# 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 Wave-Optics Mesh Multi-Iteration Imaging Stable Resonator Modeling Conclusions 2

3 Introduction and Motivation When evaluating a new laser gain medium, it is common to build a multi-mode stable resonator around the gain medium to demonstrate maximum extraction. Modeling multi-mode lasers is difficult because The modes tend to interact with the saturable gain medium and create modeling instabilities and The mesh requirements for a multi-mode 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 on-axis ray) will eventually escape from the resonator. 4

5 Hermite-Gaussian Modes HG=Hermite-Gaussian HG modes are orthogonal modes of a stable resonator with infinite internal apertures. HG modes tend to appear clearly in rectangular-aperture resonators. 5

6 Laguerre-Gaussian Modes LG=Laguerre-Gaussian LG modes are orthogonal modes of a stable resonator with infinite internal apertures. LG modes tend to appear clearly in circular-aperture resonators. 6

7 The Iterative Fourier Transform (aka Fox & Li) Technique A field is propagated through repeated round-trips 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. Gerchberg-Saxon 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. QE-4, (July 968). 7

8 Comments on Fox & Li Solutions Solution is for an instantaneous state, which is typically the steady-state 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 higher-order mode. Not generally appropriate for pulsed or timevarying solutions unless the time-varying nature is much slower than a resonator round-trip time. This is analogous to a split-time modeling techniques. 8

9 Example Resonator: RADICL Laser Output R c = 0m λ =.353e-6m 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 Water-Vapor 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 Wave-Optics Mesh Determination

12 Calculation : Half-Round-Trip Estimate based on two limiting apertures analysis [Coy/Mansell]: N 4DD z 2 4 6cm.35 m 6cm 80cm 3,688, N 2.87E8 8.8x0-6 m NOTE: This is for a half of a round trip. 3,688 points per dimension Very Difficult 2

13 Calculation 2: Full Round-Trip 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/2-L)) = ( ) 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 round-trips Graphical Method of Determining the Number of Round-Trips to Image 5

16 Number of Round-Trips to Image/Repeat Plano-Concave Stable Resonator Linearized Plano-Concave 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 re-enter 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 well-represent the minimum mesh size required. NOTE: 3,688 / 58 ~ = number of round-trips 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 ~ round-trips 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 Round-Trips to Image This problem can be addressed using ray matrices. Consider a resonator with a round-trip ray-matrix given by M. In N round-trips, the ray-matrix 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 plano-concave 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 self-replicate 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 Wave-Optics Mesh Initial Conclusions When modeling a stable resonator, the reduction in required angular bandwidth can be reduced by approximately the number of round-trips (iterations) required to image times 2/3. The number of round-trips 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 round-trip matched theory. Experiment: Launch a Gaussian beam through a single round trip and evaluate the size after one round-trip..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 5-mm 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 Cross-Section Comparison for 5-mm Aperture Cases with Different Iteration Numbers 35

36 Variation of Output Intensity with Last 99 Frames of 0,000 with 5-mm Diameter Aperture Note the repeating pattern every frames. 36

37 Variation of Output Intensity with Last 99 Frames of 50,000 with 5-mm Diameter Aperture Note the repeating pattern every frames.

38 Variation of Output Intensity with Last 99 Frames of 75,000 with 5-mm 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 round-trip iterations to image directly impacts the wave-optics mesh requirements. We showed three different techniques for determining the number of round-trips 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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