Measurement and extraction of the Giles parameters in Ytterbium-doped fibre

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(1)Measurement and extraction of the Giles parameters in Ytterbium-doped bre by. Adriaan Jacobus Hendriks Thesis presented in partial fulllment of the requirements for the degree of Masters of Science at the University of Stellenbosch. Physics department Faculty of Natural Science Supervisor: Dr Christine M. Steenkamp Co-supervisor: Mr Alexander M. Heidt. March 2009.

(2) Declaration By submitting this thesis electronically, I declare that the entirety of the work contained therein is my own, original work, that I am the owner of the copyright thereof (unless to the extent explicitly otherwise stated) and that I have not previously in its entirety or in part submitted it for obtaining any qualication.. Date: 3 March 2009. Copyright 2007 Stellenbosch University. c All rights reserved.

(3) ii. Abstract The role fullled by theoretical models is rapidly increasing due to lasers becoming application driven to satisfy certain criteria and demands. Construction of high precision lasers requires good theoretical models and consequently good approximations of the parameters that such models are based upon. Despite the dierent model formalisms, most share a common set of input parameters, including bre waveguiding properties, input powers, transition cross-sections and overlaps between guided modes and the dopand distribution. Experimental and numerical work which was aimed at obtaining the wide-band emission and absorption cross-sections of bre indirectly by means of the Giles parameters was done. The Giles parameters were used rather than the well known ionic cross-sections primarily because of the convenient encapsulation of the cumbersome overlap factors and the ionic cross-sections within the Giles parameters. The wide band spectral characteristics of the Giles parameters are indispensable in the design of bre lasers and ampliers, as they form the key parameters for laser models. These parameters are normally obtained utilizing absorption spectroscopy to obtain the absorption cross-sections and models such as the Fuchtbauer Ladenberg relation, the McCumber relation or uorescence spectroscopy to obtain the emission cross-sections. Recent research however indicates that these methods are inaccurate in certain spectral regions. An investigation was launched to extract the Giles parameters from measurements of the amplied spontaneous emission (ASE) and pump absorption in ytterbium-doped bre for several lengths of bre and subsequent computer simulations, utilizing an amplier model. The Giles parameters are extracted with a tting algorithm that adjusts the relevant numerical values to minimize the least square dierence between the numerical data obtained from the amplier model and the measured data. Using the model devised in this project on literature data, the Giles parameters were extracted and compared to the Giles parameters extracted in literature on the same data. This comparison conrms the extraction of the Giles parameters, utilizing the model devised in this project, as successful. Subsequently the model devised in this project was applied to extract the Giles parameters from experimental data measured at Stellenbosch, using a double cladding ytterbium-doped bre. Finally a bre laser was built utilizing the double cladding ytterbium-doped bre and the output was measured. The Giles parameters extracted were then used in a bre laser model to calculate the output and compare it to the measurements taken. This served as sucient verication that the Giles parameters extracted can be used to model a bre laser eciently..

