Abstract: An efficient method for controlling the spatial coherence has previously been demonstrated in a modified degenerate cavity laser. There, the degree of spatial coherence was controlled by changing the size of a circular aperture mask placed inside the cavity. In this paper, we extend the method and perform general manipulation of the spatial coherence properties of the laser, by resorting to more sophisticated intra-cavity masks. As predicted from the Van Cittert Zernike theorem, the spatial coherence is shown to depend on the geometry of the masks. This is demonstrated with different mask geometries: a variable slit which enables independent control of spatial coherence properties in one coordinate axis without affecting those in the other; a double aperture, an annular ring and a circular aperture array which generate spatial coherence functional forms of cosine, Bessel and comb, respectively.
© 2015 Optical Society of America
The ability to control and manipulate the spatial coherence of light is useful in many applications . For example, it is well known that spatially coherent monochromatic light suffers from speckle noise, which strongly affects the signal to noise ratio and results in low image quality, e.g. in holography [2,3] and laser projection . By reducing the spatial coherence it is possible to suppress speckle noise and improve image quality. Also, the deleterious effects in optical propagation through random or turbulent media , and optical communication  can be alleviated by resorting to partially coherent light. In addition, light sources with controlled spatial coherence could be useful in applications which require tailored spatial correlations, such as in advanced optical coherence tomography .
Typically, incoherent beams are formed by incorporating random time varying phases or intensities into an otherwise coherent beam. This is usually done by means of either a moving optical diffuser [8–12] or a spatial light modulator [3,13,14]. Other techniques include moving a lens  or an aperture , rotating an aperture at the Fourier plane , controlling a device with micro-mirrors [17,18], rotating or vibrating multimode fibers [19,20], and rotating a magneto optic disk . Unfortunately, all these methods involve mechanically moving components, which are relatively slow and require long acquisition times. Other methods for obtaining partially coherent beams include a tunable laser source and either a spectrometer or a multimode fiber to obtain spatial displacement of the light beam [22,23], optical feedback effects , illumination through a media with temporal and spatial variations of the refractive index generated by ultrasonic waves [25,26], and binary phase masks at the Fourier plane [27,28]. Control of either local  or global [11,12,18,29] spatial coherence properties have also been demonstrated, using moving optical diffusers, SLMs or digital micro mirrors devices.
Recently, a rapid and very efficient method for spatial coherence control was introduced . This method is based on a degenerate cavity laser, in which a variable circular aperture is inserted (see Fig. 1). The aperture serves as a spatial filter that introduces loss to higher order modes. It has been shown that by varying the size of the aperture, it is possible to control the number of lasing modes over a range of five orders of magnitude, while the output power is reduced by less than a factor of two. This method does not involve slow moving elements, yet it allows great flexibility in tuning the degree of spatial coherence. It is efficient and does not reduce the level of temporal coherence. Here we extend the method by resorting to more sophisticated intra-cavity spatial filters (masks). Accordingly, we can obtain wider control and greater manipulations of spatial coherence properties. We start with a variable slit aperture mask, with which we show that it is possible to independently control the spatial coherence properties along one coordinate axis, without affecting those in the other coordinate axis. Then, we resort to more sophisticated intra-cavity masks and demonstrate laser sources with unique spatial coherence functional forms, such as cosine, Bessel and comb.
2. Experimental arrangements
The experimental arrangements of our method are schematically presented in Fig. 1. They include a modified degenerate cavity laser and three different arrangements for measuring and characterizing the coherence properties of the laser. As shown, the degenerate cavity is comprised of a Nd:Yag gain medium, a rear mirror with high reflection and a front mirror with 50% reflection, two lenses in a 4f telescope configuration that images the rear end of the cavity onto the front end of the cavity. The 4f arrangement assures that any initial field distribution will be imaged onto itself after a single round trip, and therefore any field distribution is an eigenmode of the cavity. The cross-section area of the gain medium is large (in our experiments it was a rod of 1 cm in diameter), such that it supports many transverse modes [30–32]. This basic degenerate cavity arrangement is modified by inserting a binary amplitude mask inside the cavity at the Fourier plane between the two lenses, which serves as a spatial filter. In the extreme case where all modes are equally transmitted through the mask, they will all lase simultaneously. Accordingly, the cavity fulfils the degeneracy condition. However, if some modes are blocked or partially blocked and suffer from loss, the degeneracy would be lifted. Due to mode competition, modes with greater loss would not lase, and consequently spatial coherence of the output beam would vary.
