Laser Resonator Modeling with VirtualLab 5.8

Webinar, 06 th November 2013 Laser Resonator Modeling with VirtualLab 5.8 Daniel Asoubar, LightTrans VirtualLab UG Daniel.Asoubar@lighttrans.com

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Webinar Topics Resonator analysis and tolerancing: Transversal eigenmode calculation Influence of thermal lenses in resonators Laser output power calculation for inhomogeneous optical pump light Customized laser beam profile generation using micro-structured mirrors

Resonator analysis: Transversal eigenmode calculation

Optical Modeling Task Light distribution in source plane is given. Light distribution in target/detector plane is demanded. Light distribution is to be analyzed to allow evaluating the optical function of the system.

Optical Modeling Task Source modeling Propagation Detection

Resonator Modeling by Field Tracing Mirror Source modeling Propagation in laser cavity Mirror Detection

Tendencies: Detectors Innovative optical design often requires accurate access (merit function) to e.g. Amplitude and phase Polarization, e.g. Stokes or Jones vector Degree of polarization and coherence Poynting vector, energy flow Pulse duration, chirp, Light representation Ray bundles allow to handle some of the situations. Electromagnetic fields allow to handle all situations.

Resonator Modeling by Field Tracing Mirror Source modeling Propagation in laser cavity Mirror Detection

Field propagation in resonators Detector Mirror Mirror We can not use Ray Tracing because: Diffraction effects at apertures have significant influence on transversal resonator modes What simulation technique should be used for field propagation through resonators?

Tendencies in Optical Science and Technology Enormous variety of different types of optical surfaces: Smooth freeform surfaces

Tendencies in Optical Science and Technology Enormous variety of different types of optical surfaces: Smooth freeform surfaces Microstructured surfaces

Tendencies in Optical Science and Technology Enormous variety of different types of optical surfaces: Smooth freeform surfaces Microstructured surfaces Multilevel surfaces

Tendencies in Optical Science and Technology Enormous variety of different types of optical surfaces: Smooth freeform surfaces Microstructured surfaces Multilevel surfaces Miniaturized components Surfaces with different feature sizes are combined in optical systems. One modeling technique for all of that?

Optical Modeling by Rigorous Technique Detector Mirror Mirror Rigorous modeling in part of optical system reasonable and often necessary!

Optical Modeling by Rigorous Technique Detector Mirror Mirror Rigorous modeling of entire optical system not practical!

Resonator Modeling by Field Tracing Mirror Mirror Propagation Detection in laser cavity Field Tracing Light representation: Electromagnetic field Light propagation for single resonator round trip: Combination of rigorous and approximate solutions of Maxwell s equations in different regions of the system Iterative procedure for mode analysis

Combining Modeling Techniques Unified Modeling Geometrical Optics Finite differene time domain (FDTD) Thin element approximation (TEA) Layer matrices Beam propagation method (BPM) Fresnel integral Rigorous coupled wave approach (RCWA) more Finite element method (FEM) Spectrum of plane waves (SPW) Fourier modal method (FMM)

Resonator types In literature we can find 3 different kinds of laser resonators: Stable Unstable Ringresonator

Resonator types In literature we can find 3 different kinds of laser resonators: Stable Unstable

Resonator types In literature we can find 3 different kinds of laser resonators: Stable Unstable Ringresonator

Scenario 508 (1.2) Mode Analysis of Laser Ring Resonator Author: Daniel Asoubar (LightTrans) Requirements: VirtualLab 5.8 Starter and Laser Resonator Toolbox License: CC-BY-SA 3.0

Calculation procedure We will use the Fox&Li Algorithm for the mode analysis of a laser ring resonator using. Therefore a single resonator round trip is built up in the Light Path Diagram of VirtualLab TM. Then the external module Scenario_508_LPD_Module_Fox-Li.cs is applied to calculate the most dominant transversal resonator eigenmode.

The simulation task Calculation of the most dominant transversal eigenmode of the following laser ring resonator. Top view of laser ring resonator 10mm 6 mm M 1 ideal plane mirror reflectance R = 0.95 Circular aperture with diameter D = 400 µm M 3 ideal plane mirror reflectance R = 1 10mm 4 mm M 2 ideal plane mirror reflectance R = 1

The simulation task To build up a single resonator roundtrip we have to unfold the ring resonator setup at the position we like to calculate the eigenmode. Top view of laser ring resonator M 3 ideal plane mirror reflectance R = 1 3mm Position for unfolding resonator 7 mm 10mm 6 mm M 1 ideal plane mirror reflectance R = 0.95 Circular aperture with diameter D = 400 µm 4 mm M 2 ideal plane mirror reflectance R = 1

Building up single resonator roundtrip In the Light Path Diagram the single roundtrip will look like in the following way. After the first round trip the field obtained in the Virtual Screen will be stored in the start field component by the Fox-Li module and the next round trip will be calculated. This step will be repeated iteratively.