(4) iii. Opsomming Die rol gevul deur teoretiese modelle neem daagliks toe, namate lasers meer doelgerig ontwerp word om sekere kriteria en versoeke te bevredig. Konstruksie van hoë presisie lasers vereis goeie teoretiese modelle en gevolglik ook goeie benaderings van die parameters waarop so 'n model berus. Ongeag die verskillende model formalismes wat gebruik word, deel hulle 'n stel inset-parameters, insluitend die vesel se golfbegelydings-eienskappe, inset drywing, oorgangs-deursnitte en die oorvleuelings tussen begelyde modusse en die doteerdistribusie. Eksperimentele en numeriese werk gemik om die wyeband emissie- en absorpsie-deursnitte van 'n vesel indirek deur middel van die Giles parameters te verkry is gedoen. Die Giles parameters word eerder gebruik as die welbekende oorgangs-deursnitte, hoofsaaklik omrede die oorvleuelings-parameters sowel as die oorgangs-deursnitte saamgevat word in die Giles parameters. Die wyeband spektrale eienskappe van die Giles parameters, is onontbeerlik in die ontwerp van vesellasers en versterkers, omrede hul die sleutel parameters vorm van lasermodelle. Hierdie parameters word gewoonlik verkry deur gebruik te maak van absorpsie spektroskopie vir die bepaling van die absorpsie-deursnitte. Modelle soos die Fuchtbauer Ladenberg verhouding, die McCumber verhouding of uoresensie spektroskopie word gebruik vir die bepaling van die emissie-deursnitte. Onlangse navorsing toon egter dat hierdie metodes nie akkurraat is in sekere spektraal gebiede nie. Ondersoek is ingestel om die Giles parameters te ekstraheer deur gebruik te maak van metings van die versterkte spontane emissie (VSE) en die pomp absorpsie in ytterbium-gedoteerde vesel, vir 'n aantal lengtes van vesel en daaropvolgende rekenaar simulasies wat gebruik maak van 'n versterkermodel. Die Giles parameters word geëkstraheer deur 'n passingsalgoritme wat aanpassings maak aan die relevante numeriese waardes om die verskil tussen die numeriese data bepaal deur gebruik te maak van die versterkermodel en die gemete data te minimeer volgens kleinste kwadraat verskil metode. Die model ontwerp in hierdie studie, is gebruik om die Giles parameters te ekstraheer vir literatuur data, en die resultate is vergelyk met die geëkstraheerde Giles parameters vanaf literatuur vir dieselfde eksperimentele data. Hierdie vergelyking dien as bevestiging dat die Giles parameters geëkstraheer, deur gebruik te maak van die model ontwerp in hierdie studie, 'n sukses was. Vervolgens is die model ontwerp in hierdie studie gebruik om die Giles parameters te ekstraheer vir eksperimentele data gemeet op Stellenbosch. Daar is gebruik gemaak van 'n dubbel jas ytterbium-gedoteerde vesel. Laastens is daar 'n vesellaser gebou en die uitset gemeet. Die geëkstraheerde Giles parameters is gebruik in 'n vesellaser model en is vergelyk.

(5) iv. met die gemete data. Dit bewys dat die geëkstraheerde Giles parameters gebruik kan word om 'n vesellaser eektief te modelleer..

(6) v. Acknowledgments I would like to thank the following people: •. Dr. C.M Steenkamp for her guidance, assistance and supervision during the project. •. Mr. A.M. Heidt for guiding and assisting in the project. •. Brunet et. al. for providing experimental data towards this project. •. Dr. Burger for guidance and assistance during initial phase of my studies. •. Dr J-N Maran for help and assistance on numerous aspects in this project. •. Mr. U. Deutschla¨nder, E. Shields and J. Germishuizen for the technical assistance they contributed towards this project. •. Mr G. Hager and J.D. Blanckenberg for reading through this thesis. •. Miss F. H. Mountfort for her moral support. •. My family and friends for their faith and support. •. Last but not least the Lord my God for carrying me through my studies. My studies were funded by the National Research Foundation (NRF) and DPSS (CSIR).

(7) Contents 1 Problem statement. 1. 1.1. Introduction and problem statement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1.2. Aim. 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2. 2 Theory of Light-matter interaction. 4. 2.1. Light: Classical description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2. Matter: Quantum mechanical description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2.1. Time-dependent Schr¨ odinger equation. 2.2.2. Two-state quantum system with a monochromatic plane wave as source. . . . . . . . .. 7. 2.2.3. The Density matrix and decay processes . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 2.2.4. Ionic cross-sections and Einstein A & B coecients . . . . . . . . . . . . . . . . . . . .. 10. 2.2.5. The population rate equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. 2.2.6. Light amplication and Gain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 Fundamental Properties of Optical bres 3.1. Fibres 3.1.1. 3.2. Optical coupling in waveguides. 4.2. 17. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 21. Methods for obtaining the ionic cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. Transitions in active media. 4.4. 28. Amplier model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28. 4.1.1. Rate equations. 28. 4.1.2. Propagation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 29. 4.1.3. The Giles parameters. 31. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Obtaining the Giles parameters 4.2.1. 4.3. 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 Numerical models 4.1. 5. 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Dopand and hosts 3.2.1. 3.3. 4 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31. Simulation algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32. Model testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39. Fibre lasers 4.4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41. Laser model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 41. 5 Experimental setups. 45. 5.1. Diode Laser output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 45. 5.2. ASE spectra and residual pump power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46. 5.2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. Fibre Laser output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50. 5.3. Component correction factor. vi.

