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Abstract
A rapid and efficient all-optical method for forming propagation invariant shaped beams by exploiting the optical feedback of a laser cavity is presented. The method is based on the modified degenerate cavity laser (MDCL), which is a highly incoherent cavity laser. The MDCL has a very large number of degrees of freedom (320,000 modes in our system) that can be coupled and controlled, and allows direct access to both the real space and Fourier space of the laser beam. By inserting amplitude masks into the cavity, constraints can be imposed on the laser in order to obtain minimal loss solutions that would optimally lead to a superposition of Bessel-Gauss beams forming a desired shaped beam. The resulting beam maintains its transverse intensity distribution for relatively long propagation distances.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Diffraction causes light beams to diverge with propagation, thereby placing a limit on the depth of focus where the transverse intensity distribution remains essentially constant. Over the past decades several methods have been proposed to overcome this limit [1–6]. One of these methods involves beams whose transverse field distributions can be described by Bessel functions [1,2]. Such Bessel beams can maintain their transverse intensity distributions, independent of propagation distance. Moreover, any superposition of such beams is propagation invariant as well.
Several Bessel beams were superimposed to form a shaped beam with a complex transverse intensity distribution that remains invariant as they propagate [7,8]. For example, propagation invariant beams with random intensity distributions (speckles) were achieved by illuminating an optical diffuser with coherent light and applying an annular Fourier filter [9]. Similar results were later achieved by replacing the diffusers with a liquid crystal display of random patterns [10]. Also, propagation invariant arrays of intensity spots were demonstrated by using a diffractive element and a thin annular aperture [11]. More general arbitrary shaped propagation invariant beams were demonstrated by using an amplitude mask and an annular Fourier filter [8], as well as by using a spatial light modulator [12,13]. Unfortunately, in these methods the light efficiency is relatively low and/or the time for forming the shaped beam is long.
In this paper we present a method to efficiently and rapidly form propagation invariant shaped beams with a modified degenerate cavity laser (MDCL), which has an extremely large number of degrees of freedom that can be coupled and controlled [14–16]. Specifically, we show that with intra-cavity amplitude masks it is possible to force the MDCL to rapidly find minimal loss solutions of a desired transverse intensity distribution as superposition of Bessel-Gauss beams. Consequently, due to the unique propagation properties of such beams, the desired laser output would maintain its transverse intensity distribution over relatively long propagation distances.
2. Method and experimental arrangement
The field distribution of a superposition of Bessel-Gauss beams can be described as
where Jn is a Bessel function of the first kind of order n, r and θ are the radial and azimuthal coordinates, kz and kr are the longitudinal and radial wavevectors, k = (kz2 + kr2)1/2 = 2π/λ with λ the wavelength, and An is a complex weight for each Bessel distribution that is tailored to form the desired transverse shape of the output beam and w0 is the width of the Gaussian envelope. Notice that all Bessel-Gauss beams in the sum have the same value of kr, and therefore they do not constitute a complete basis. Consequently, it is not likely that there is a set of weights An which would accurately result in the desired transverse intensity distribution. Generally, it is only possible to find a superposition of Bessel-Gauss beams that would approximate the desired intensity distribution. Finding the weights An that would provide the best match to the desired intensity distribution is a difficult and time consuming task [13]. In the following, we present a method that rapidly and efficiently finds the An weights, by resorting to optical feedback in the MDCL.The MDCL arrangement and typical beam distributions at the Fourier plane and output plane are presented in Fig. 1. Figure 1(a) schematically shows the experimental MDCL arrangement. It is comprised of an 11cm long Nd:YAG gain medium, a high reflective mirror, a partially reflective output coupler mirror and two lenses in a 4f telescope arrangement. The 4f telescope inside the cavity assures that any field distribution is accurately imaged onto itself after every round trip, so any field distribution is an eigenmode of the cavity. In our experiments, the gain medium was pumped by a double Xenon flash lamp with quasi-CW 100µsec long pulses at a repetition rate of 1Hz, so as to avoid deleterious thermal effects. The focal length of each of the two lenses was f = 25cm, and the diameter of the gain medium was 9.5mm, much larger than the diffraction spot size of the telescope. Consequently, the cavity could support a large number of transverse modes (320,000 modes [14]). As all modes are accurately degenerate in loss, they all lase simultaneously, despite mode competition.
Fig. 1 Experimental arrangement and results. (a) Experimental arrangement of the modified degenerate cavity. (b) Experimental intensity distributions of the MDCL at the Fourier plane for circular or annular apertures. The inset shows a radial crossection of the intensity distribution along the dashed white line. (c) Experimental intensity distributions at the output plane of the MDCL for circular or annular apertures.
