Design and performance of diffractive optics for custom laser resonators

Transcription

1 Design and performance of diffractive optics for custom laser resonators James R. Leger, Diana Chen, and Greg Mowry Diffractive optical elements are used as end mirrors and internal phase plates in an optical resonator. A single diffractive end mirror is used to produce an arbitrary real-mode profile, and two diffractive mirrors are used to produce complex profiles. Diffractive mirror feature size and phase quantization are shown to affect the shape of the fundamental mode, the fundamental-mode loss, and the discrimination against higher-order modes. Additional transparent phase plates are shown to enhance the modal discrimination of the resonator at the cost of reduced fabrication tolerances of the diffractive optics. A 10-cm-long diffractive resonator design is shown that supports an 8.5-mm-wide fundamental mode with a theoretical second-order mode discrimination of 25% and a negligible loss to the fundamental mode. 1. Introduction Conventional spherical-mirror laser resonators differ in design, depending on the characteristics of the gain medium. Low-gain systems, such as He Ne gas lasers, require a resonator with a very low fundamental-mode loss. Higher-gain lasers can tolerate some loss to the fundamental mode in exchange for more desirable mode discrimination. Establishing a common mode across an array of lasers 1such as a diode laser array2 requires special optics for producing a fundamental mode with peaks that match the individual waveguides. The majority of low-gain lasers utilize a stable Fabry Perot resonator to establish the laser mode. Although this resonator design has a low fundamentalmode loss, it has several inherent disadvantages: 112 To achieve adequate modal discrimination, long resonator lengths and small-diameter modes and mode-selecting apertures must be used; 122 the small modal diameter results in limited interaction with the gain medium, reducing the amount of power that can be extracted; 132 the Gaussian shape of the fundamental mode promotes spatial hole burning, resulting in excitation of higher-order modes at high-power levels; and 142 although the Gaussian profile of the fundamental mode has many desirable properties, there are many applications that could benefit from The authors are with the Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota Received 8 August 1994; revised manuscript received 23 December @95@ $06.00@0. r 1995 Optical Society of America. an alternative mode shape. In particular, a fundamental mode with uniform intensity may find application in a variety of laser systems. Many of the problems with stable Fabry Perot resonators can be circumvented by the use of unstable resonator geometries. 1,2 Unstable resonators can support a large-diameter fundamental mode while simultaneously preserving adequate higher-order mode discrimination. These resonators have been used with great success in high-gain laser systems such as high-power Nd:YAG, CO 2, and excimer. However, the unstable geometry has an inherently lossy fundamental mode and is not suitable for lowand medium-gain laser systems. In addition, unstable resonators sometimes have an obstructed output aperture that produces an undesirable near-field pattern. Resonators that use unconventional mirrors have been employed to improve the modal properties of lasers. Variable reflectivity mirrors have been used in unstable resonators to reduce the diffraction from the hard edges of a conventional unstable resonator and to increase the uniformity of the near-field intensity. 3 9 This idea has been extended to variable phase mirrors to provide flat near-field patterns in CO 2 lasers. Finally, a spatial filter in the form of a grid of opaque strips has been used to flatten the beam profile from a Nd:YAG laser. 13 External resonators have also been applied to arrays of semiconductor lasers. Fourier spatial filtering, as well as Talbot cavities and Talbot filters, has been demonstrated to generate substantial powers from both one-dimensional 21,23 and two-dimensional arrays APPLIED Vol. 34, No. 10 May 1995

2 Many of the characteristics of the above laser cavities can be greatly enhanced if the conventional mirrors are replaced with diffractive optical elements. A single diffractive element has been used to produce a square flattop Nd:YAG laser beam and to provide enhanced discrimination against higher-order spatial modes. 25 This configuration has also been applied to arrays of semiconductor lasers 26,27 and to large-area semiconductor amplifiers. 28,29 Additional diffractive elements placed inside the resonator can provide further modal discrimination, allowing the cavity length to be reduced and the mode size increased. 