Noise Reduction Statistical Analysis in Microchip Lasers

Transcription

1 Nose Reducton Statstcal Analyss n Mcrochp Lasers Marus Ghta Portland State Unversty, Electrcal and Computer Engneerng Department, Learnng From Data Course Abstract - Frequency doubled mcrochp laser output s not constant due to competton between multple longtudnal modes. A clam that a specal drvng of the pump nfluences the output unformty of the laser s studed here. The study bulds on exstng theoretcal models for the frequency doubled Nd:YAG lasers. It s the purpose of ths work to statstcally analyze the output of theoretcal models of frequency doubled lasers. Key words - Mcrochp Laser, Green Problem, Laser Nose, Frequency-Doubled Laser, Second-Harmonc Generaton. I. INTRODUCTION Frequency-doubled mcrochp lasers (FDML) are small lasers (a few mllmeters n length) that ncorporate a frequency doublng crystal n the laser cavty [][][3][4]. Fgure descrbes the structure of such a laser. Fgure. Mcrochp laser dagram These lasers are a subclass of lasers that employ frequency doublng schemes n generatng an output of a desred wavelength [7][8]. It has been observed expermentally, and shown theoretcally that the output of these devces can be very non-unform. Snce green laser lght was easest to obtan usng mcrochp lasers, the nose problem of the laser became known as the Green Problem. The nose n the output made mcrochp lasers hard or even mpossble to use n applcatons where the unformty of the output s mportant. Effort was put n the search for methods of reducng the nose n the output of FDMLs, wth varous degrees of success [5][6][0]. Engneers at Laser Vson (LV) have expermentally observed that a certan method of drvng FDML wll cause the nose n the output to be reduced to less than % of the average ntensty. The purpose of ths work s to verfy the ft between current mcrochp laser models and the nose reducton method dscovered by Laser Vson. II. METHODOLOGY A. Obtanng the dataset. The ntal ntent was to use real data collected by measurng the operaton of an actual laser. However, ths was not possble due to problems wth the laser drver. Snce these problems were not solved n a tmely manner, a theoretcal study of the problem was pursued. There exst a large number of research papers on dfferent aspects of mcrochp lasers, snce they were studed snce 970 s. A number of authors studed the subject n detal [][][3]. Many researchers n ths feld bult on the theory put forth by Baer [3]. Ths was also the strategy used to obtan the data set for ths project. Baer s model was adapted to accommodate the laser drvng scheme suggested by Laser Vson [0]. Snce the equatons descrbed n the models studed do not have a closed soluton, numercal methods were used to solve them. The numercal solutons generated provded the data set used to analyze the problem n more detal. Once ntensty solutons were obtaned for each mode n the runs performed, these ntenstes were added to obtan the total ntensty. Ths s approprate snce the frequences of each mode are sgnfcantly dfferent from each other. Should they have been the same, the electrc felds of each mode would have been to be added, and ths sum squared to obtan the total ntensty. The process of generatng the data set conssted of two parts. Ths dvson was necessary n order to evaluate the valdty of the method used to generate the data. The parts are Verfcaton of complance wth prevous work by other authors and Introducton of the new pumpng scheme nto the SHG theory. ) Verfcaton of prevous work. I have used as startng pont n ths experment the work done by Baer [3], and Mandel & Wu [][4]. Ths part was necessary n order to ensure that further work would be vald. Ths verfcaton s qualtatve n nature, and t was not used drectly n drawng conclusons about the model.

