COMPUTER MODELING OF LASER SYSTEMS

Transcription

1 University of Ljubljana Faculty of mathematics and physics Department of physics COMPUTER MODELING OF LASER SYSTEMS POSTGRADUATE SEMINAR Dejan Škrabelj adviser: dr. Marko Marinček adviser: prof. dr. Irena Drevenšek-Olenik february 2007 Abstract Pulses produced by a Q-switched solid-state laser are found very effective in tatoo removal application. To assure equal illumination of a treating tissue an unstable laser cavity output coupled with a supergaussian mirror producing a top-hat near field intensity profile is preferred to a stable one. Construction of a laser system, especially one operating with an unstable cavity, usually begins with computer simulations. In the seminar the model developed for treating a Q-switched laser is present. Its feature is a short computational time of one-pulse-buildup cycle, although it is based on a wave optics and predicts parameters as pulse energy, temporal pulse width, pulse divergence and transverse intensity profiles of the pulse at different distances inside and outside of a laser cavity. The results of simulations were found to be in good agreement with measurements performed on a ruby laser system running in a stable as well as in unstable cavity configuration. The ruby system operating with an unstable cavity, planned to be used for tatoo removal, has to be further optimized in order to produce a top-hat near field intensity profile. Fotona d.d. Institut Jožef Stefan Faculty of mathematics and physics

2 Contents 1 Introduction Tatoo removal with laser Basic laser physics Q-switch technique Laser model for Q-switched lasers Free space propagation Propagation through a lens Thermally induced lensing of the laser rod neto heating Effect of a mirror Q-switch element, population inversion and gain Simulation of a ruby laser system 11 4 Conclusion 13 2

3 1 Introduction Laser light as being powerful and nearly monochromatic it is not surprising that it has found widespread use in many areas of modern world. Lasers are present in metrology, range finding, biology, material processing, science research, medicine and information technology. Fotona, a Slovenian company, is one of the leaders in the development and production of medical laser systems. One of the latest projects, a ruby laser system, which will be used for tattoo removal, is under active investigation at the moment. In the early development phase of a new laser system it is very important to select appropriate optical components of the system as quick as possible. A good computer simulation model able to predict properties of a laser beam produced by a real laser can enormously reduce development time and development expenses. In the first part of this seminar we will briefly describe desirable characteristics of the laser light that is appropriate for tattoo removal application. Through this description we will find the parameters that should be predicted by a simulation model in order to be helpful in development process. After that, a short description of a general solid state laser will be provided to get an idea about model input parameters. The central part of this seminar will be devoted to the description of the simulation model for a Q-switched pulsed laser that was developed at Fotona in cooperation with the University of Ljubljana. In the last part we will focus on a development of the new ruby laser. Comparison between the simulations and experiments will be provided. 1.1 Tatoo removal with laser Tattoos are constituted from pigment molecules inserted into the skin. Unfortunately, after time many tattoos become a burden for their owners and they want to get rid of them. The most effective non-destructive method for tattoo removal is a laser treatment. Recent market analysis has identified 41 pigments [9] whose absorption peaks depend on the wavelength of the incident light. For example, the blue pigment can be successfully removed with the ruby laser of wavelength λ = nm. To induce the pigment break-down, the high-power laser pulse must be delivered to the pigment. Decayed pieces of the pigment are then removed from the body with its own immune system. High power pulses are usually obtained with Q- switch technique; typical width of the pulse produced by a Q-switched ruby laser is t 30 ns which means the average pulse power of 33 MW at the pulse energy E 1 J. Another strongly desirable property of the tatoo-removing light is a uniformly distributed intensity in the whole illuminated spot area in the near field of the laser beam. Such pattern assures that all parts of the skin are equally illuminated. The uniformly distributed profile in the near field called a top-hat is usually obtained with an unstable laser cavity composed of a supergaussian mirror output coupler. So far we have talked only about the laser light characteristics that should be fulfilled 3