30 In this paper we discuss some of the design aspects of this new type of resonator. In the first section, we review the technique of diffractive mode-selecting mirrors 1DMSM s2 in a laser cavity. In Section 2 the design of the DMSM for optimizing mode shape and mode discrimination is described. In particular, we are concerned with the feature-size requirements and the number of phase levels needed to achieve the desired performance. Finally, we analyze the effect of a second diffractive element in the cavity and describe some of the design considerations for optimizing performance. We explore several possible selections for the phase mask and report on the impact of this selection on the feature-size requirements of the DMSM. 2. Concept of Diffractive Optic Resonators A diffractive optic resonator that consists of two reflective diffractive optical elements with transparent diffractive elements placed between the mirors is shown in Fig. 1. In addition, various apertures may be included inside the cavity to provide selective filtering of the higher-order modes. Because diffractive optics can be fabricated to approximate any required phase reflectance, the resulting optical resonator can have any desired fundamental-mode shape. The designer starts by specifying the desired geometric shape and complex-mode profile of the fundamental mode just to the left of the output mirror. The diffraction pattern of this mode at the diffractive end mirror is then calculated numerically with the scalar nonparaxial diffraction formula and with the propagation through intracavity phase plates and apertures taken into account. An end-mirror phase reflectance that will return the complex conjugate of this specific wave front is then calculated. It is easy to show 11,25 that the return wave that is incident upon the output mirror is the complex conjugate of the original desired distribution. If we design this sec- Fig. 1. General design of a diffractive optical resonator. ond mirror to reflect the complex conjugate of this wave front and if both mirrors and apertures are sufficiently large, the resulting field is identical to the original desired field. This, by definition, is a mode of the cavity. For a mirror to produce the complex-conjugate wave front, it need only modify the phase of the wave front. Thus both mirrors are phase only and can be fabricated as diffractive optical elements if integer multiples of a wavelength are subtracted from the phase-conjugating surface and if the remaining surface-relief structure is represented as a series of quantized phase steps. By careful choice of other system parameters, these mirrors can also be used to discriminate against all higher-order modes. For this reason, we have called these mirrors DMSM s and the resonator based on them a DMSM resonator. The theoretical performance of this resonator can be assessed by the solution of the eigenvalue equation, ek1x, x82u n 1x82d 2 x8 5g n U n 1x2, 112 where the integral kernel K1x, x82 describes the roundtrip propagation in the cavity, U n 1x2 are the eigenfunctions of the equation and g n their corresponding eigenvalues. The squared magnitude of an eigenvalue associated with a particular mode gives the round-trip attenuation of that mode, which is due to diffraction. The gain required for overcoming this attenuation is called the modal threshold gain and is defined as G n 5 1@g n The modal threshold gain is used exclusively in this paper as a measure of modal discrimination and fundamental-mode loss, in which a threshold gain of unity corresponds to a lossless cavity. The eigenfunctions of Eq. 112 describe the amplitude and the phase of the various modes. The kernel K1x, x82 contains all the information regarding the specific implementation of the diffractive optic. Hence solutions to this equation can be used to analyze the effect of feature-size limits, phase quantization, etc., on the resonator performance. 3. Design of a Single-Element Diffractive Mode-Selecting Mirror Resonator The simplest form of the DMSM resonator consists of a single DMSM for an end reflector and a flat mirror for an output mirror. This requires the mode at the output mirror to be a real function 1so that a conventional mirror can return the complex conjugate2. There are several goals in the design of a DMSM resonator. First, the fundamental mode must be acceptably close to the desired mode shape and must have a sufficiently low loss for the particular laser system. Second, the separation in threshold gain between the fundamental mode and the higher-order 10 May Vol. 34, No. APPLIED OPTICS 2499

3 modes must be sufficient to prevent multimode oscillation. Third, the fabrication of the DMSM must be within the state of the art, and losses introduced by quantization and fabrication errors must be minimized. The design parameters consist of cavity length, aperture size and placement, number of DMSM phase-quantization levels, and feature-size limitations of the DMSM. In this section, we address each of these issues by modeling a particular DMSM cavity and solving for the eigenfunctions and the eigenvalues of Eq Two different mode shapes are studied in this paper. The first is a single beam with a super- Gaussian profile, chosen to be very close to a uniformly illuminated square. Its amplitude is given by A 1 1x, y2 5 exp321x@v exp321y@v , 132 where v 0 is the half-width of the square. The second is a mode from a laser array consisting of M 1 1 Gaussian beams, expressed by A 2 1x, y2 5 M@2 o m52m@2 exp3 2 1x 2 ma22 1 y 2 v , 142 where v 0 8 is the width of a single Gaussian beam and a is the separation between the beams. A. Optimum Cavity Length and Aperture Sizes It is well known that the cavity length has a great effect on the mode discrimination in conventional Fabry Perot optical resonators. Because the length and the mode size are dependent on one another, the cavity is often desribed by a dimensionless parameter. The approximate parameter is different for a single beam and for an array of beams. We describe the cavity characteristics of each of these fundamental modes below. 