Start field Any monochromatic light source which is available in the light path diagram can be used as the starting condition of the Fox&Li Algorithm. To ensure fast convergence of the eigenmode a good initial guess should be used. This guess can be e.g. the expected dominant eigenmode of the resonator system.

Advanced start fields Random phase field can be loaded in the Light Path Diagram using the Stored Complete Field Light Source. Phase of plane wave multiplied with random-phase only transmission function

Sampling of round trip operator To ensure a correct sampling please always place directly in front of the Virtual Screen a sampling component and manually choose a sufficient sampling distance and computational domain size.

Results Evolution of eigenvalue amplitude Dominant transversal eigenmode of ringresonator setup

Tolerancing VirtualLab allows position tolerancing of all components. Top view of laser ring resonator M 3 ideal plane mirror reflectance R = 1 3mm Position for unfolding resonator 7 mm 10mm 6 mm M 1 ideal plane mirror reflectance R = 0.95 SHIFTED, circular aperture with diameter D = 400 µm, shift x=0.1mm 4 mm M 2 ideal plane mirror reflectance R = 1

Tolerancing

Results: Tolerancing Dominant mode of resonator with centric aperture Dominant mode of resonator with off-centric aperture

Summary I VirtualLab can do a wave-optical simulation of any kind of resonator using Fox-Li Algorithm. Therefor the resonator round trip operator can be easily build up in the Light Path Diagram. The field tracing concept allows a plenty of different propagation techniques for calculation of the round trip operator. Inclusion of position and fabrication tolerancing.

Transversal Mode Analysis of Unstable Resonators

Modeling Task Mirror 1 Mirror 2 D = 58.8 mm L = 500 mm D = 20 mm Radius of curvature mirror 1: R1 = -1500 mm Radius of curvature mirror 2: R2 = 500 mm Reflectance of mirrors: 1 Transmittance at outcoupling mirror 2 (indicated by blue arrows): 1

Unstable Resonators Step 1 Mirror 1 Mirror 2 Aperture D = 58.8 mm L = 500 mm D = 20 mm Radius of curvature mirror 1: R1 = -1500 mm Radius of curvature mirror 2: R2 = 500 mm Reflectance of mirrors: 1 Transmittance at outcoupling mirror 2 (indicated by blue arrows): 1

Unstable Resonators Step 1 Mirror 1 Mirror 2 Aperture D = 58.8 mm L = 500 mm D = 20 mm Unfolding position

Unstable Resonators Step 1 Dominant mode after step 1

Unstable Resonators Step 2 Mirror 1 Replace Mirror 2 by stop D = 58.8 mm L = 500mm Stop D = 20 mm Unfolding position

Unstable Resonators Result Outcoupled mode

Influence of thermal lenses in resonators

The simulation task Let s take the ring resonator setup again. Inclusion of active medium (axial pumped) Top view of laser ring resonator M 3 ideal plane mirror reflectance R = 1 3mm Position for unfolding resonator 7 mm 10mm 6mm 6 mm M 1 ideal plane mirror reflectance R = 0.95 Circular aperture with diameter D = 400 µm 4 mm Pump Laser M 2 ideal plane mirror reflectance R = 1

Refractive Index Distribution In optical pumped laser crystals: Temperature gradient due to inhomogeneous pump light distribution. Heat distribution causes spatial refractive index distribution Thermal lensing Heat conduction in crystals can be modeled by external thermodynamics software. VirtualLab allows the data import of the refractive index distribution n(x,y,z).

Refractive Index Distribution Refractive index distribution in transversal plane (x,y) at z-position of 1 mm from the front surface.

Surface Deformation Temperature gradient causes thermal stress surface deformation h(x,y) at front- and back-side. VirtualLab allows the data import h(x,y).

Results Dominant transversal eigenmode of ringresonator setup without thermal lens. Dominant transversal eigenmode of ringresonator setup with thermal lens.