2 a) Two mode laser wth constant pumpng. The most basc descrpton of a mult-mode second harmonc generatng (SHG) mcrochp laser or FDML found n the lterature was that of a two mode SHG laser. The SHG process for ths laser s descrbed by the followng dfferental equatons: Mode : I G Mode : I G = ( G c = f = ( G c = f α ε I ε I ) I [( β I β I + ) G + G ] α ε I ε I ) I [( β I β I + ) G + G ] 0 0 () (3) Where: I, I : mode ntensty; G, G : mode gan; α, α : mode losses; ε: mod losses; β: nter-modal losses; G 0, G 0 : mode pumpng. These equatons form a set of coupled lnear dfferental equatons. The system was solved usng the 4 th order Runge-Kutta method, both by programmng t n C and n MATLAB. The results obtaned by the two methods were smlar. Snce a large number of runs had to be executed, the C program was chosen for further consderaton due to ts speed advantage. b) Three or more modes wth constant pumpng. The equatons for the SHG n a mcrochp laser that runs n more than modes are gven below: I G = ( G α ε I c = f j ε I j ) I [( β I β I + G + G ] Where and j are laser modes. ) 0 () (4) (5) (6) Fgure. Pumpng term - G 0 For the purposes of ths experment the G 0 term was made constant across the equatons. Ths s n accordance wth work performed by other researchers n the feld [3][4], who kept the G 0 constant across the equatons. To ensure that ntal condtons dd not affect sgnfcantly the solutons to the system, the frequency of the pump - f and the break length bl were vared n the lmts of the actual laser drver provded by Laser Vson. These lmts were: MHz MHz for the frequency, and 5 ns 50 ns for the break length. After all the runs were recorded, the ntensty waveforms were processed and plotted usng MATLAB scrpts developed especally for ths purpose. To ensure that the data obtaned s correct, the system consstng of equatons 4-5 was numercally solved for other ntal condtons. These ntal condtons were generated pseudo-randomly n the nterval (0, 0). It was the runs wth varyng ntal condtons that were used later n the study to draw conclusons from. B. Processng the data Once the data was generated, methods presented n the Learnng From Data [] course were used to analyze these numercal solutons ) Complance wth physcal observatons To ensure that the data generated s generally complant wth results obtaned from a real laser, the trend test was appled on the data. The runs to whch the trend test was appled to were selected at random. From each of these runs, the frst 000 ponts were gnored. Ths was done to dmnsh the nfluence of transent effects present n the begnnng of each run. Ths system of coupled lnear dfferental equatons was also solved usng the 4 th order Runge-Kutta method, by adaptng the C program used n the prevous secton. ) Introducton of the new pumpng scheme nto the SHG theory. Researchers at Laser Vson [0] have found expermental evdence that a pumpng scheme wth short nterruptons reduces the nose of a mcrochp laser. Therefore, a pumpng functon wth a very hgh duty cycle was ntroduced, to model the very short breaks n the sgnal that was appled n the lab (Fg. ).

3 ) Analyss of applcaton of Laser Vson pumpng scheme to the data. For each run performed, the average and standard devaton were computed. The standard devatons for each run s partcularly mportant snce t s an ndcator of how stable/unform the value of the ntensty n that partcular run was. Further processng was appled to ths data. a) Gaussan Kernel smoothng of the ntensty average and ntensty standard devaton. Snce t s mpractcal to try to solve the system of equatons at every possble pont, kernel smoothng was used to generate the ntensty average and ntensty standard devaton surfaces. A 3D gaussan kernel smoothng algorthm was used. Ths algorthm was an extenson of the D kernel smoothng algorthm presented n the Learnng from Data course. b) Analyss of the Average to Standard Devaton rato The Average to Standard Devaton rato was analyzed because ths would yeld a very convenent ndcator of regons of stablty. Fg. 3. Two mode ntensty Fg. 4 shows the average ntenstes and ther sum for one three mode run wth constant pumpng. A vsual nspecton of fg. 4 shows that the ntensty sum has no perodc structure. Ths too, s n agreement wth already publshed work. c) Constant Intensty analyss Data smoothed by the methods mentoned above was used to obtan constant ntensty curves. The ntensty tolerance used to extract the constant ntensty curves was ± %. Once the (break length, break frequency) pars of nterest were obtaned, the standard devatons for these pars were extracted from the smoothed data. The nose n ths data was then compared to a dstrbuton for the nose that a stable output was expected to produce. III. RESULTS A. Obtanng the dataset The systems of equatons were solved and the data recorded n ASCII format. For the two mode model, the modes ntenstes are shown n Fgure 3. Ths matched the results obtaned by Baer [3]. Fg. 4 Three mode ntensty Solvng the system after the LV pumpng scheme was ntroduced n the equaton generated numercal solutons smlar to multple mode solutons obtaned prevously wthout the LV pumpng scheme. One set of such solutons s presented n fg. 5. The values obtaned for the mean and standard devaton for runs wthout LV pumpng scheme were: µ constant pump = 6 a.u. and σ constant pump = 7. a.u.