1. Super-Gaussian Fundamental Mode For the single super-gaussian mode of Eq. 132, the cavity Fresnel numbers can be used to describe the effects of cavity length and aperture sizes. The Fresnel numbers N 1 and N 2 that correspond to the two apertures are defined as Fig. 2. Threshold gain of fundamental and second-order modes as a function of two cavity Fresnel numbers N 1 and N 2. The DMSM is adjusted to be phase conjugate to a super-gaussian that is slightly smaller than the output aperture. of the super-gaussian. It is clear from the graph that a sufficiently large value of N 2 ensures a low-loss fundamental mode for all cavity lengths. This results simply from using a DMSM with sufficient size to collect and phase conjugate virtually all the diffracted light from the output aperture. In addition, using a smaller value of N 2 does not appreciably increase the discrimination between the fundamental and the second-order modes, whereas the resulting cavity mode shape can differ significantly from the desired one because of clipping at the mirror. Hence there is no advantage to using a limiting aperture in the DMSM plane. The Fresnel number governed by the output aperture N 1 has a dramatic effect on modal discrimination. The discrimination increases with decreasing N 1 until N 1 reaches a value of approximately 1@p, whereupon it tends to saturate 1for low N 2 2 or decrease 1for high N 2 2. This implies that the optimum cavity length is roughly given by the Rayleigh range z 0 5 pv defined by a conventional Gaussian with beam waist v 0. Thus, for large beam widths, the optimal cavity length can be quite large. This issue is addressed in Section 4. Figure 2 was based on the fabrication of a DMSM with a mode size approximately equal to the outputaperture size. We now consider the effect of using an output aperture that is not matched to the fundamental mode of the cavity. Figure 3 shows the losses N i 5 d i 2 4lz, 152 where d 1 is the width of the aperture covering the output mirror, d 2 is the width of the DMSM aperture, z is the cavity length, and l is the wavelength of light. The modal threshold gain required for overcoming the losses to the fundamental and the second-order cavity modes as a function of these two Fresnel numbers is shown in Fig. 2. For every point on this curve, the DMSM has been adjusted to produce a super-gaussian whose width v 0 is slightly smaller than the output aperture, such that d v 0. This value was chosen to produce negligible clipping Fig. 3. Threshold gains of the fundamental and the second-order modes as a function of normalized output-mirror-aperture size. Normalization is with respect to the super-gaussian full width APPLIED Vol. 34, No. 10 May 1995

4 to the first two modes as functions of the normalized aperture width d8 5 d 0. The cavity length is chosen to be one Rayleigh range of a conventional Gaussian 1pv and the Fresnel number of the DMSM aperture 1N 2 2 is infinite. From this result, it is apparent that the output aperture is very important for modal discrimination. Normalized aperture sizes of d8 : 1 reduce the modal discrimination significantly, whereas sizes of d8 9 1 result in a substantial fundamental-mode loss. When d8 5 1, the super-gaussian fundamental uniformly illuminates the aperture with very little clipping, whereas the higher-order modes 1with substantal light outside the aperture2 are efficiently filtered out. The apertures also affect the shape of the fundamental mode, and care must be taken to avoid significant modal distortion. Figure 41a2 shows the effect of changing the Fresnel number N 2 by the use of different aperture sizes at the DMSM. For N 2 $ 20, the effect on the mode is minimal, and the modal shape is very close to the expected super-gaussian. As the Fresnel number is decreased, the clipping of the fundamental mode becomes significant and the mode is distorted. The increased loss is also evident in the slightly smaller mode height. From this result, it is apparent that the DMSM should be fabricated with a size sufficient to result in a Fresnel number of N 2 $ 20. The effect of the output-mirror-aperture size on modal shape is shown in Fig. 41b2. For maximum modal discrimination, it is desirable to have the aperture approximately equal to the mode size. However, the slight clipping that results distorts the mode. The solid curve corresponds to an aperture that is 1.08 times the mode size. There is essentially no clipping, and the mode shape is almost a perfect super-gaussian of the 20th order. The dashed curve corresponds to an aperture that is equal to the mode size. A small amount of ripple that is due to the clipping of the fundamental mode by the output aperture can be observed. Although the distortion is small in this example, lower-order super-gaussians will have more clipping, and the distortion may become significant. In these cases, the aperture must be enlarged to preserve the modal shape at the expense of the modal discrimination. 2. Gaussian Array Fundamental Mode The DMSM cavity has been studied for arrays of Gaussian beams suitable for application with diode laser arrays. In this case, the output aperture is formed by the waveguides on the laser array itself. The dimensionless design parameters are the fill factor of the array 3 f 5 2v 0 8@a from Eq and the cavity length normalized to a Talbot distance Z t 5 2a The modal threshold gain of the second-order mode as a function of normalized cavity length Z@Z t is shown in Fig. 51a2. As the DMSM size was chosen to be large, the fundamental-mode loss was negligible and is not shown in the figure. It is apparent that the cavity length affects the modal behavior in a much more complex manner than the single beam result of Fig. 2. The lengths corresponding to large modal discrimination are at one-half Talbot distance and one-quarter Talbot distance. The actual heights of the peaks and the optimum choice between these two cavity lengths are functions of the fill factor of the array. 