One of the main advantages of the MDCL is that it allows direct access to both the real space and Fourier space of the laser beam. To exploit this advantage and generate propagation invariant shaped beams, we added two amplitude masks inside the cavity, as shown in Fig. 1(a). One mask was placed near the back mirror (Mask 1), and its image was formed at the surface of the gain medium adjacent to the output coupler mirror. Due to the 4f telescope, any diffraction limited spot on one of the mirrors is independent of any other, and consequently the entire region within Mask 1 lases. Accordingly, Mask 1 determines the transverse intensity distribution of the output beam, up to a scaling ratio that depends on the magnification of the 4f intra-cavity telescope (which was equal to unity in our cavity). A second mask was placed at the Fourier plane between the two lenses (Mask 2), and served as a spatial filter in Fourier space that introduces loss to undesired modes. Since the filtering is done at the Fourier plane, far from the gain medium, it has a negligible effect on the output power. Indeed, the number of lasing transverse modes in a modified degenerate cavity laser (MDCL) can be varied over five orders of magnitude, while the output power is reduced by only a factor of two [14]. Moreover, such an intra-cavity Fourier filter can affect the functional form of the spatial coherence [15]. In our experiments, Mask 2 controlled the numerical aperture of the system and therefore it determined the propagation properties of the output beam.
Propagation invariant shaped beams can be formed in the MDCL by letting Mask 2 be an annular aperture. Such an aperture ensures that any lasing beam is composed of k-vectors all lying on a cone. It is well known that the representation of a Bessel beam of order n in Fourier space is an annulus with azimuthal phase exp(inθ). Consequently, the MDCL with an annular aperture at the Fourier plane generates a beam that is a superposition of Bessel beams, and is expected to be propagation invariant.
To obtain propagation invariant shaped beams, it is necessary to control both the shape of the output beam as well as its Fourier components. In our experiments we placed an amplitude mask in the form of the letter “R” at the back mirror (Mask1), and an annular aperture at the Fourier plane between the lenses (Mask2). The intensity distributions at the Fourier and output planes are shown in Fig. 1(b-c), along with a comparison to the intensity distributions when a circular aperture, rather than an annular aperture, is used.
The circular aperture was 2.5mm in diameter. The annular aperture was obtained by blocking the central part of the circular aperture with a 2.1mm metallic disk, such that the annular aperture had a mean diameter of 2.3mm and width of 0.2mm. Although the metallic disk blocked 95% of the energy, the output power was reduced by only 30%, demonstrating the ability of the laser to minimize loss, even under such strenuous conditions as blocking of the central region (DC order).
3. Results
Figures 1(b-c) show the detected intensity distributions at the Fourier and output planes of the MDCL, for a circular and annular apertures in the Fourier plane (Mask 2). As evident, the output beam closely resembles the desired beam (i.e. the shape of Mask 1, as shown in Fig. 1(a)) for both the circular and annular apertures, although the similarity is reduced for the annular aperture. This reduction is expected, because the MDCL with the annular aperture can only find an approximate solution which satisfies the constraints of both the shaped mask in the form of an “R” (Mask 1) and the annular aperture (Mask 2).
For the case of an annular aperture, Fig. 1(b) shows that the intensity distribution at the Fourier plane is also annular. To further characterize the expected propagation properties of the output beam, we analyzed the beam divergence in the radial direction at the Fourier plane. Specifically, we considered radial crossections of the Fourier plane intensity distribution at different propagation distances z’ (z’ is the propagation distance from the Fourier plane, see Fig. 1(a)); a typical radial crossection is shown in the inset of Fig. 1(b) for z’ = 0. For each propagation distance z’ we calculated the average width of 200 radial crossections of the annular shaped intensity distribution. From these we plotted the average radial crossection width as a function of propagation distance z’ (not shown), and determined the beam quality factor M2 = 1.04 along the radial direction. This results indicates that at the Fourier plane there is a single transverse mode along the radial direction (however the beam is multimode in the azimuthal direction). Consequently, the lasing modes can be well approximated by a superposition of Bessel-Gauss beams [17], namely a superposition of Bessel beams with Gaussian envelope.