(8) CONTENTS. vii. 6 Results and Discussions. 51. 6.1. 6.2. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51. 6.1.1. Diode laser output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51. 6.1.2. ASE spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53. 6.1.3. Residual and injected pump power . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 57. 6.1.4. Fibre laser output. 60. 6.2.1. . . . . . . . . . . . . . . . . . . . . . . . . . .. Testing of the numerical model . . . . . . . . . . . . . . . . . . . . . . . . . .. Giles parameters utilizing backward ASE. 60 61 66. . . . . . . . . . . . . . . . . . . . . . . . . .. 79. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82. Simulated data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82. Results obtained for the Laser 6.3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Giles parameters utilizing forward ASE 6.2.1.1. 6.2.2 6.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Results obtained for the Giles parameters. 7 Summary and Conclusions. 86.

(9) List of Figures 2.1. Possible transitions between two energy levels. 2.2. N adjacent bre increments representing the bre as a whole. . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.1. Dierent sections of a double cladding optical waveguide.. 3.2. Fibre face-front of the optical waveguide used at Stellenbosch with courtesy of FIBERCORE. 12. . . . . . . . . . . . . . . . . . .. 14. . . . . . . . . . . . . . . . . . . . .. 16. [21]. One can observe quite a dierence between cross-sectional areas of the core and the inner cladding. This is expected since the number of allowed modes relates to the cross-sectional area of each section. 3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Acceptance cone of injected pump power. Any light entering the bre with a angle than the acceptance angle angle of. 3.4. (θi1 ).. (θc ). will be coupled to the outer cladding, but not light with an. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18. The ASE propagates inside the bre core and can be seen to exhibit some intensity distribution which is of signicance if one wants to calculate the overlap factors.. 3.5. 17. (θi2 ) greater. . . . . . . . . . . . . . .. 19. As the pump is injected into the inner cladding one can once again expect some intensity distribution conned to the core and inner cladding. What can be seen is a top hat prole considered as the intensity distribution as well as the rst three modes of the inner cladding.. 3.6. Particularly important are the emission lines. One can observe four emission lines i.e. 977 1020. nm, 1032 nm and 1069 nm for ytterbium-doped bre.. nm,. The rst emission line corresponds. to a quasi-three level system and the rest to a quasi-four level system. 3.7. 19. The energy level diagram for ytterbium-doped bre obtained from Newell et. al.[22] is shown.. . . . . . . . . . . . . .. 21. Making use of absorption spectroscopy and the McCumber relation the emission cross-sections are obtained for Erbium doped in Al/Ge-SiO2 . This is the best McCumber t for Erbiumdoped bre obtained from Digonnet et. al. [9].. 3.8. . . . . . . . . . . . . . . . . . . . . . . . . . .. 23. Making use of absorption spectroscopy and the McCumber relation the emission cross-sections are obtained for Erbium doped in NA as host. This is the worst McCumber t for Erbiumdoped bre obtained from Digonnet et. al. [9].. 3.9. . . . . . . . . . . . . . . . . . . . . . . . . . .. 24. Absorption cross-sections obtained making use of absorption spectroscopy and emission crosssections using uorescence saturation spectroscopy for ytterbium-doped bre with a doping concentration of 1.23x10. 26. −3. m. . Obtained from reference [12]. . . . . . . . . . . . . . . . . . .. 3.10 Emission cross-sections obtained using uorescence spectroscopy at dierent temperatures [22].. 25 26. 3.11 (Red) Absorption cross-sections measured for the ytterbium-doped bre used in the present. 26. study (with a doping concentration of 2.5x10. m. −3. ) with uorescence spectroscopy by FIBER-. CORE. (Green) The emission cross-sections obtained making use of gure 3.9 and normalizing with respect to the FIBERCORE measurements of the absorption cross-sections. (Blue) The emission cross-sections obtained using FIBERCORE measurements of the absorption cross-. 4.1. sections and the approximate McCumber relations. . . . . . . . . . . . . . . . . . . . . . . . .. 27. Schematic of numerical procedure followed to extract the Giles parameters.. 33. viii. . . . . . . . . . ..