27 Figure 51b2 shows the effect of the fill factor on the second-order mode at a constant cavity length of one-half Talbot distance. A larger fill factor reduces the coupling between the individual Gaussian beams and hence reduces the discrimination of the second-order mode. However, unlike a simple Talbot cavity, 31 very small fill factors also reduce the modal discrimination. 27 Thus there is an optimal fill factor for an array of lasers that maximizes the modal discrimination. B. Fabrication Tolerances and Requirements The above resonator models assumed perfect diffractive optics. We have shown that, for sufficiently large DMSM s, the loss to the fundamental mode is negligible. However, because of fabrication limitations, the effects of finite feature size and phase quantization must be taken into account. In most other diffractive optics applications, we are interested primarily in the efficiency and the possible distortion of the designed response that are due to these quantizations. In a resonator application, however, the efficiency of both the fundamental and the secondorder modes must be considered 1where it is desirable to minimize the efficiency of the latter2. In this Fig. 4. Fundamental-mode shape as a function of aperture size: 1a2 aperture is applied to DMSM, and curves correspond to Fresnel numbers N 2 5 4, 8, and 20; 1b2 aperture is applied to output mirror, normalized to the super-gaussian full width. 10 May Vol. 34, No. APPLIED OPTICS 2501

5 Fig. 5. Modal threshold gain of second-order mode for an array of eight Gaussian sources: 1a2 threshold gain as a function of normalized cavity length for an array fill factor of 0.12, 1b2 threshold gain as a function of the fill factor for a cavity length of z section, we consider the effects of these fabrication tolerances on the modal behavior of the resonator. 1. Feature-Size Requirements The feature-size effects were studied by the application of a size quantization across the phase pattern at the DMSM. The phase was held at a constant average value across the width of the feature. Figure 6 shows the modal threshold gain of the fundamental and the second-order modes as functions of this feature width quantization for the single super- Gaussian beam of Eq The cavity is assumed to have a Fresnel number N 1 5 1@p to maximize the modal discrimination and an output aperture slightly larger than the mode size d v 0. The second Fresnel number N 2 is sufficientlylarge so that the clipping at the DMSM is negligible. The feature size Dl is expressed in dimensionless units Dy, where Dy 5 Dl Œlz and z is the cavity length. Because N 1 5 1@p 1corresponding to a cavity length z 5pv the reduced units are given simply by Dy 5 Dl@Œpv 0. Thus, for a beam half-width of 0.6 mm, the reduced feature size corresponds to the approximate actual feature size in millimeters. It is apparent that the 162 feature-size requirements are very lax for this application. For example, a square beam with a width of 1.2 mm 1v mm2 can be fabricated with a mode loss of less than 1% by the use of a minimum feature size of 40 µm. This is easily achievable with simple lithography and wet chemical etching. A feeling for the effect of finite feature size on the loss to the fundamental mode can be obtained when the requirements of a spherical diffractive mirror used to establish a simple Gaussian beam of the same beam waist v 0 are considered. We first calculate the radius of curvature R1z2 at the mirror by propagating the beam the length of the cavity 1one Rayleigh range2. We then have R1z 5 z z1 1 1 z 0 2 z z The focal length of the mirror used to reflect this wave must be equal to half the radius of curvature of the beam, or z 0. The mirror diameter must be chosen to collect the majority of the light. We choose a mirror with Fresnel number N 2. Thus, at a Rayleigh range, we have a mirror diameter d of and an f-number for this mirror of d 5 2v 0 ŒpN z 0 f@# v 0 ŒpN 2 It is well known that the minimum feature size at the edge of a two-level diffractive optic mirror is given by l1 f@#2. Thus the minimum feature size Dl at the edge of our M-level mirror is 2l f@# Dl 5 M 5 v 0 MŒ p, 1102 N 2 or, in dimensionless units, Fig. 6. Threshold gains of super-gaussian fundamental and second-order modes as functions of minimum DMSM feature size. Dy 5 1 MŒN 2, APPLIED Vol. 34, No. 10 May 1995