4 to make tattoo removal effective. Of course, there are also some other important demands which must be taken into consideration during the laser system development. Because of its coherence properties, laser light propagation often leads to very high intensity maxima present on a small part of the spot area. Thus special care must be devoted to the analysis of such spiking behavior, because a high local intensity can exceed the damage threshold of a laser component and thus leads to its destruction. The same problem can occur if the effective beam radius is too small. For the propagation purposes we have to know the effective beam radius and its divergence inside and outside of the resonator. Effective radius should not exceed dimensions of the optical elements, as otherwise the propagating beam becomes deformed. Another important effect that has to be taken into account is thermal lensing of the laser rod, which can seriously deform the shape of laser profile. It follows that a good model should predict parameters for a given cavity configuration such as: pulse energy, pulse width, intensity distribution in a plane perpendicular to the propagation direction, effective beam radius at different distances, etc. 1.2 Basic laser physics Laser is an optical oscillator constituted with two mirrors. Optical wave is reflected back and forth between those mirrors. In order to couple some of the light out of the resonator, one of the mirrors must be partially transmittive. At each round-trip the losses in the resonator are compensated by the amplification process based on stimulated emission. In the case of a solid state laser, the process takes place in a laser rod, made of a doped crystal. Population inversion 1 in the active medium, which is a necessary condition for the amplification, is obtained by optical pumping with an external light source that has an appropriate spectrum for exciting the dopant ions. In the excited state they spend a limited amount of time, before they decay back to the ground state. In order to achieve population inversion, the lifetime of the upper laser level must be long compared to the typical transient time from the ground to the upper laser level Q-switch technique The model presented here considers Q-switched pulsed lasers. In the Q-switch technique an additional element is put into the resonator which mediates the resonator losses. At first the losses Λ have to be high, unity. During this period, all of the produced radiation is filtered out of the resonator. Because a flash lamp continuously delivers energy to the laser medium through this part of the cycle the population inversion builds up. When the losses are reduced to the minimal value, photons begin to bounce between back and front resonator mirrors. Because of the high population inversion the photon flux inside of the resonator starts to increase. During the laser pulse formation the population inversion density n and 1 The state where there are more ions in excited than in ground state. 4

5 Figure 1: Q-switch theory: laser pulse formation [2]. the photon density Φ obey the equations [2] n t Φ t = σcφγn, (1) = (σcn l l Λ t R )Φ, (2) where σ is the stimulated emission cross section, c is the speed of light in the laser rod, l is the length of the laser rod, l is the length of the resonator, t R = 2l /c is the round-trip time inside the resonator and γ is 1 + g 2 /g 1, with g 2 /g 1 being the ratio between upper and lower laser level degeneracies. This factor is important in three-level lasers; in the ruby case it is γ ruby =2. In four-level lasers it is γ 4 = 1. The process of laser pulse formation is present in Fig. 1. The Q-switch element in the ruby laser system is constituted of an electrooptic modulator (Pockels cell) and a polarizer, which makes the laser light linearly polarized. The Q-switch element is placed between the laser rod and the back mirror. From the general laser description we got an idea which should be typical input parameters. We see that the influence of all the individual resonator elements on the general form of the EM field has to be determined. We have to simulate the initialization of the lasing process, optical gain and thermal lensing effect of the laser rod, Q-switch element, etc. 5