(10) LIST OF FIGURES. 4.2. ix. Procedure to reach numerical stability between the upper state population and the propagation equations labeled loop 2 in gure 4.1. This step is specically important because it is directly derived from the amplier models.. 4.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38. Numerical stability determined from upper state population for loop 1. After 17 iterations a numerical stability of -60dB is reached. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.6. 35. Numerical stability determined from upper state population for loop 2. After 15 iterations a numerical stability of -80 dB is reached.. 4.5. 34. Non-linear curve tting function utilizing an interior trust region approach to extract the Giles. Laser resonator with gain media of length mirror at the front and back of the cavity.. `. and resonator length of. L. 38. created by placing a. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. m2. 5.1. Diode laser output to be injected into the gain bre measured with power meter,. 5.2. Optical setup used to measure the backward ASE at the front of the bre and residual pump. . . . . .. 42 46. power at the back of the bre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47. 5.3. Setup used to determine the mirror losses and determine the compensation spectrum . . . . .. 49. 5.4. The laser conguration is identical to that used to measure the ASE spectra, but with the gain. ◦. bre cleaved at 90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. M1. Diode laser output measured after mirrors. 6.2. Diode laser spectrum for three dierent temperatures.. 6.3. Central wavelength of diode laser output at the diode temperatures as seen in gure 6.2. 6.4. Backward ASE measured for several bre lengths and a xed injected pump power of 429.2. mW can be observed.. and. M2. 6.1. 50. for dierent diode currents used. . . .. 52. . . . . . . . . . . . . . . . . . . . . . .. 52. . . .. 53. In the peak at the right one can observe a red shift, the lowest measured. emitted power as a result of the shortest bre and the highest measured emitted power as a result of the longest bre. 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. 54. Measurements conducted for the mirror-combination and glass plate, using the setup in gure 55. The glass plate measurements of gure 6.5 are normalized with respect to the mirror-combination powers. The ratio between these spectra provides the compensation spectrum for the mirrorcombinations to incident light.. 6.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55. The mirror compensation spectrum is obtained using gure 6.6. This spectrum was obtained by subtracting the mirror-combination spectrum from the glass plate spectrum (in log scale). On a linear scale this will yield a multiplication factor with which the ASE spectra should be multiplied by, to obtain the compensated spectra.. 6.8. . . . . . . . . . . . . . . . . . . . . . . . .. 56. Compensated ASE spectra obtained by using gure 6.7. The compensated spectra were obtained by adding the graph of gure 6.7 to that of gure 6.4 (on a log scale). This corresponds on a linear scale to multiplication with the compensation spectrum.. 6.9. . . . . . . . . . . . . . .. 56. The ratios between measured and corrected ASE, using gure 6.7, is illustrated. In (a) a bre length of 4.003 m and current of 702. mA. is used, resulting in a correctional factor of 1.163 for. the measured power. In (b) a bre length of 6.109 m and a current of 903. mA is used resulting. in a correctional factor of 6.322 for the measured power. . . . . . . . . . . . . . . . . . . . . .. 57. 6.10 Measured residual pump power for dierent lengths of bre using an injected pump power of 429.2. mW.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58. 6.11 Measured residual pump powers including extrapolated injected pump using quadratic and cubic non-linear tting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Ratio of extrapolated injected powers to that of the measured optical power after mirror and. M2 .. The mean percentage obtained by using the quadratic t is 87 %.. 59. M1. . . . . . . . . . .. 59. 6.13 Fibre laser output at the front (counter-propagation) and back (propagation) of the gain bre. 60. 6.14 Measured ASE spectra at the back of the bre for dierent bre lengths. Courtesy of Brunet et. al. [13]. A shift to higher wavelengths can be seen for longer bres. . . . . . . . . . . . . .. 61.