In our experiments the coherence properties of the laser output were characterized by three different methods. In the first method the spatial coherence was quantified using the complex coherence factor μ, defined as Fig. 1(a). In one arm the output was imaged directly onto a CCD camera, and in the second arm it was imaged onto a small 50 µm pinhole aperture, that served as a selected reference point for the complex coherence factor. Light transmitted through the pinhole aperture was collimated and expanded so that the size of the resultant beam at the CCD camera would overlap the beam from the first arm. A slight angle wasintroduced between the two arms, so as to form a fringe pattern which was detected by the CCD. The visibility of the fringes as a function of space, v(x,y), indicates the degree of coherence at every point (x,y) with respect to the selected point (x’,y’), according to
Recall that for a fully incoherent source, the Van Cittert Zernike theorem  indicates that
In the second method, the number of modes was determined by measuring the beam quality factor M2 of the output beam, with the arrangement shown in Fig. 1(b). The beam quality was calculated as the product of the waists of the near field Wnf and the divergence (waist of the far field over the focal length) Wff/f of the output beam, normalized by that of a diffraction limited beam λ/π. In 1D the M2 is equal to the number of modes N, and in 2D it is necessary to calculate Mx2 and My2 separately before finding the total number of modes N = M4 = Mx2⋅My2.
In the third method, the spatial coherence properties are characterized by analyzing the speckle contrast of the laser output. A polarization maintaining optical diffuser was placed outside the cavity and the speckle pattern was detected with a CCD camera, as shown in Fig. 1(c). The speckle contrast C was calculated by C = σ/˂I>, where σ is the standard deviation of the intensity, and ˂I> is the mean intensity. The relation between the speckle contrast C and the number of lasing modes N is assumed to be C = N-1/2.
3. Experimental and calculated results
We first considered a circular pinhole aperture as a spatial filter at the Fourier plane, similar to . The experimental and calculated results are presented in Fig. 2. Figure 2(a) shows the speckle contrast C and the calculated speckle contrast N-1/2 as a function of aperture diameter for all three characterization methods. As evident, the results for the three methods are in good agreement, and indicate that the degree of coherence depends on the aperture diameter. There are, however, small differences between the results of the three methods. We attribute these differences to experimental limitations, such as aberrations of the optical elements, laser noise, finite resolution of the imaging systems and non-uniform illuminations. For an aperture diameter of 0.12 mm the number of modes was N = M4 = 1.12, and for an aperture diameter of 6 mm it was N = M4 = 130,000. It should be noted that control over the number of lasing modes is achieved without varying the effective size of the gain medium, so the output power of the laser should not be greatly affected, if at all. Indeed, power measurements as a function of aperture size indicated that, although the number of modes can be varied over five orders of magnitude, the output power changes by less than a factor of two. This is a unique property of the degenerate cavity that cannot be obtained with conventional stable resonators .
The number of modes determined from M4 measurements was compared to those calculated from the complex coherence factor measurements. For a small aperture diameter of0.12 mm, there was nearly perfect overlap between the near field image and |μ|, yielding N = 1.18, in excellent agreement with the M4 measurement. For large aperture diameters, the area of |μ| was very small, comparable to the resolution limit of the measurements, so the results are not accurate. For example, |μ| measurements for an aperture size of 6 mm yielded N = 11,000, which is significantly smaller than the N = 130,000 obtained from the M4 measurements.
Speckle contrast images were obtained using an external engineered light shaping 2D diffuser with no zero order transmittance and a random phase correlation length of 5 μm. When performing the speckle contrast measurements, we noticed that variations in spatial distribution of intensity induce significant errors in speckle contrast values. To overcome this, we divided the speckle pattern image to smaller subimages – which are still significantly larger than the speckle size, and calculated the speckle contrast by averaging the results of all subimages. The error bars of the blue line in Fig. 2(a) denotes the standard deviation of speckle contrast measurements of all individual subimages. As evident, speckle contrast measurements yielded contrast levels that varied from 0.86 to 0.02 by increasing the aperture diameter. For an aperture diameter of 0.12 mm, the measured speckle contrast was 0.86, which is lower than expected. Yet, it was comparable to that measured in our arrangement from a separate single mode CW Nd:Yag laser, indicating that the contrast measurements were limited by the imaging system. As the aperture diameter was increased, the speckle contrast decreased. For an aperture diameter of 6 mm, a speckle contrast of 0.02 was measured. This value, which is higher than expected, is attributed to non-uniform intensity distribution.
Figure 2(b) shows speckle images and experimental and simulation images of absolute values of complex coherence factors |μ| at three specific diameters of the circular aperture, 0.12 mm, 0.5 mm and 6 mm, corresponding to A, B and C in Fig. 2(a). The experimental images illustrate the dependence of speckle contrast and |μ| on aperture size, as indicated by Fig. 2(a). The simulation images of |μ|, which are based on a Lee-Fox algorithm , are in good agreement with the experimental results.
Next, we replaced the circular aperture by a variable slit as the spatial filter at the Fourier plane, and characterized the spatial coherence properties of the output beam. The experimental and calculated results are presented in Fig. 3. Figure 3(a) shows the speckle contrast C and calculated speckle contrast N-1/2 as a function of slit width for all three characterization methods. As evident, the results for all three methods are in reasonable agreement. Again, as in Fig. 2(a), there are small differences between the results of the three methods, and we attribute them, as before, to experimental limitations. It should be noted that in 1D there is a weaker dependence of contrast on aperture size than in 2D, and therefore the differences between the results of the three methods are more apparent.