Next, we determined the overlap integral C of the desired and the actually measured output intensity distributions as a function of propagation distance z, when using the circular or annular apertures. The overlap integral is defined as
where Im(x,y;z) and Id(x,y;z) are the measured and desired intensity distributions at propagation distance z (the desired intensity distribution is the shape of mask 1 (see Fig. 1(a)), and integration is taken over the entire detected image. Notice C(z) is normalized, such that it varies between 0 to 1. Clearly, if the output beam diffracts strongly even after a short propagation distances, C(z) will decay rapidly; whereas if the output beam maintains its original transverse intensity distribution after long propagation distances, C(z) will decay slowly. The measured results for C(z) are presented in Fig. 2 for circular and annular apertures, together with representative detected intensity distributions at different propagation distances. As evident, with the circular aperture the overlap at z = 0 is high, but then it quickly deteriorates after a relatively short propagation distance. With the annular aperture, on the other hand, the overlap is lower at z = 0 but is maintained for relatively long propagation distances, indicating that the shape of the output beam will remain essentially the same over this distance. The insets show detected images of the output at different propagation distances, for both the circular and annular apertures.Fig. 2 Experimental and simulation results for diffraction of the MDCL’s output beam as a function of propagation distance. (a) Detected intensity distributions at the output, and after propagation distances of 10mm, 20mm and 30mm, for circular and annular apertures. (b) Experimental and simulation results for the overlap integral of the desired and laser output intensity distributions as a function of propagation distance. Solid lines denote calculated results for a circular (blue) and annular (red) apertures, Blue squares denote measured results obtained with a circular aperture, and red circles denote measured results obtained with an annular aperture.
The maximal propagation distance zmax, over which a Bessel-Gauss beam is expected to maintain the same transverse intensity distribution, is determined by the width of the Gaussian envelope w0, normalized by the propagation angle of the Bessel beam [17],
Note, the width of the Gaussian envelope of a single Bessel-Gauss beam w0, is inversely proportional to the width of the annular aperture, and kr is proportional to the mean diameter of the annular ring. Therefore, the maximal propagation distance zmax is inversely proportional to both the width and mean diameter of the annular aperture.For the parameters used in our system, zmax = 37mm, in reasonable agreement with the experimental results shown in Fig. 2(b). Similarly, the beam generated with the circular aperture of diameter D, can be considered as a superposition of Airy disks, and can be well approximated by a superposition of narrow Gaussian beams of width w0 = 0.59λf/D. For the parameters in our system, w0 = 63µm. The beam generated with the circular aperture is thus expected to maintain its transverse intensity distribution over a distance zmax = πw02/λ = 12mm, in agreement with the experimental results of Fig. 2(b).
With the annular aperture, there is an inherent tradeoff between resolution and the maximal distance zmax over which propagation invariance holds. Specifically, larger annular apertures (i.e. larger kr) increase the numerical aperture of the system, and therefore better resolution is obtained, albeit at the price of reduced zmax. To demonstrate this, we placed a US Air Force resolution target as Mask 1 near the back mirror of the MDCL, and compared the propagation properties of the output beams for two annular apertures, each with a different mean diameter. The first annular aperture had a mean diameter of 1.8mm and the second a mean diameter of 2.8mm, and the width of both annular apertures was 0.6mm. Accordingly, the maximal diameters of these apertures were 2.1mm and 3.1 mm, respectively.
The results are presented in Fig, 3. Figure 3(a) shows the maximal resolution as a function of propagation distance for both annular apertures. The maximal resolution is defined as the inverse of the minimal pair line spacing on the resolution target, for which the visibility is higher than 1/e. Generally, the resolution at the output plane is determined by the highest k-vector (kr = πD/λf), and is therefore expected to decrease linearly with the diameter of the annular aperture (independently of the width of the annular aperture). Indeed, from the data of Fig. 3(a) we found that the ratio between the maximal resolution at z = 0 for the small and large annular apertures is 7.13/11.3 = 0.63, close to the ratio between the diameters of the annular apertures, 1.8/2.8 = 0.64. As expected, with the larger aperture the resolution over a short propagation distance is better, but then it deteriorates fast. The propagation distance zmax is 33mm for the large aperture, and 49mm for the small aperture. The ratio between zmax for the small and the large annular apertures is 33/49 = 0.67, close to the ratio between the diameters of the annular apertures, as predicted from Eq. (3). Figure 3(b) shows the desired intensity distribution, obtained by illuminating a resolution target outside the laser cavity, as well as the detected intensity distributions at the output plane (z = 0) for a circular aperture, and for a large and small annular apertures. As expected, the laser with the large annular aperture has better resolution than with the small annular aperture, but still not as good as with the circular aperture. Of course, with the circular aperture there is no propagation invariance.
Fig. 3 Tradeoff between resolution and maximal distance for propagation invariance. (a) Maximal resolution as a function of propagation distance zmax, for annular apertures with mean diameters of 1.8mm (maximal diameter of 2.1mm, blue curve with circles) and 2.8mm (maximal diameter of 3.1mm, red curve with squares). (b) The desired intensity distribution (first from the left), obtained by illuminating a resolution target outside the cavity, and the intensity distribution obtained in the MDCL laser output (z = 0), when placing at the Fourier plane a large annular aperture (second from the left), a large annular aperture (second from the right) and a small annular aperture (first from the right).
