Stable Laser Resonator Modeling: Mesh Parameter Determination and Empty Cavity Modeling


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1 Stable Laser Resonator Modeling: Mesh Parameter Determination and Empty Cavity Modeling Justin Mansell, Steve Coy, Kavita Chand, Steve Rose, Morris Maynard, and Liyang Xu MZA Associates Corporation
2 Outline Introduction & Motivation Choosing a WaveOptics Mesh MultiIteration Imaging Stable Resonator Modeling Conclusions 2
3 Introduction and Motivation When evaluating a new laser gain medium, it is common to build a multimode stable resonator around the gain medium to demonstrate maximum extraction. Modeling multimode lasers is difficult because The modes tend to interact with the saturable gain medium and create modeling instabilities and The mesh requirements for a multimode stable resonator are very resource intensive. 3
4 Laser Resonator Architectures: Stable vs. Unstable Stable Unstable Rays captured by a stable resonator will never escape geometrically. All rays launched in an unstable resonator (except the onaxis ray) will eventually escape from the resonator. 4
5 HermiteGaussian Modes HG=HermiteGaussian HG modes are orthogonal modes of a stable resonator with infinite internal apertures. HG modes tend to appear clearly in rectangularaperture resonators. 5
6 LaguerreGaussian Modes LG=LaguerreGaussian LG modes are orthogonal modes of a stable resonator with infinite internal apertures. LG modes tend to appear clearly in circularaperture resonators. 6
7 The Iterative Fourier Transform (aka Fox & Li) Technique A field is propagated through repeated roundtrips until the field has converged to a stable field distribution. This is a commonly used technique for simplifying the 3D solution of Maxwell s Equations into a 2D problem (i.e. GerchbergSaxon technique) A. G. Fox and T. Li. Resonant modes in a maser interferometer, Bell Sys. Tech. J. 40, (March 96). A. G. Fox and T. Li, Computation of optical resonator modes by the method of resonance excitation, IEEE J. Quantum Electronics. QE4, (July 968). 7
8 Comments on Fox & Li Solutions Solution is for an instantaneous state, which is typically the steadystate of the laser. Stable resonators are much more computationally intensive to model than unstable resonators. We attribute this to the geometric output coupling of an unstable resonator. Larger eigenvalue difference between fundamental mode and the next higherorder mode. Not generally appropriate for pulsed or timevarying solutions unless the timevarying nature is much slower than a resonator roundtrip time. This is analogous to a splittime modeling techniques. 8
9 Example Resonator: RADICL Laser Output R c = 0m λ =.353e6m Saturable Gain Medium L = 0.254m Flat Mirror Other Parameters L resonator = 0.8 m D ap = 0.06 m (mirrors and gain) g ss = %/cm I sat = kw/cm 2 Curved Mirror Data source: Eppard, M., McGrory, W., and Applebaum, M. "The Effects of WaterVapor Condensation and Surface Catalysis on COIL Performance", AIAA Paper No , May
10 Predicted Gaussian Radius TEM00 Mode Radius for Plano Concave Resonator L = 0.8m L( R 2 L) ω ω 0 ω R 2 R 2 L L.35 m R = R 2 =0m 0 2.mm.0656mm 0
11 WaveOptics Mesh Determination
12 Calculation : HalfRoundTrip Estimate based on two limiting apertures analysis [Coy/Mansell]: N 4DD z 2 4 6cm.35 m 6cm 80cm 3,688, N 2.87E8 8.8x06 m NOTE: This is for a half of a round trip. 3,688 points per dimension Very Difficult 2
13 Calculation 2: Full RoundTrip Mesh points = 6 * Fresnel number (Mansell, SPIE 2007) a = half limiting aperture size = 3 cm λ =.35 μm 2 a N f L L =.7524 m = (L + (L*R/2)/(R/2L)) = ( ) m N f = 39 for the 6 cm aperture, 6*N f = 6256 mesh points in each dimension Still Difficult 3
14 Calculation 3: Maximum Angular Content Ray Optics Analysis of a Stable Laser Resonator (D) A ray launched parallel to the optical axis will oscillate back and forth across the resonator cavity The rays slope increases until it crosses the center of the cavity, then decreases. This gives us an estimate of max. Using ray optics we estimate max~= 0.0 for this resonator. Obtained with ray tracing with a ray at the edge of the aperture. 4
15 radius (m) / angle (radians) Rays and Angles Unwrapped radius theta roundtrips Graphical Method of Determining the Number of RoundTrips to Image 5
16 Number of RoundTrips to Image/Repeat PlanoConcave Stable Resonator Linearized PlanoConcave Stable Resonator Image Planes 6
17 Calculation 3: Maximum Angular Content (2) Now that we have, we can immediately obtain the constraint on the mesh spacing by applying the Nyquist criterion: 2 max.35 m m Finally, we can obtain the constraint on N by imposing the requirement that no rays leaving the cavity from one side should be able to wraparound and reenter the cavity from the other side: N D max z 6cm m 80cm 58 This method makes the greatest use of resonator information, so we believe it to wellrepresent the minimum mesh size required. NOTE: 3,688 / 58 ~ = number of roundtrips to image 7
18 Examining the Ray Tracing As is typical of a stable resonator, a ray launched at the edge of the resonator walks to the other side of the resonator and then back to the starting side. For this resonator, this process takes ~ roundtrips Therefore, the angle required is approximately reduced by a factor of. How can we find the number of round trips to image? 8