Scenario 510 (1.1) Laser Resonators with Gain Author: Daniel Asoubar (LightTrans) Related Application Scenarios: 508 Requirements: VirtualLab Advanced 5.8.0 Starter and Resonator Toolboxes License: CC-BY-SA 3.0

Modeling Task Calculation of dominant transversal resonator mode: Resonator includes saturable active medium, which is axialpumped by Gaussian-shaped pump light. The mirrors have circular aperture. Spherical Mirror M1 Radius of curvature = - 141.3 m Aperture Diameter = 1 mm Reflectance R 1 = 1 Spherical Mirror M2 Radius of curvature = -6 km Aperture Diameter = 0.8 mm Reflectance R 2 = 0.95 6 mm 10 mm Active medium (e.g. Nd:YAG)

Overview The following simulations have to be done to tackle the task: 1. Simulation of pump light distribution inside the active medium using Scenario_510_Pump_setup_1.lpd 2. The resonator setup in the file Scenario_510_Roundtrip_2.lpd defines a single round trip through the resonator. For further details of the round trip concept in VirtualLab please check Scenario_508. 3. Power method (Fox-Li Algorithm), Rate Equations and Field Tracing Operators are used to calculate dominant transversal mode with correct output power. Therefore the Fox-Li-Algorithm is applied by Scenario_510_Fox- Li_3.cs.

Theory of simple 4-level system The light amplification inside the active medium can be described by the rate equations. If we only consider the transitions given below we can write the rate equations in the following way: E 3 E 2 N 3 N 2 γ 32 dn 3 dd W p N 3 N 0 N 3 γ 32 dn 2 dd W sss ΔN N 2 γ 21 + N 3 γ 32 W sss γ 21 W p dn 1 dd W sss ΔN + N 2 γ 21 N 1 γ 10 dn 0 dd W p N 3 N 0 + N 1 γ 10 E 1 N 1 N = N 0 + N 1 + N 2 +N 3 N 0 E 0 γ 10 N 0 with N j being the population of energy level j W p the transition probability of the pump W sss the transition probability due to stim. emission γ jj the decay rate due to spontaneous emission.

Theory of simple 4-level system In steady state there is no change of population of energy levels, and we can write the gain coefficient in the form: g 0 E p (x, y, z) 2 g = 1 + E+ sss(x, y, z) 2 + E sss(x, y, z) 2 I sss with the electric field of the pump beam E p and the forward and backward running electric field of the laser beam to be amplified E + sss x, y, z and E sss x, y, z. All relevant material constants of the active medium can be included in g 0 and I sss. For a simple 4-level system the constants are given by: g 0 = 1 β n pττττε 0 n(v p ) I 4hv sss = 4hv sss (2a) p τττε 0 n(v sss ) (2b) with the material constants according to: β - heat efficiency factor σ - stim. Emission cross -section n p - pump efficiency α - absorption coefficient τ - spon. fluo. life time of upper laser level c - speed of light ε 0 - vacuum permittivity h - Planck s constant v p/sss - light frequency of pump/laser light n - refractive index of active medium (1)

Active medium parameter The following parameter have been used for the active medium: Parameter Value β 0.02 σ n p 0.8 2.8E-17 1/mm α 350 1/m τ 230 µs λ sss λ p 1064 nm 600 nm n 1.823

Simulation 1: Pump light distribution Parameters of pump light setup: - Axial optical pump setup - Pump source: Gaussian diode laser with Waist radius: 1.5 mm Wavelength: 600 nm Pump power: 20 W Light distribution of diode laser used for optical pump - Absorp. coefficient α of active medium: 350 1 m - Field distribution inside the active medium is calculated by beam propagation method (BPM). Number of BPM steps: 41 - Please ensure that the Pump Beam medium is already embedded in the active medium refractive index (here n=1.823). Scheme of optical pump setup in Light Path Diagram

Simulation 2: Resonator round trip operator After calculation of pump field distribution, the resonator round trip operator is build up in Scenario_510_Round-trip.lpd file. Therefore we have to unfold the resonator in the following way: Spherical Mirror M2 Spherical Mirror M1 10 mm 6 mm 6 mm 10 mm Active medium (Nd:YAG) Active medium (Nd:YAG) Backward Propagation Forward Propagation

Initial guess transversal field distribution Dominant eigenmode and eigenvalue evolution of cold cavity analysis:

Initial guess field power Once we have a good initial transversal field distribution we can calculate an energy correction factor ε which has to be multiplied with the cold cavity dominant mode. For small resonator losses we can write the correction factor by: ε = 4g 0 E p 2 zhvsss R 2 ττ with z 2 2 E p exp αz ddd E p = 0 z 2 and E p = max Ep (x,y,0) 2 = E p 2 1 exp αα zz

Initial guess field power In our example we have and E p 2 = 978705421.1 V 2 ε = 1.9064 10 5 which has to be multiplied the cold cavity dominant mode using Manipulation > Operation with constant > Multiplication > m 2

Initial guess field power Finally our initial guess is: ε =

Results After 170 iterations (approx. 3 hour calculation time) the Fox-Li algorithm will reach the given convergence criteria simulation.tolerance = 1E-6 and the finale results will be shown: Power of field inside the cavity for different iteration steps. Eigenvalue for different iteration steps.

Results transversal mode distribution Furthermore the dominant transversal mode pattern inside the cavity directly in front of the outcoupling Mirror M2 is given by VirtualScreen #603: Dominant transversal resonator mode of cavity including saturable gain which is inhomogenously pumped

Results output power The output power of the laser resonator can be calculated by multiplying the power of the dominant transversal mode pattern with the mirror reflectance R 2. There are several ways to display the power of the dominant mode: First option: Click on View > Show Property Browser Second option: Go to Table in the Power of field inside the cavity for different iteration steps window and look at the power value at the last iteration

Results output power After multiplication of the cavity field power with the reflectance of the mirror M 2 our final output power will be P ooo = 11.16286847 W 0.05 = 0.5581 W

Summary II VirtualLab can simulate active media with gain saturation including output power calculation. Wave-optical modeling of pump light distribution. Beam propagation method (BPM) enables waveoptical propagation inside nonlinear active media. Nonlinear effects like spatial gain broadening can be modeled. Fox-Li algorithm enables calculation of dominant transversal modes of stable, unstable and ring resonators.

Scenario 12.01: Using a microstructured mirror for the design of the Eigenmode of a resonator. In this application scenario we show how a resonator with a micro-structured mirror can be designed such that the Eigenmode has a pre-defined amplitude. In particular we design an Eigenmode with a top-hat shape. Keywords: laser resonator, eigenmodes, eigenvalues, micro-structured mirror, beam control, design, top-hat Required Toolboxes: Laser Resonator Toolbox Related Tutorials: FS.009 Related Application Scenarios: 08.01, 09.01

Modeling Task 100 mm Micro-structured mirror = Plane Mirror + Phase Function Plane mirror Outcoupling mode (Top hat) Wavelength 532 nm

Design of the phase function We know from free-space propagation: P SSS

Design of the phase function We know from free-space propagation: cccc. P SSS

Modeling Task P SSS 100 mm cccc. P SSS

Design of the phase function Step 1: Generate a field representing the outcoupling mode. E.g. use a plane wave with a diameter of 300 µm. It represents a top hat.

Design of the phase function Step 2: Propagate the field (use SPW) by the distance of the two mirrors, 100 mm in our case (left picture). Embed (via Sampling Manipulation) to 700x700 pixels (right picture). Sampling distance and embedding is problem dependent.

Design of the phase function Step 3: Conjugate the phase via Phase manipulation. conjugate E x, y = t x, y E(x, y)

Design of the phase function Step 4: Divide the conjugate phase by the original phase function via Array-Array-Operations (left picture). Convert to a Jones Matrix Transmission via Conversion (right picture). t x, y = E x, y E(x, y)

Using the phase function The phase function can be used in a Stored Mirror Function. The phase function is designed such that: For the phase of the design outcoupling mode the reflected phase is the conjugate of the incident phase.

Simulation Results Simulated fundamental mode of the laser resonator

Summary III Laser Resonator Toolbox can model microstructured mirrors and components. Diffraction effects at micro-structured elements and apertures are included in eigenmode calculation. Micro-structured mirrors can be used to design fundamental mode shape inside and outside of resonator.

Webinar Summary VirtualLab can do a wave-optical simulation of standingwave, ring and unstable resonators using Fox-Li or Arnoldi Algorithm. Therefor the resonator round trip operator can be easily build up in the Light Path Diagram, enabling a plenty of different propagation techniques. Inclusion of position and fabrication tolerancing. VirtualLab can simulate active media with gain saturation including output power calculation. Wave-optical modeling of pump light distribution. Beam propagation method (BPM) enables wave-optical propagation inside nonlinear active media. VirtualLab can simulate thermal lens effects.

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