6 where the Rayleigh range z 0 5 pv has been substituted. This indicates that, for a given efficiency 1corresponding to a certain number of levels at the edge of the mirror2 and a given diffractive loss at the DMSM 1corresponding to a specific Fresnel number N 2 2, the required feature size is directly proportional to the beam size v 0. So, for example, if a 1.2-mm beam is desired 1v mm2 and a Fresnel number N is sufficient to keep diffractive losses to an acceptable level, the required 16-phase-level mirror would have a feature size at the edge of the mirror or Dl 5 15 µm. The local diffraction efficiency of the mirror at its edge h edge can be approximated by the relationship h edge 5 0 sin1p@m2 p@m 0 2 < p M2 2, 1122 where the approximation is true for large M. Solving Eq for M, substituting this value into Eq. 1122, and defining the loss at the edge of the mirror L edge that is due to the finite feature size as L edge 5 1 2h edge results in L edge < pn Dl v In terms of reduced coordinates, the loss can be simply expressed as L edge < p2 N 2 3 Dy Finally, approximation 1142 can be expressed in terms of the fundamental modal threshold gain G 1 as G p2 N 2 3 Dy < 1 1 p2 N 2 3 Dy2, 1152 where the approximation holds for Dy 9 11@p2 Œ3@N 2. The actual threshold gain of the element shown in Fig. 6 consists of an average across the entire mirror and is thus expected to be considerably smaller. The effect of feature size on the second-order mode is similar to the fundamental mode, so that the mode discrimination stays approximately constant, independent of feature size. The feature-size constraints of the DMSM fabricated for the array of Gaussians from Eq. 142 can be far more demanding. Figure 7 shows the modal threshold gain of the first two modes for an array of eight lasers with a fill factor of f 5 2v 0 8@a , where a is the spacing between Gaussian beams. The mirror is placed at a distance of one-half Talbot length to maximize the modal discrimination. The Fig. 7. Threshold gains of Gaussian-array fundamental and second-order modes as functions of minimum DMSM feature size. normalized feature size Dy now becomes Dy 5 Dl Œlz 5 Dl a 5 fdl 2v From Eq. 1162, we see that the smaller the Gaussian beam waist v 0, the smaller the required feature size Dl. This is as expected, as beams with smaller waists have larger divergences, and thus the angular plane-wave spectrum incident upon the DMSM is larger. This, in turn, requires finer pitch gratings for phase conjugating the higher-frequency plane waves. As an example, a standard single-mode waveguide in a laser array has a lateral beam waist v 0 of 3 µm. From Fig. 7, we see that the fundamental-mode threshold gain becomes substantial for values of Dy This corresponds to a minimum feature size Dl of 3 µm. 2. Phase-Quantization Requirements Because diffractive optics are often fabricated with discrete phase levels, it is of interest to determine the effect of this phase quantization on modal threshold gain and mode shape. Figure 81a2 shows the result of quantizing the phase of a diffractive optic to various numbers of phase levels. As before, the diffractive optic is designed to produce a 20th-order super-gaussian mode with a cavity length of one Rayleigh range 1based on a conventional Gaussian beam2. No feature-size restrictions are used in this model, and the aperture size d 1 at the output mirror is adjusted to be 1.08 times the fundamental-mode width 2v 0. The modal threshold gain of the fundamental mode is seen to be similar to that expected from a conventional multiphase lens or grating 1shown as a dotted curve in the figure2. The modal threshold gain of the second-order mode also increases with a reduced number of phase levels. Hence the overall modal discrimination is large even with a low number of phase levels. The mode shape is also influenced by the number of phase levels. Figure 81b2 shows the resultant funda- 10 May Vol. 34, No. APPLIED OPTICS 2503

7 Fig. 8. Effect of phase quantization on laser resonator performance: 1a2 threshold gain of fundamental and second-order modes as functions of number of phase-quantization levels; 1b2 mode shapes for 4, 8, and 32 phase-quantization levels. mental mode when 32, 8, and 4 phase levels are used to produce the 20th-order super-gaussian. It is clear that eight phase levels produce significant distortion of the fundamental mode. In addition, there is a significant cavity loss, as seen by the smaller mode heights. However, as the feature sizes are very large, even when many phase levels are used, it is relatively easy to fabricate mirrors with many phase levels, resulting in low-loss fundamental modes and accurate mode shapes. Figure 9 shows the four mask patterns required for fabricating a 16-level DMSM for a 20th-order square super-gaussian with a beam width of 1.2 mm. 4. Laser Resonators With Internal Phase Plates A simple two-mirror cavity design can establish an arbitrary real-mode profile with a single DMSM and an arbitrary complex profile with two mode-selecting mirrors. When simple mode profiles such as super- Gaussians are desired, however, large discrimination between spatial modes occurs only when the cavity length is approximately one Rayleigh range. Thus, for large beam diameters, these methods can result in very large cavity lengths, compromising mechanical stability and increasing the pulse length for Q- switched laser operation. In this section we describe a diffractive laser cavity that contains a phase plate inside a single DMSM resonator. The setup is as shown in Fig. 1, with a simple flat mirror used as the output mirror. The design of the cavity proceeds in the same way as above. The mode at the output mirror can be selected to be any desired real function, but the diffraction pattern incident upon the DMSM is modified by the phase plate. As in the above, the DMSM is designed to return the complex conjugate of this pattern. As the light passes through the phase plate on the return trip, it recreates the desired mode. Thus, if a laser crystal is placed between the phase plate and the output mirror, the mode in the crystal can be simple 1e.g., super-gaussian2 and can extract power from the medium in an optimum