The measurements of M2 along the coordinate axis perpendicular to the slit corresponded well with the M4 measurements for the circular aperture. A slit width of 80 μm yielded M2 = 1.21, and a slit width of 6 mm yielded M2 = 320. The measurement of the complex coherence factor indicated that for a slit width of 80μm, the calculated number of modes was N = 1.02, in good agreement with the M2 measurements. An aperture of 5 mmyielded N = 100. Again, the discrepancy with the results of the M2 measurements can be attributed to the limited resolution of the complex coherence factor measurements. The measurements of the speckle contrast were performed with an external 1D optical diffuser, where the scattering is along one direction only, and the random phase correlation length along that direction is 3µm. Here the speckle contrast varied from 0.78 for 80 μm slit width to 0.05 for 6 mm slit width. It should be noted that the difference between the measured speckle contrast of the minimal sized circular aperture and that of the minimal sized slit width is below the resolution of our measurements, as is indicated by the error bars in Figs. 2(a) and 3(a). Therefore, we attribute the discrepancy between the measurements mainly to non-uniform illumination.
The complex coherence factor images, shown in Fig. 3(b) with the corresponding speckle images at three specific widths of the slit aperture, illustrate our ability to generate coherence functions with highly tunable anisotropy. To illustrate the usefulness of such an anisotropic coherence we also performed a related experiment where we varied the orientation of the slit relative to the 1D diffuser. The results are presented in Fig. 4. These show the speckle images when the slit is either parallel or perpendicular to the scattering direction of the diffuser. As evident, there is strong speckle contrast when the slit and the 1D diffuser are parallel to one another, and low speckle contrast when they are perpendicular to one another. This clearly indicates that it is possible to obtain high brightness along one coordinate axis, and yet low speckle noise along the other coordinate axis. Such an anisotropic coherence of the laser source can be useful in certain applications.
Due to modal degeneracy in the cavity, the light propagating through the transmitting regions of the spatial filter is uniform (I(x,y) = const), and its transverse coherence length at the Fourier plane is determined by the optical resolution of the intra-cavity telescope (measured <100 μm). Therefore, if the total transmitting area in the spatial filter is large enough, the light propagating through the spatial filter is incoherent. Under these conditions, the Van Cittert Zernike theorem (Eq. (3)) predicts that the complex coherence factor μ will be proportional to the Fourier transform of the spatial filter. Accordingly, it should be possible to generate unique functional forms of μ, by varying the intra-cavity spatial filter geometry.
We considered four different geometries for the spatial filter: circular aperture, double circular apertures, array of circular apertures and annular aperture, and determined the corresponding |μ| for each geometry. The diameter of the circular aperture was 0.5 mm; the diameter of each of the double apertures was 0.2 mm and the spacing between them was 1mm; the diameter of each of the circular apertures in the array was 0.25 mm and the period of the array was 1 mm; the mean diameter of the annular aperture was 1.5 mm and the annular width was 0.1 mm. The results are presented in Fig. 5, which shows the experimental images of |μ| and their cross-sections. With the circular aperture, μ resulted in a jinc function J1(αr)/αr (Fig. 5(a)); with the annular aperture μ resulted in a zero order Bessel function J0 (Fig. 5(b)); with the double circular apertures μ resulted in cosine function (Fig. 5(c)); and with the array of circular apertures μ resulted in a 2D comb (Fig. 5(d)). The imagery of Fig. 5(b) was obtained after taking 16 sequential images and performing averaging. The corresponding experimental and calculated cross-sections of |μ| are also shown in Fig. 5. The calculations were based on the Van Cittert Zernike theorem, according to Eq. (3), and are in good agreement with the experimental results. The discrepancies between the calculated and experimental results are attributed to laser noise, such as spontaneous emission and thermal lensing.
4. Concluding remarks
We showed that the spatial coherence in a modified degenerate cavity laser can be efficiently controlled, and that different global spatial coherence can be manipulated. This was accomplished with intra-cavity binary amplitude masks which determine the functional form of the complex coherence factor, in accordance to the Van Cittert Zernike theorem. The effect of aperture size was thoroughly investigated for circular and slit masks, and the effect of aperture geometry and shape was exemplified by obtaining four different functional forms of the spatial coherence. As opposed to previously suggested methods for spatial coherence control, our method does not involve an SLM or moving elements, and can therefore generate sources which are incoherent on much shorter timescales . We believe that the spatial coherence manipulation in the modified degenerate cavity can be advantageous for speckle free wide field imaging systems, and can be employed in applications which require tailored spatial coherence properties.
We gratefully acknowledge Prof. Hui Cao (Yale University) for many fruitful discussions and helpful comments.
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