6 2 Laser model for Q-switched lasers The EM field of the laser light is expressed as E(r, t) = A(r, t)e iϕ(r,t), (3) where A(r, t) is the amplitude and ϕ(r, t) the phase of the EM field at the position r and time t. Let us assume that the laser light is propagating along z axis of the resonator. At particular planes perpendicular to the z axis the EM field is changed due to the optical components presented at those planes. Before we can discuss how a particular optical element influences the EM field, we have to know how it is modified when it propagates through free space for a distance d. 2.1 Free space propagation If the field E 1 (x 1, y 1 ; z 1 ) in the z 1 plane is known the field E 2 (x 2, y 2 ; z 2 ) can be determined by the diffraction integral. In our model a method based on the 2D Fourier transform approach is used instead. At first we have to evaluate the transversal spatial spectrum of the EM field e t at the plane z e t (s x, s y ; z) = E(x, y, z)e 2πi(sxx+syy) dxdy, (4) where s x and s y are the transversal spatial frequencies, defined as s x,y = kx,y 2π and k x and k y are the x and y component of the wave vector k. Spatial frequencies measure the number of cycles per unit length of the waves which constitute the EM field. The propagating field must obey the amplitude wave equation 2 E(r) + k 2 E(r) = 0. We put the transversal spectrum (4) into the amplitude equation and get d 2 dz 2 e t(s x, s y ; z) + ( 2π λ )2 (1 λ 2 s 2 x λ 2 s 2 y) e t (s x, s y ; z) = 0, (5) where λ is the wavelength of the light. When the field E(x, y; z 1 ) is propagated for a distance d = z 2 z 1, its transversal spatial spectrum e t (s x, s y ; z 1 ) is changed to e t (s x, s y ; z 2 ) = e t (s x, s y ; z 1 ) e i 2π λ 1 λ 2 s 2 x λ2 s 2 y d. (6) The inverse Fourier transform of (6) is the EM field at position z 2 E(x, y; z 2 ) = e t (s x, s y ; z 2 )e 2πi(sxx+syy) ds x ds y. (7) In the numerical calculation we cover the x-y plane of the EM field with n n mesh points. Therefore we have to calculate 2D discrete Fourier transform and its inverse on the mesh points. Usually a fast Fourier transform algorithm is used. 6

7 2.2 Propagation through a lens Optical lens is characterized by its focal length f, which is given as 1 f = (n 1)( 1 r r 2 ), (8) where r 1 and r 2 are the radii of curvatures and n is the refractive index of the lens material. From equation (8) it can be seen that a curved partially transmitting mirror acts as a lens on the transmitted part of the wavefront. If optical wave passes through the slice of a medium with refractive index n and of thickness d(x, y) its phase is modified as: ϕ(x, y) = nk 0 d(x, y) + k 0 (d 0 d(x, y)) = (9) = nk 0 d 0 + (n 1)k 0 d(x, y), (10) where k 0 = 2π λ 0 is the magnitude of the wave vector in vacuum and d 0 is the maximal width of the medium in the propagation direction. The terms in equation (9) represent phase addition inside and outside the medium, respectively. The first term in (10) is constant and thus of no importance. The spherical profile d(x, y) for a thin lens leads to the lens transmittivity factor t L = E tr = e ik(x 2 +y 2 ) 2f, (11) E in which presents the ratio between the transmitted (E tr ) and incident (E in ) EM field Thermally induced lensing of the laser rod neto heating Laser rod acts like a lens, too. This effect has a strong influence on the EM field development inside the resonator, so it has to be considered in the real laser system design. Let us assume that the heat produced by a flash lamp is uniformly absorbed inside the laser rod. The radial temperature distribution in a cylindrical rod with thermal conductivity D can be obtained from the heat conduction equation. If the temperature on the laser rod surface is equal to T (r 0 ) the radial temperature profile is expressed as T (r) = T (r 0 ) + Q 4D (r2 0 r 2 ). (12) Q = P b /(πr 2 0 L) is the heat flow generated per unit volume of the laser rod. P b is the absorbed power in the laser rod, r 0 is the radius and L length of the rod. The temperature-dependent change of the refractive index is expressed as From equation (12) we get n(r) = [T (r) T (r = 0)] dn dt. (13) n(r) = Q dn 4D n(r) = n(0) Q 4D 7 dt r2, (14) dn dt r2. (15)