fashion. The mode to the left of the phase plate is far more complex, however. Below we show that this increased complexity greatly increases the modal discrimination of the cavity. The following sections describe the performance enhancement offered by both periodic-grating phase plates and pseudorandom phase plates. A. DMSM with a Sine-Wave Phase Plate We first describe the performance of a sine-wave phase plate placed 20 cm from the output mirror and 30 cm from the DMSM, for a total cavity length of 50 cm. The same fundamental mode and output aperture were chosen as in the above simulation 12v mm, d mm2. However, the total cavity length of 50 cm corresponds to less than one-half of a conventional Gaussian Rayleigh range at the operating wavelength of 1.06 µm. The phase plate studied was a simple phase grating with amplitude transmittance t1x, y2 given by t1x, y2 5 exp3 jm sin12p f g x 1f24, 1172 where m is the modulation index, f g is the frequency, and f is the shift of the phase grating. The choice of the aperture size ensured that the loss to the fundamental mode was negligible 1,0.1%2. Figure 10 shows the loss to the second-order TEM 10 mode as a function of grating frequency. The modulation depth m was set to unity, and the grating phase f was set to zero. A grating frequency of f g 5 0 corresponds to a simple DMSM cavity and has a second-order modal threshold gain g This value can be improved remarkably if the grating frequency is increased to 3.75 mm 21, where the second-order modal threshold gain is 3.7 1corresponding to a loss of 73%2. Increasing the frequency past this point leads to a rapid decrease in modal discrimination, followed by a more gradual reduction. For sufficiently high grating frequencies, the modal discrimination reduces to the value obtained with no phase plate present 1g This is to be expected, as at these high spatial frequencies there is very little overlap between the grating orders. The DMSM is then illuminated by mulitiple copies of the same distribution from a simple DMSM resonator, and there is no improvement in performance. The high modal discrimination exhibited by the diffractive resonator can be utilized to reduce the 2504 APPLIED Vol. 34, No. 10 May 1995

8 Fig. 9. Four mask patterns for fabricating 16-level DMSM. The smallest feature size is 50 µm. cavity length or to increase the mode size. For this example, if a modal threshold gain of 1.15 is adequate to discriminate against higher-order modes, the mode diameter of the diffractive resonator can be increased Fig. 10. Second-order TEM 10 modal threshold gain as a function of sinusoidal-grating frequency. The loss to the fundamental mode is less than 0.1%. to5mm1from 1.2 mm2 while still ensuring singlespatial-mode operation. The optimum phase of the sinusoidal grating was studied next. Figure 11 shows the modal threshold gain of the second-order TEM 10 mode as a function of grating shift f. For each grating shift, a new DMSM was calculated to establish a low-loss fundamental mode. The loss to the second-order mode was then determined. For our cavity parameters, a sinusoidal pattern 1f 502 has a considerably larger modediscrimination ability than the corresponding cosinusoidal pattern 1f However, the actual optimum value for f appears to be a function of the specific cavity parameters, such as modulation depth m and grating frequency f g. Figure 12 shows the second-order TEM 10 modal threshold gain as a function of grating modulation depth m. The solid curve corresponds to a grating frequency of 2 mm 21, and the dashed curve corre- 10 May Vol. 34, No. APPLIED OPTICS 2505

9 Fig. 11. Second-order TEM 10 modal threshold gain for sinusoidal phase grating with different phase shifts. sponds to a frequency of 3.75 mm 21. For the lower frequency, an increase in discrimination is observed with a higher modulation depth of up to approximately m This increase is expected, as phase gratings with larger modulation indices contain more diffraction orders. As above, these higher orders interfere and increase the complexity of the modeselecting mirror, thereby increasing the modal discrimination. The higher-frequency grating 1corresponding to the optimum point in Fig. 102 achieves a maximum at a much lower modulation index. A modulation depth greater than unity does not appear to increase the modal discrimination. One possible explanation for this is that the overlap of the diffraction orders at higher grating frequencies is reduced. Consequently the higher orders generated by a larger modulation index do not interfere with the lower orders and so do not improve the mode-discrimination ability of the DMSM. B. Selection of Phase-Plate Pattern The phase function encoded on the phase plate has an impact on the mode-discrimination ability of the cavity as well as on the fabrication and the alignment tolerances of the DMSM. We have investigated both periodic gratings and pseudorandom phase arrays for this element. A periodic phase grating can have several practical advantages. First, it may be easier to fabricate, as sine-wave and square-wave gratings can be fabricated by conventional means. In addition, the initial gross alignment is easier, as the absolute center of the plate does not have to be determined. The disadvantage of a grating structure was seen in Subsection 4.A. The pattern incident upon the DMSM results from interference between discrete grating orders, and high grating frequencies prevent these orders from overlapping. Thus there is a limit to the mode discrimination achievable with a periodic grating. A pseudorandom phase plate has a continuous angular plane-wave spectrum and hence does not suffer from this problem. The effect of these two phase plates on the resonator properties is examined below. 