=4.2cm$, so Eq. (1) yields $f'=1m$.The experimental near-field and far-field intensity distributions for the three different DCL arrangements of Fig. 1 are presented in Fig. 2. The near-field intensity distributions are quite similar, all resulting from the same mask array, where each hole of the mask corresponds to an independent laser. The near-field in Fig. 2 contains $\approx 250$ lasers, where the fill factor in the near-field is $\approx 1/5$. Such a fill factor and corresponding loss are readily supported by the degenerate cavity laser. Note that in Fig. 2 (a) the intensities of all the lasers in the array are essentially the same, but those in Figs. 2 (b) and 2 (c) are lower when away from the center. This is mostly due to edge effect: the lasers located near the edges (away from the center) cannot coupled to all its neighboring lasers so they suffer from higher loss compared with lasers near the center [10].
Fig. 2. Experimental and simulated near-field and far-field intensity distributions in the three different DCL arrangements shown in Fig. 1. No significant differences are observed in the near-field intensity distributions of the three arrangements, whereas differences are observed in the far-field intensity distributions. The far-field intensity distributions in the top row (a) indicate that there is no coupling between individual lasers, whereas in the lower rows (b) and (c) there are couplings, which are essentially identical.
The experimental and simulated far-field intensity distribution in Fig. 2 (a) are broad Gaussian distributions, corresponding to uncoupled lasers [2,10]. The far-field intensity distributions in Fig. 2 (b) correspond to phase-locked coupled lasers ( out of phase coupling) for a square array [10]. Around $120$ lasers are present in the near-field intensity distribution, where most, if not all of them, are phase-locked [10]. The far-field intensity distributions in Fig. 2 (c) also correspond to phase-locked lasers, similarly to the results in Fig. 2 (b). These results indicate that the couplings in the arrangements in Figs. 2 (b) and (c) are essentially equivalent. For the simulations, we resorted to an algorithm that combines the Fox-Li [20] and the Gerchberg-Saxton [21] algorithms [10]. As evident in Fig. 2, there is good agreement with the experimental results.
3. Controlled one-dimensional coupling with a cylindrical lens
Our results are similar to those of the Talbot diffraction, but they also offer control on the selection of the lasers to couple. For example, it is possible to obtain one-dimensional coupling in a two-dimensional square array of lasers. This was achieved by replacing the spherical lens, in Fig. 1 (c) with a cylindrical lens, whose focal length $f'$ was determined by Eq. (1). Figure 3 shows the experimental and simulated near-field and far-field intensity distributions of the DCL arrangement with a cylindrical lens in the far-field plane.
Fig. 3. Experimental and simulated near-field and far-field intensity distributions for the DCL arrangement shown in Fig. 1 (c), with an intracavity cylindrical lens in the far-field plane, at different angular orientation. Two different lenses were used, one with focal length of $f'=1m$ to couple lasers vertically and horizontally and the other with $f'=0.5m$ to couple lasers diagonally. As evident, the coupling is along one coordinate. By rotating the cylindrical lens, it is possible to select which lasers are coupled.
As indicated by the stripes patterns in the far-field intensity distributions, the cylindrical lens only couples lasers in one dimension. Around $15$ lasers along one dimension are present in the near-field intensity distribution, most of which are phase-locked. By rotating the cylindrical lens, it is possible to choose the orientation of the lasers to couple. Note that for an angular orientation at $0^{\circ }$ the lasers are coupled vertically, whereas for an angular rotation of $90^{\circ }$ they are coupled horizontally.
Coupling the lasers diagonally is equivalent to the next nearest neighbors coupling in a square array. With an angular orientation of $45^{\circ }$ or $135^{\circ }$, the period of the lasers (i.e. distance between them) becomes $\sqrt {2}a$, so the focal length of the cylindrical lens was reduced to $f'=0.5m$. As evident in Fig. 3, it is also possible to phase-lock lasers diagonally along one dimension with a cylindrical lens, thereby obtaining next nearest neighbors coupling.
4. Controlled and versatile coupling with BOE
We also investigated intracavity binary phase elements (BPEs) for controlling the coupling and selecting the lasers to couple. Specifically, we replaced the far-field spherical lens with a BPE in Fig. 2 (c). A typical BPE can introduce a $0$ or a $\pi$ phase shift to different areas of an incident light beam. It was used in the past as a beam combiner to actively phase-locked an array of fiber lasers [22]. When placed in the far-field plane of a DCL, a BPE imposes an intensity distribution, which can be used for controlling coupling of lasers in an array.
Representative experimental far-field intensity distributions from a DCL with different BPEs are shown in Figs. 4 and 5. Figure 4 shows three intensity distributions with three different BPEs whose centers were aligned along the axis of the DCL. Figure 4 (a) shows the results for a BPE comprised of two sectors, one introducing $0$ phase shift (white sector) and the other $\pi$ phase shift (black sector).
Fig. 4. Representative experimental far-field intensity distributions for the DCL arrangement shown in Fig. 1 (c), with different intracavity binary phase elements (BPEs) centered along the axis of the DCL.