8 The phase change produced by the laser rod is equal to ϕ(r) = k 0 n(r). The transmittivity factor of the rod introduced by heating effect is thus where e ik 0n(0) e ik 0 Q 4K dn dt r2 = (16) = e ik0n(0) e ik 0 (x 2 +y 2 ) 2f rod, (17) f rod = 2K Q 1. (18) dn dt The first term in (17) is constant and of no importance. The second has the form of lens transmittivity factor (11). 2.3 Effect of a mirror Laser resonator is bounded with two mirrors. Back resonator mirror has reflectance R = 1 and a radius of curvature r, so it changes only the phase of the reflected wavefront. The phase change introduced by back resonator mirror can be obtained with similar analysis as for the lens. It is equal to t M = e ik(x2 +y 2 ) r. (19) The radii r < 0 and r > 0 represent concave and convex mirrors, respectively. Back mirrors are usually concave. The outcoupling mirror transmits part of the wavefront out of the resonator and reflects other part back into the cavity. For the transmitted part the front mirror acts like a lens, so the phase addition is obtained from the equation (11). The phase modification of the reflected part has to be multiplied with (19). The amplitudes of the transmitted and reflected EM field are changed, in addition. They are equal to A t = 1 R A i and A r = R A i where R represents the reflectance of the front mirror and E i the magnitude of the incident field. Before we introduce a (super)gaussian mirror some additional laser resonator optics has to be explained. The curvatures of the mirrors (r 1, r 2 ) and the resonator length L determine the type of the resonator, which are generally speaking divided into stable and unstable family. Beam supported by the resonator is characterized with the M 2 factor, which measures its quality. The quality is determined by the beam parameter product (BPP), which is defined as the product of the beam radius at its waist w and its divergence ϑ: BP P = wϑ. For a given beam the BPP represents an invariant quantity, which means that even if the beam passes through an optical system, its BPP is preserved. The minimal BP P = λ/π is characteristic for the fundamental Gaussian beam. For non-gaussian beams the BPP is increased by the factor M 2 meaning that a Gaussian and non-gaussian beams with the same effective radii at the waist point the Gaussian beam will have lower divergence than the non-gaussian one. The Gaussian beam is thus often called the diffraction limited beam and is required for some applications. The EM field of a stable resonator is often constituted of the sum of Hermite-Gaussian and Laguerre-Gaussian beams of higher orders and not only from the zero order Gaussian beam. 8

9 In such a case the M 2 parameter of the beam increases because higher order modes have greater effective radii and greater effective divergences. If we allow only the Gaussian beam to lase inside the resonator, the laser rod becomes poorly illuminated and the efficiency of the energy extraction is low. A basic difference between a stable and an unstable resonator is that unstable resonator produces beams with low M 2 factor connected with considerable illumination of the laser rod. With use of confocal unstable resonators powerful beams which are nearly diffraction limited can be obtained. Unstable resonators can be output-coupled with (super)gaussian mirrors, which have a non-uniform reflectance profile: R(r) = R 0 e 2( r w )o, (20) where R 0 is the central reflectance, r is the radial distance, w is the effective radius of the mirror reflectance profile and o is the order of the mirror. The center of the mirror reflects more light back into the resonator than its edge. With the proper combination of the resonator and supergaussian mirror parameters we can get a near field profile with uniform intensity distribution. As mentioned earlier, in computer simulation EM field is approximated with n n mesh points. Consequently the resonator elements have to be presented with n n discrete values obtained with given equations. For example, if the supergaussian mirror is used, its reflectance is described with n n array of values obtained from the equation (20). If the mirror with uniform reflectance R is used the array is filled with equal numbers, namely R. 2.4 Q-switch element, population inversion and gain The Q-switch element is approximated with a nearly step function (see Fig. 4) which considers a realistic transit between high and low losses level of the Q-switch element. The initial population inversion is the model input parameter and is equal to n i = n 0 [α + (γ 1)(α 1)], where n 0 represents the number density of dopant ions and parameter α = n iu /n 0 measures initial portion of atoms present on the upper laser level. We see that n i = n iu in the case of four-level laser system. Spatial distribution of the population inversion is a model input parameter, too; usually the uniform distribution is assumed. At the initialization step at first photon flux density Φ i with randomly distributed phases over the mesh points is generated. It represents the spontaneous emission effect in the real laser system which is the origin for the laser light formation. Their number is predicted using the relation Φ i = a n iu [1 exp( t/τ)]. Constant a measures the portion of the photons emitted in an appropriate solid angle, τ is the lifetime of the upper laser level and t = l/c is the estimation for the time in which the initial photon flux Φ i is generated. Once we determine Φ i and n i their values are modifying due to the equations (1) and (2). The flux density addition represents the gain mechanism. In our model laser resonator is divided into five planes, which are perpendicular to the resonator axis. The first plane represents front mirror, the second the rod entrance plane on 9