1. Periodic-Grating Phase Plate The ideal grating pattern provides very high modal discrimination without requiring excessively small features on the DMSM. We performed a comparative study between sine-wave and square-wave phase gratings of similar frequency, phase, and depth of modulation. Although the square-wave grating may be simpler to fabricate, it appears to have an inherent disadvantage compared with the sine-wave grating. Figure 13 shows the calculated modal threshold gains of the fundamental and the second-order modes that are due to finite feature size on the DMSM. The cavity configuration is the same as in Subsection 4.A 120th-order super-gaussian, 2v mm, d mm2, the frequency of the grating is 3.75 mm 21, and the peak-to-peak grating modulation is p rad. The modal threshold gain is plotted for various minimum feature sizes on the DMSM. As expected, the required modal gain for the fundamental mode increases with increasing feature size. However, the required gain is always higher for the square wave than for the sine wave. Thus, for a particular fundamental-mode loss, the feature-size requirements for the square-wave phase pattern are more stringent than for the sine-wave pattern. Because the discrimination between the fundamental and the secondorder modes is similar for the two grating types, there is an advantage to using a sine-wave grating instead of using a square-wave grating. The reason for the improved performance of the sine-wave grating is apparent when the angular Fig. 12. Second-order TEM 10 modal threshold gain for a sinusoidal phase grating with different modulation depths. Fig. 13. Fundamental and second-order TEM 10 modal threshold gains versus feature size for square-wave and sine-wave grating phase plates APPLIED Vol. 34, No. 10 May 1995

10 plane-wave spectra of the two gratings are compared. The operation of the mode-selecting mirror can be viewed as a linear superposition of blazed gratings in which each grating retroreflects a particular incident plane-wave component back along its original angle of incidence. The larger the angular plane-wave spectrum, the smaller the features required on the DMSM. For a lossless fundamental mode, these gratings must efficiently phase conjugate all components of the incident angular plane-wave spectrum. The squarewave phase pattern contains many more high spatial frequencies than does the sine-wave phase pattern with a similar modulation depth. This can be seen by comparing the asymptotic behavior of the diffraction order intensity. The power in the qth diffraction order of a square wave is proportional to 1@q 2. However, the power in the qth diffraction order from a sine-wave phase grating 1assuming unity area and unity illumination intensity2 is given by P q 5 0 J q 1m@220 2, 1182 where J q is the qth-order Bessel function and m is the modulation index. The asymptotic behavior of this function for large q and q. m is given by 0 J q 1m@220 2, 2pq1 1 em 4q2 2q, q : 1, 1192 where e is the irrational number The falloff of the higher-order harmonics is much sharper for the sine-wave grating, and there is very little power contained in these high-frequency terms. Hence the feature-size requirement on the DMSM is less demanding than for the square-wave grating. 2. Pseudorandom Phase Plate The mode-discrimination ability of the periodic grating is limited by the discrete nature of the diffraction orders. Grating frequencies that do not produce overlap of the diffraction orders have poor modal discrimination. A pseudorandom phase plate, however, does not have discrete orders and so arbitrarily large bandwidths can be used. It was postulated that the degree of modal discrimination was related to the angular plane-wave spectrum incident upon the mode-selecting mirror. To test this, we performed a series of experiments by using random phase plates with different angular plane-wave spectra. A Gerchberg Saxton algorithm was used to apply the phase-only constraint and the Gaussian angular spectrum constraint simultaneously. 32 The resultant phase-only data had an approximately Gaussian angular plane-wave spectrum with the power spectral bandwidth defined as the 1@e 2 point of the Gaussian. Figure 14 shows the increase of threshold gain to the second-order mode with increasing phase-plate bandwidth. The error bars show the statistical variation in the simulation 161 standard deviation2. Very high modal discrimination can be obtained if the DMSM is presented with a sufficiently complex light field. It appears from Fig. 14. Modal threshold gain of second-order mode as a function of bandwidth of a pseudorandom phase plate. this test that, in theory, arbitrarily large modal discrimination can be obtained with a pseudorandom phase plate of sufficient spatial frequency bandwidth. There are some practical considerations that limit the use of arbitrarily high bandwidths, however. Higher bandwidth phase plates result in more complex DMSM s with finer feature-size requirements. Figure 15 shows the effect of mode-selecting-mirror feature-size quantization on the modal gains of the fundamental and the second-order modes. For fundamental mode losses of approximately 1%, features as large as 3 µm can be used for the low-bandwidth phase mask 118 mm 21 2, whereas 1.0-µm features are required for a high-bandwidth phase mask 153 mm At these fine feature sizes, scalar diffraction theory is also not appropriate, and better estimates of diffraction efficiency must account for vector diffraction theory effects. 