To gain better understanding of the underlying mechanism for generating propagation invariant shaped beams, we performed several simulations, which yielded a field matrix that describes the output of the MDCL. We began by running a Gerchberg-Saxton (GS) algorithm [18], similar to that used in Ref [13]. In this algorithm the initial field has randomly distributed phases and uniform amplitude at every transverse point on one side of the cavity. The field repeatedly propagates through the cavity until it finally reaches a steady state solution. After every round trip, the phases of the propagating field are taken as inputs for the next round trip, while the amplitudes are replaced with the square root of the desired intensity distribution.
We then compared this basic algorithm to variants of the Fox-Li simulation [19], which better simulate the lasing mechanism. In these simulations, the initial field is generated by distributing random phases and amplitudes to the field at every transverse point on one side of the cavity, and repeatedly propagating the field through the cavity until finally reaching a steady state. In every round trip, the simulation field amplitudes were normalized to account for loss and gain and to prevent numerical instabilities. In the simulations, we used two methods to normalize the field amplitudes. In one method, a cold-cavity was simulated by requiring that the total energy be conserved in the cavity after every round trip, namely the field was multiplied after every round trip by a factor I0/∑Ii,j, where I0 is a normalization constant of the total energy in the cavity, Ii,j the simulated intensity at point (i,j) and the summation is taken at the output plane of the cavity. This method is expected to converge to the cavity eignenmode that has minimal loss. In the other method, an active-cavity was simulated by normalizing the field at each point (i,j) on the gain medium by a factor of G/(1 + Ii,j/Isat), where G is the gain factor,, and Isat the saturation intensity; in our simulations, G = 5 and Isat = 2000.
The modified Fox-Li simulations yielded the results shown in Fig. 4. Figures 4(a-b) show the simulated intensity distribution at the output plane of the MDCL for just one realization of the GS cold-cavity simulations. As evident, the simulated intensities differ significantly from the desired output intensity distribution of Fig. 1(a) (shape of mask 1), indicating poor overlap between the two. In practice the MDCL supports many independent longitudinal modes, whereby the output beam is composed of many independent lasing realizations. Accordingly, we repeated the simulations 100 times, with different random initial conditions, and summed their intensity distributions (ensemble averaging). The result is shown in Figs. 4(d-e). As evident, such a summation significantly improves the overlap with the desired transverse intensity distribution. Finally, Figs. 4(c, f) show the simulated intensity distribution, calculated from an incoherent sum of 1 and 100 realizations in an active-cavity. As evident, the effect of the gain saturation further improves the overlap.
Fig. 4 Calculated output intensity distributions from the MDCL using Gerchberg-Saxton (GS) and modified Fox-Li simulations. (a, d) Intensity distribution of 1 and 100 realizations of a GS simulation. (b, e) Intensity distribution of 1 and 100 realizations of a cold cavity simulation. (c, f) Intensity distribution of 1 and 100 realizations of an active-cavity simulation.
The advantage of gain saturation in obtaining better overlap with the desired profile can be readily explained. In the cold-cavity simulation and near steady-state, the total intensity is relatively constant and does not vary substantially from one round trip to another. Therefore, in the regime Ii,j~Isat, the active-cavity simulation converges to the cold-cavity simulation near threshold. However, in the regime Ii,j~Isat, the gain may vary from one point to another, enabling an additional degree of freedom for the simulated field to comply with the constraints of both Mask 1 and Mask2. With an active-cavity, the field at spatial points with low intensities will be amplified, while that at spatial points with high intensities will be suppressed. As a result, a better overlap with the desired intensity distribution is obtained.
Based on the active-cavity simulations, we calculated the expected overlap integrals at the output of the MDCL for both circular and annular apertures. The results are presented in Fig. 2(b). As evident, the calculated simulation results agree well with experimental results for small propagation distances (z≤15mm). However, there are discrepancies for large propagation distances, which we attribute to the fact that the camera did not collect the entire diffracted beam.
4. Conclusion
We demonstrated a novel method for generating propagation invariant shaped beams, using a MDCL. The MDCL supports many transverse modes and selects the minimum loss solution that satisfies the constraints of the amplitude masks placed inside the cavity. With an annular aperture at the Fourier plane inside the cavity, the laser finds a superposition of Bessel beams that result in an optimal overlap with the desired transverse intensity distribution. In this sense, the lasing process performs a computational task [20, 21]. Since the method is based on all-optical feedback, it is extremely efficient as compared to other Fourier filtering methods [8-9], and extremely rapid as compared to iterative computer algorithms [13]. For example, the buildup time of the lasing mode in a Q-switched laser is in the nanosecond regime. As we showed, the ensemble averaging of intensity distributions, which results from the many longitudinal modes in the cavity and active feedback, improved the overlap between the intensity distribution of the output beam and the desired intensity distribution.
Funding
Israel Science Foundation; the Israel Ministry of Industry; Israel-US Binational Science foundation.
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