10 Figure 2: Schematic drawing of the five effective planes of the laser resonator. Plane 1 represents the outcoupling mirror, planes 2 4 describe the laser rod, and plane 5 describes a Q-switch element and the back resonator mirror. [5]. the front mirror side, the third is the rod center plane, the fourth the rod entrance plane on the back mirror side and the fifth plane represents the back mirror. The EM field formed from spontaneously generated photons 2 is propagated with the Fourier transform technique along the laser rod. For the propagating flux laser rod ends (planes 2 and 4) represent diaphragms with radii equal to the laser rod radius. Therefore the propagating flux has to be appropriately cut, whenever it passes the laser rod end. Transmittivity of the Q-switch element determines the portion of the flux that is propagated to the back mirror (plane 5). At the beginning no flux is transmitted. When losses of the Q-switch element are set to 0, all flux is allowed to propagate to the back resonator mirror where it is multiplied with its phase factor (19). After propagation, the amplification process occurs in the center of the laser rod. Portion of photons with random phases is added to the coherent flux to account for the spontaneous emission contribution. In the plane 3 the flux is also multiplied with the thermal lens transmittivity factor. It propagates further to the front mirror (plane 1). Its reflectivity determines the ratio between reflected and transmitted part. Currently we have two flux planes presented in the resonator that are propagating in the opposite directions. A photon flux plane propagation and its calculations at the resonator planes are repeated until the upper laser level becomes depleted. At the times t j = j t F (j = 1, 2, 3,...) part of the photon flux is transmitted through the outcoupling mirror. t F is equal to the time of flight of one photon through the half resonator optical distance. The pulse intensity is a sum of all individual contributions 2 Magnitude of the EM field component A is related to the photon flux density Φ through the relation A Φ. Phase of the EM field can be assigned also to the photon flux. Representations E and Φ are thus equivalent. 10

11 I(x, y) = j=0 I(x, y, t j). From the intensity of the generated pulse we determine the effective width in the x direction as [4] r x = 2 I(x, y)x2 dxdy. (21) I(x, y) dxdy If we replace x with y we obtain effective beam width in the y direction. The two values are expected to differ in case of a non-uniform population inversion. Of course, double integration translates to double summation in the discrete space. The effective pulse width is estimated as t = j=0 I(t j) t F j=0 I(t j), (22) where I(t j ) represents the intensity contribution summed over the whole plane transmitted through output mirror at the time t j. With the knowledge of the magnitude and phase of the output pulse field it can be propagated further through space. From the far field intensity distribution its divergence and M 2 factor can then be determined. 3 Simulation of a ruby laser system We plan to construct the unstable cavity with a supergaussian mirror output coupler. First step was a construction of a test Q-switch ruby laser with a stable cavity in order to estimate some model parameters as thermal lensing of the rod f rod and population inversion parameter α. The parameters were optimized with comparison between pulse energies and their transversal dimensions obtained with the model and experimentally in the laboratory. We estimated the lensing parameter to be f rod 12 m (model parameter). The estimation of parameter α was meaningless because we decided to reshape both, the laser rod and reflector of the illumination system in order to get better energy efficiency of the system. We did several simulations and chose a supergaussian mirror with parameters P 1 in cavity with parameters C 1 3. The predicted NF profile and pulse shape are present in Figs. 3 and 4. The pulse begins to build up when the resonator losses are reduced. The pulse energy is estimated to be 1 J and the pulse width is t 26 ns, which is for a factor of 3 larger than in Nd:YAG Q-switch laser. We constructed the resonator with reshaped rod and reshaped reflector. Before mounting the supergaussian mirror in the resonator we measured some parameters of the stable resonator constituted from a classical mirror with reflectance R = 40%. In the measured intensity profile structure an unusual pattern occurred, which was not expected. It had the shape of a TEM 0,10 mode as it can be seen in Fig. 5, (A). We anticipated that the pronounced pattern occurred because of the reflection from coatings on the rod end surfaces resulting in an additional resonator formation. A second resonator could be formed between front resonator 3 Exact parameter values are not important for the discussion. 11