4 Fg. 5 Three mode ntensty sum wth LV pumpng scheme. Fg. 6: 3D kernel smoothng of ntensty wth constant ntal condtons. Weght w = B. Processng the data As mentoned n the Methodology secton, the trend test presented n the Learnng From Data course was appled to the data. Ths was appled to all of the tree types of solutons obtaned: two modes, three modes, and three modes wth LV pumpng scheme. The data used was randomly selected from these runs. The null hypothess was that the sum of the ntenstes does not have an ncreasng or decreasng trend for a partcular run. The results of applyng ths test are presented n Table. Table. Trend test appled to the data. Run ID Cox & Stewart P Value Run ID Cox & Stewart P Value 7 No trend No trend No trend No trend No trend No trend No trend No trend No trend No trend No trend No trend No trend No trend No trend No trend No trend No trend No trend No trend Fg. 7: 3D kernel smoothng of the standard devaton wth varyng ntal condtons. Weght w = All the p-values for the trend tests performed are greater than 0., therefore the evdence aganst the null hypothess of no trend s very weak. Further processng of the data conssted n applyng 3D methods of smoothng to the runs that made use of the LV pumpng method. Fgures 6 and 7 are typcal results of usng 3D Gaussan kernel smoothng, for constant and varyng ntal condtons, and dfferent bass weghts appled to the ntensty. Fg. 8: 3D kernel smoothng of the standard devaton wth varyng ntal condtons. Weght w = 0..

5 Another attempt to see the nose n the output was by dvdng the average of a run by the standard devaton. Fg. 9 presents the result of the method appled to a run n whch the ntal condtons were vared. A very smlar result was obtaned for the run n whch the ntal condtons were not vared. It should be mentoned that the sharp drop n one of the corners of the surface dsplayed n Fg. 9 s due to a very low value for the average ntensty. A decson was made to set the rato to zero when the average dropped below 0.5. For these values the standard devaton was large compared to the average ntensty. Ths corresponds n real lfe wth a laser functonng around the lasng threshold, a regme n whch the laser output s nstable. (b) Standard Devaton vs. Break Length for run 0j Fg. 9: Average/Standard Devaton plot used to detect ponts of low standard devaton. The standard devaton of constant ntensty runs was also analyzed to determne f they have a local mnmum. A constant ntensty run was defned by the target ntensty ±%. Fg. 0 presents typcal standard devaton values of constant ntensty data. (a) Standard Devaton vs. Frequency for run 0j (c) Constant Intensty Curves n (freq. break length) space Fg. 0. Standard Devaton of constant ntensty runs For each pont n the constant ntensty runs, the nose level was computed by dvdng the standard devaton to the average ntensty. These values were then tested to see how close they get to an deal nose dstrbuton. The deal nose dstrbuton was defned to be the postve sde of a Gaussan wth mean µ = 0 and standard devaton σ = 0.0. The test appled was the Kolmogorov-Smrnov test for ft between two dstrbutons. The use of ths test s approprate because t does not make any assumptons about the dstrbutons tested. The null hypothess used when applyng ths test was that the dstrbutons are the same. Table. Kolmogorov-Smrnov Test for two populatons. Constant Intensty (a.u.) Kolmogorov-Smrnov Test for two dstrbutons