19 Analytical Method to Determine the Number of RoundTrips to Image This problem can be addressed using ray matrices. Consider a resonator with a roundtrip raymatrix given by M. In N roundtrips, the raymatrix will be given by M N. To reproduce any input ray, we need to determine the number of round trips to make the identity matrix, or M N =I. A C B D N M N This can be determined numerically, but can also be addressed using eigenvalue analysis. The approach on the right derives an equation for N assuming a planoconcave resonator with length L and the end mirror radius of curvature equal to 2f. 9 I 0 0 M M N 0 2k M L X M L / / X tan N N v f i N f / tan e L / e f v 2L j jn N N f i X X X 2k 0 L L / 2 2 j 2 0 / f X X 2 f L
20 Example of Analytical Method Resonator Setup M L / / f f 2L L 2 L / / f f L = 0.8 m R = 0 m f eff = 5 m X N tan 2 L / f M 0 0 I 20
21 Numerical Method of Determining the Number of RTs to Image A ray launched parallel to the optical axis will oscillate back and forth across the resonator cavity Round  Trip ABCD Matrix M rt M rt 0 0 Rays selfreplicate every ~ round trips through the cavity 2
22 maximum resonator angle / maximum ray angle(radians) Angular Bandwidth Requirement Dependence on RTs to Image Data y=0.66 * x Varying the end mirror radius of curvature (ROC) allows us to study the relationship between the iterations to image and the reduction in the maximum angular content N  iterations to image 22
23 WaveOptics Mesh Initial Conclusions When modeling a stable resonator, the reduction in required angular bandwidth can be reduced by approximately the number of roundtrips (iterations) required to image times 2/3. The number of roundtrips required to reimage can be determined numerically, graphically, or analytically (through eigen analysis). 23
24 Stable Resonator Without Gain 24
25 Example Laser Resonator Setup Secondary Mirror Flat / R nominal =95% Primary Mirror ROC = 0m R nominal =00% 0.8 m 25
26 26 Wave Train Model
27 Gaussian that reproduced itself in a roundtrip matched theory. Experiment: Launch a Gaussian beam through a single round trip and evaluate the size after one roundtrip..067 mm Theory : mm Note: mesh spacing was 0.02 mm 27
28 Parameters R c (Radius of curvature of secondary mirror) = 0 m Cavity Length = 0.8 m Propagation grid = 256 by 56 µm (4.3 mm diameter) Wavelength = µm Reflectivity of output mirror = 95 % Initial Field = BwomikTopHat field of 6 cm initial Radius and 5000 amplitude Normalization = Iterations = 0000 Varying Aperture diameter 28
29 Seeded all laser modes initially with a Bwomik field. Bwomik Field is a plane wave with random phase that tends to seed all the modes of a resonator. This is implemented in LaserGridInitializers.h. 29
30 Converged Resonator Field Dependence on Internal Aperture Diameter 30
31 Larger aperture results approximate the theory more accurately. Iterations = 0000 Aperture diameter = 2.0 mm Aperture diameter = 4.0 mm NOTE: Theory is TEM 00 shape. 3
32 Bigger apertures require more iterations to converge. Iterations = 0000 Aperture diameter = 5.0 mm Aperture diameter = 7.0 mm NOTE: Theory is TEM 00 shape. 32
33 The 5mm case begins to approach convergence after 50,000 iterations. Iterations = Aperture diameter = 5.0 mm Iterations = Aperture diameter = 5.0 mm NOTE: Theory is TEM 00 shape. 33
34 The 5mm diameter aperture was reasonably well converged after 75,000 iterations. Iterations = Aperture diameter = 5.0 mm 75,000 iterations takes about 3 hours NOTE: Theory is TEM 00 shape. 34
35 Intensity CrossSection Comparison for 5mm Aperture Cases with Different Iteration Numbers 35
36 Variation of Output Intensity with Last 99 Frames of 0,000 with 5mm Diameter Aperture Note the repeating pattern every frames. 36
37 Variation of Output Intensity with Last 99 Frames of 50,000 with 5mm Diameter Aperture Note the repeating pattern every frames.
38 Variation of Output Intensity with Last 99 Frames of 75,000 with 5mm Diameter Aperture All these frames look good
39 Stable Resonator Model Conclusions The WaveTrain stable resonator model showed that the models with larger aperture (~5 times w 0 ) converged very close to the theoretical shape in many (75,000) iterations. Even fairly early in the iterations, the beam intensity shape repeats itself every iterations, as is predicted by theory. 39
40 Conclusions In stable resonator modeling, the number of roundtrip iterations to image directly impacts the waveoptics mesh requirements. We showed three different techniques for determining the number of roundtrips to image: analytical, graphical, and numerical. Using this theory, we have shown that our stable resonator model without gain matches well with the theoretical predictions. 40
41 Future Work and Acknowledgements We need to perform more anchoring to experimental data to complete the verification of this new technique. Experiment: Aperture in a small stable resonator. We need to consider how the different frequencies impact model performance. Acknowledgements A. Paxton and A. E. Siegman for technical discussions Funded via the LADERA contract 4
42 Questions? (505) x22 42
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