12 Figure 3: Predicted NF top hat profile. Figure 4: Predicted temporal development of the pulse and losses inside the resonator. mirror (plane 1) and the laser rod end surface on the back mirror resonator side (plane 4) or between back resonator mirror (plane 5) and the rod surface on the front resonator mirror side (plane 2). It has turned out that the internal resonator formed between planes 1 and 4. At every passage of the photon flux plane through the plane 4 part of it is reflected back toward the front mirror. The flux incident on the outcoupling mirror is the sum of the flux reflected from the back mirror and the flux reflected from the laser rod surface. We modeled the internal resonator in the stable cavity and obtained the profile presented in Fig. 5, (B). Parameter T f = Φ t /Φ i defines the portion of the transmitted flux. In order to obtain the elliptic shape we proposed a non-uniform population inversion. With higher flux transmission value T f 0.99, which is probably more realistic value, the pattern inside of the spot becomes Figure 5: NF profile of the ruby system operating in a stable cavity (A) and profile obtained with the model by setting T=0.99 (B). 12

13 Figure 6: Measured profile obtained with unstable cavity. Figure 7: Model predicted profile with unstable cavity; f rod = 40 m and T f = less intense. Its shape is preserved, however. That is the reason why NF profile obtained with T f = 0.95 is present. Finally we mounted a supergaussian mirror in the resonator. Instead of expected top-hat profile we observed profile presented in Fig. 6. The edge of the spot is more intense than its center. With the help of the model it has turned out that the thermal lensing parameter of the laser rod was wrongly estimated. When a model was run with f rod = 40 m and T f = 0.99, the NF profile shown in Fig. 7 was obtained. Because of a good agreement between the experiment and the simulation we concluded that the thermal lensing parameter has to be equal to f rod = 40 m instead of 12 m. We have tested our system not only with single shots but also with higher repetition rate pulses. During the initial tests an optical damage in the laser rod occurred. From equations (18) follows that f rod 1/Q. So higher repetition rates of laser pulses produce higher average accumulation of the energy inside of the laser rod and thus stronger thermal lensing behavior. Our model proposed spiking behavior at f rod = 20 m and a consequent damage. All obtained results will be considered in the next development iteration. 4 Conclusion Computer simulations are indispensable tool in the early development phase of a new laser device. They help engineer to quantitatively estimate parameters of the developing system. Of course, modeling is just the first step in the development of a new device. After adequate estimation of appropriate parameters is accomplished, many hours of laboratory work are needed to get laser system ready for the market. 13

14 References [1] B. E. A. Saleh, Fundamentals of photonics, University Science Books, 1986 [2] W. Koechner, Solid State Laser Engineering, Springer, 1999 [3] A. E. Siegman, Lasers, University Science Books, 1986 [4] N. Hodgson, H. Weber, Laser Resonators and Beam Propagation, Springer, 2004 [5] M. Marinček, Razvoj elektromagnetnega polja v sunkovnih trdninskih laserjih, disertacija, 1999 [6] M. Morin, Graded Reflectivity mirror unstable laser resonators, Optical and Quantum Electronics 29 (1997), [7] L. Yajun, Propagation and focusing of Gaussian beams generated by Gaussian mirror resonator, J. Opt. Soc. Am. A 19 (2002), [8] S. De Silvestri, V. Magni, S. Taccheo, G. Valentini, Q-switched Nd:YAG laser with super-gaussian resonator, Optics Letters 16 (1990), [9] http : // attoodet.html 14