6 Values close to for the result of the Kolmogorov-Smrnov test ndcate rejecton of the null hypothess. Based on the results presented n table, there s very strong evdence that that the nose dstrbuton for the constant ntensty data and stable output nose are dfferent. IV. DISCUSSION The results presented n table, fgures 3 and 4 qualtatvely agree wth the operaton of mcrochp lasers n contnuous wave mode. Ths agreement conssts n the fact that once the lasng actvty starts, the output of the laser fluctuates about a specfc value. Another way to say ths s that real lasers output s bound and usually has no trend (assumng no decay of the laser system components). The type of results obtaned for runs that have a constant pump term are n agreement wth results of other authors. The runs that were modfed to accommodate the specal LV pumpng scheme yeld, upon nspecton, ntensty values that resemble both theoretcal results of runs wth constant pump values and observed behavor of a real laser n regons of nosy output. The three dmensonal nterpolaton technques appled to the standard devaton data obtaned was desgned to make t easer to dentfy regons of stablty. However, ths technque dd not lead to fndng any regons of sgnfcant stablty. The threshold standard devaton value for whch the output would have been accepted as stable was % of the average ntensty. The plot of the rato of the average to the standard devaton (fg. 9) was expected to yeld an ndcator of regons of stable laser output. It was expected that the regons of stable output to be represented by peaks on the plot n Fg. 9. Vsual nspecton however showed that the standard devaton of the ntensty dd not fall below % of the average ntensty value of any of the runs. Ths method of searchng for regons of stablty can be appled n the future to real laser output measurements. The fnal step n the search for stablty n the results was to look at the nose n the constant ntensty curves. Examnaton of the results of the Kolmogorov-Smrnov test revealed that the nose dstrbuton n the data comes s dfferent than the expected nose dstrbuton for a stable output laser. lmts for the LV drvng parameters not wde enough to capture the desred effect. () Falure for the model to account for thermal coeffcents. (3) The approxmatons that were used n dervng the model do not hold under the LV pumpng scheme. Out of these three, the most lkely canddates are () and (3). Even though the present analyss dd not reveal the sought for effects, the methods appled here are of nterest to the author n the study of a real FDML laser. VI. REFERENCE: [] C.J. Kennedy and J.D. Barry, Stablty of an ntracavty frequency-doubled Nd:YAG Laser, IEEE Journal of quantum Electroncs, Vol. qe-0, No. 8, August 974. [] P. Mandel and X.G. Wu, Second-harmonc generaton n a laser cavty, Journal of the Optcal Socety of Amerca, Vol. 3, No., p , July 986. [3] T. Baer, Large-ampltude fluctuatons due to longtudnal mode couplng n dode-pumped ntracavty-doubled Nd:YAG lasers, Journal of the Optcal Socety of Amerca B, Vol. 3, No. 9, p , September 986. [4] X.G. Wu and P. Mandel, Second-harmonc generaton n a multmode laser cavty, Journal of the Optcal Socety of Amerca, vol. 4 Issue, p , November 987 [5] M. Oka and S. Kubota, Stable ntracavty doublng of orthogonal lnearly polarzed modes n dode-pumped Nd:YAG lasers, Optcs Letters, Vol. 3, No. 0, p [6] M. Tsunekane and N. Taguch, Elmnaton of chaos n a multlongtudnal-mode, dode-pumped, 6-W contnuous-wave, ntracavty-doubled Nd:YAG laser, Optcs Letters, Vol, No. 3, p [7] A. Yarv, Quantum electroncs, thrd edton, p , [8] L. Casperson, Laser system desgn, course at Portland State Unversty. [9] Laser Vson, Full color laser televson, not publshed, crca 998. [0] Ed Mesak and Mke Overton of Laser Vson Technologes, Method to sgnfcantly reduce the ampltude-nose of a dode pumped ntracavty doubled laser, not publshed, crca 998. [] J. McNames, Learnng from data, course at Portland State Unversty, sprng 00. V. CONCLUSION The man result that emerged from ths study s that the laser characterstcs dscovered by LV were not reproducble by the theoretcal models studed here. There are at least three reasons why the approach presented n ths paper faled to account for regons of stable output: () The Ths s not a hard value, rather just a convenent threshold put forward by the Laser Vson engneerng and desgn team