Fig. 5. Representative experimental far-field intensity distributions for the DCL arrangement shown in Fig. 1 (c), obtained with an eight sectors BPE at different angular orientations and/or different displacements from axis of DCL.
As evident, the far-field intensity distribution corresponds to one-dimensional array of horizontally coupled lasers. By changing the angular orientation of the BPE, it is possible to change the direction of coupling. Figure 4 (b) shows the results for a BPE comprised of four sectors, two introducing $0$ phase shift and the other two a $\pi$ phase shift. Here, the far-field intensity distribution, with dark light in its center, corresponds to two-dimensional square array of coupled lasers (in out-of-phase solution). Figure 4 (c) shows the results for a BPE comprised of eight sectors, four introducing $0$ phase shift and the others $\pi$ phase shift. As evident, the far-field intensity distribution is more complicated and seems to correspond to a ring of coupled lasers [3]. There is a singularity in the center of the BPE, where due to loss minimization (mode competition) there is no lasing. Accordingly, when the BPE is centered along the axis of the DCL, an out-of-phase solution (negative coupling), with zero light in the center, is the most favorable. Specifically, around the center, the light is distributed according to the eight sectors of the BPE, as a ring of $8$ coupled lasers having a radial distribution.
We also performed experiments to illustrate that specific BPEs can provide versatile control on the coupling so as to obtain different far-field intensity distributions. Some representative illustrations with a BPE comprised of eight sectors are shown in Fig. 5. Each intensity distribution was obtained for a different displacement and/or angular orientation of the BPE from the axis of the DCL. When displacing the BPE from the axis, there is no longer symmetry in the far-field. For each round-trip, the intracavity light passes twice through the BPE, and is rotated by $180^{\circ }$ by the DCL. Then, the total phase shift introduced by the BPE to the intracavity light is the sum of the phase of the BPE and the phase of the BPE after $180^{\circ }$ rotation. By controlling the position of the BPE in Fig. 5, it is possible to control the total phase shift introduced to the intracavity light, thereby the coupling. The intensity distribution in Fig. 5 (a) corresponds to that from a ring of coupled lasers, where in our experiment the BPE was centered along the axis of the DCL. The distribution in Fig. 5 (b) corresponds to that from an array of uncoupled lasers, where in our experiment the BPE was displaced from the axis so the intracavity light will pass through a single sector. The distribution in Fig. 5 (c) corresponds to that from a vertical one-dimensional array of coupled lasers, where in our experiment the BPE was displaced so that the intracavity light will pass through two adjacent sectors above and below the horizontal edge between them. The distribution in Fig. 5 (d) corresponds to that from a square array of coupled lasers, where in our experiment the BPE was slightly displaced both horizontally and vertically. The distributions in Fig. 5 (e) and Fig. 5 (f) correspond respectively to that from a triangular (or Kagome) and a cross array of coupled lasers, where in our experiment the BPE was displaced both horizontally and vertically as well as rotated.
5. Concluding remarks
To conclude, we presented a new and simple technique for controlling the coupling of lasers arrays. In this technique, a degenerate cavity laser with a near-field mask formed an array of independent lasers and different intracavity optical elements in the far-field plane coupled the lasers. The results revealed that it is possible to obtain controlled coupling, independently from the array geometry. With an intracavity spherical lens, approximately $120$ lasers were phase-locked in the out-of-phase solution, identically to the phase-locking of lasers with Talbot diffraction. One-dimensional nearest-neighbors and next nearest-neighbors coupling were achieved for a two dimensional array of lasers by means of an intracavity cylindrical lens. We also demonstrated that it is possible to obtain versatile coupling control and a variety of far-field intensity distributions with intracavity binary phase elements. It should be possible to obtain arbitrary control of the coupling of lasers by designing specific intracavity elements. The controlled coupling can be useful for variety of applications, including studying temporal dynamics of phase-locking by using next-nearest neighbors coupling, investigating small world networks with specific shortcut connections, investigating hard computational problems with a variety of couplings and investigating chaos synchronization and crowd synchrony. More generally, any space-invariant coupling function that depends only on the position difference $(x_{2}-x_{1},y_{2}-y_{1})$ between the lasers can be obtained by placing an intracavity element, whose amplitude and phase are the Fourier transform of the coupling function, in the far-field plane.
Appendix
Derivation of Eq. (1): assuming that the gain media in Fig. 1 (b) only changes the intensity of the input field ($z=0$ in Fig. 1), i.e. $A_{b}(x,y)E(x,y,z=0)=E_{b}(x,y,z=z_{G})$ with $A(x,y)$ a scalar function, then the inverted input field after passing the $4f$ telescope ($z=4f$ in Fig. 1), is $E_{b}(x,y,4f)=A_{b}(x,y)E(-x,-y,0)$ [16](chapter $5$). The diffraction at quarter Talbot distance from the near-field ($z=4f+Z_{T/4}=z_{F}$) can be analytically approximated using the Fresnel approximation [16](chapter $4$) as:
Funding
Israel Science Foundation (ISF); United States-Israel Binational Science Foundation (BSF).
Acknowledgements
We wish to acknowledge the Israel Science Foundation and the Israel-U.S. Binational Science foundation for their support. We thank Assaf Voliovich for his help in the measurements.
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