33 Finally, the alignment of the phase plate to the DMSM becomes more critical with a higher phase-plate bandwidth. To illustrate the potential of this technique, we have considered a case study of a 10-cm-long laser cavity containing a random phase plate with a bandwidth of 44 mm 21. The phase mask was placed in the center of the cavity, and a 20th-order super- Gaussian was chosen as the fundamental-mode profile. A mode-selecting mirror was designed with a minimum feature size of 2 µm and 16 phase-quantization levels. The resulting theoretical fundamental-cavitymode profile is shown in Fig. 161a2 for a 1.2-mm beam size. The finite feature size and phase quantization Fig. 15. Modal threshold gain of fundamental and second-order modes as functions of minimum DMSM feature size when a pseudorandom phase plate is used. 10 May Vol. 34, No. APPLIED OPTICS 2507

11 Fig. 16. Theoretical performance of a 10-cm laser cavity containing a random phase plate: 1a2 two-dimensional fundamental-mode intensity profile, 1b2 laser gain required for overcoming the diffractive losses to the second-order mode for various fundamental beam sizes. The result from a conventional Fabry Perot cavity is shown for comparison. of the mode-selecting mirror produce small nonuniformities in the beam profile and result in a fundamental mode loss of approximately 1.3%. The gain required for overcoming the losses to the second-order mode was 5.1 1corresponding to a loss of greater than 80%2. For comparison, a stable Fabry Perot cavity with the same cavity length, beam size, and fundamentalmode loss has a second-order modal gain of only 1.08, corresponding to a loss of just 7.2%. Figure 161b2 shows the second-order modal gain as a function of output beam spot size for this cavity. If a modal gain of 1.33 is sufficient to discriminate against the second-order mode 1corresponding to a loss of 25%2, beam diameters of up to 8.5 mm can be used in this 10-cm-long cavity. It is therefore possible to extract power from a wide gain medium while still maintaining a very small cavity length and small fundamental-mode loss. 5. Conclusion Diffractive optical elements were used as the end mirrors of an optical resonator. Because these elements can produce any phase profile, many desirable properties that are impossible with conventional optical elements can be incorporated into the optical resonator. We have shown that any real fundamental-mode profile can be produced by a single diffractive mirror and that complex-mode profiles are possible with two diffractive mirrors. In addition, the cavity can be designed to have a negligible theoretical fundamental-mode loss while simultaneously providing high modal discrimination against higher-order modes. Finite phase-quantization and minimum feature-size constraints were shown to affect both the shape of the fundamental mode and the loss. For the mode profiles studied, the minimum feature size was found to be quite large, making fabrication of the diffractive elements relatively easy. The modal discrimination was enhanced when an additional transparent phase plate was included in the cavity. It was shown that there is an optimum frequency, phase, and modulation depth for a sinusoidal phase grating used to discriminate between cavity modes. The modal discrimination of pseudorandom phase plates, however, appeared to increase as the spatial frequency bandwidth of the phase plate was increased. The limitation on modal discrimination in this case was dictated by practical considerations. The higher-bandwidth plates require smaller DMSM feature sizes and thus have a higher fundamentalmode loss for a given DMSM feature size. This work was supported by the National Science Foundation under grant ECS References 1. A. E. Siegman, Unstable optical resonators for laser applications, Proc. IEEE 53, Y. A. Anan ev, Unstable resonators and their applications, Sov. J. Quantum Electron. 1, G. Giuliani, Y. K. Park, and R. L. Byer, Radial birefringent element and its application to laser resonator design, Opt. Lett. 5, P. Lavigne, N. McCarthy, and J.-G. Demers, Design and characterization of complementary Gaussian reflectivity mirrors, Appl. Opt. 24, D. J. Harter and J. C. Walling, Low-magnification unstable resonators used with ruby and alexandrite lasers, Opt. Lett. 11, S. De Silvestri, P. Laporta, V. Magni, O. Svelto, and B. Majocchi, Radially variable reflectivity output coupler of novel design for unstable resonators, Opt. Lett. 12, K. J. Snell, N. McCarthy, M. Piché, and P. Lavigne, Single transverse mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror, Opt. Commun. 65, S. De Silvestri, P. Laporta, V. Magni, G. Valentini, and G. Cerullo, Comparative analysis of Nd:YAG unstable resonators with super-gaussian variable reflectivity mirrors, Opt. Commun. 77, A. Parent and P. Lavigne, Variable reflectivity unstable resonators for coherent laser radar emitters, Appl. Opt. 28, P.-A. Bélanger and C. Paré, Optical resonators using gradedphase mirrors, Opt. Lett. 16, P.-A. Bélanger, R. L. Lachance, and C. Paré, Super-Gaussian 2508 APPLIED Vol. 34, No. 10 May 1995

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