(11) LIST OF FIGURES. x. 6.15 Measured residual pump power at the back of the bre with the injected pump power at the. th. 0. position. Courtesy of Brunet et. al. [13] . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6.16 Measurements of the residual pump powers taken by Brunet et. al.. 62. and calculated residual. pump power obtained from the model devised in this project can be seen. There is a good . . . . . . . . . . . . . . . . . . . . . . .. 63. 6.17 Normalized upper state population calculated by the numerical model devised in this project.. correlation between measurements and calculations.. 64. 6.18 Forward ASE measurements collected by Brunet et. al. and ASE calculated by the numerical model devised in this project can be seen. Binning was conducted and data grouped into 60. th. equally spaced wavelength bins where every 10. bin is selected and viewed. Good correlation. can be seen between the measured and calculated forward ASE. . . . . . . . . . . . . . . . . . 6.19 Small signal absorption extracted for Brunet et.. 64. al.'s measured data using the numerical. model devised in this project can be seen. The red line indicates the cut-o introduced due to low signal to noise ratios making it impossible for the model to accurately extract the Giles parameters below 1050. nm.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65. 6.20 Small signal gain at complete inversion obtained for Brunet et. al.'s measured data using the numerical model devised in this project. The red line is the same as in gure 6.19.. . . . . . .. 65. 6.21 Utilizing the extracted Giles parameters shown in gure 6.19 and 6.20 the measured ASE in gure 6.14 was numerically reproduced using the model developed in this study. The longest bre length coincide with the highest ASE measurement and a red shift can be observed. This is similar to what was obtained for the measurements.. . . . . . . . . . . . . . . . . . . . . . .. 66. 6.22 Giles parameters extracted for (a) small signal absorption and (b) small signal gain at complete inversion for dierent initial arrays as shown in table 6.2. The close agreement indicates that the parameters play no signicant role in extracting the Giles parameters.. . . . . . . . . . .. 67. 6.23 Giles parameters extracted for (a) small signal absorption and (b) small signal gain at complete inversion for dierent initial arrays for the Giles parameters according to table 6.3. Obtaining no dierence between extracted Giles parameters entail the extraction is independent on the initial values of the Giles parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 68. 6.24 For several values of MaxFunEvals the extracted small signal absorption (a) and the calculated relative error (b) can be seen. The legend in (a) shows the dierent values of MaxFunEvals used.. In gure (b) the dierence between the Giles parameters extracted relative to the. maximum value of MaxFunEvals becomes so small (for MaxFunEvals 400 and 600) that matlab is incapable of calculating the error.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69. 6.25 Presented here is similar to that of gure 6.24. Matlab is once again incapable of calculating the error for MaxFunEvals of 200, 400 or 600.. . . . . . . . . . . . . . . . . . . . . . . . . . .. 70. 6.26 For a change in the number of wavelength bins the extracted small signal absorption (a) and the error calculated (b). The legend of gure (a) gives the number of bins into which the range 1050-1100. nm. relative error.. was divided. The results for 100 bins were used as reference to calculate the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71. 6.27 For a change in the number of wavelength bins the small signal gain at complete inversion was extracted (a) and the error calculated (b). The legend of gure (a) gives the number of bins into which the range of 1050-1100. nm. was divided. . . . . . . . . . . . . . . . . . . . . . . . .. 71. 6.28 For a change in the number of length increments, according to the legend in (a), the small signal gain was extracted (a) and the error calculated (b) with 80 increments as reference. A negative downward trend can be observed for the errors. . . . . . . . . . . . . . . . . . . . . .. 72. 6.29 For a change in the number of wavelength bins the small signal absorption at complete inversion was extracted (a) and the error calculated (b) for each value of the legend in (a) with 80 increments as reference. A negative trend can be observed for the calculated errors. . . . . . .. 72. 6.30 Small signal absorption for the ASE obtained using various injected pump powers from table 6.5 74 6.31 The small signal gain at complete inversion for the ASE obtained using various injected pump powers from table 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74.

(12) LIST OF FIGURES. xi. 6.32 The small signal absorption obtained at the pump wavelength using various injected pump powers from table 6.5. For a change in the injected pump power a small dierence is observed for the extracted small signal gain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75. 6.33 The small signal gain at complete inversion obtained at the pump wavelength using various injected pump powers. A big dierence is obtained for the extraction of the small signal gain at complete inversion for the pump in the 700. mW. regime.. . . . . . . . . . . . . . . . . . . .. 75. 6.34 The ratio between small signal gain at complete inversion and and small signal absorption at the pump wavelength.. To be able to obtain an population inversion this ratio needs to be. smaller that 1, otherwise no absorption will take place.. . . . . . . . . . . . . . . . . . . . . .. 76. 6.35 Comparison between the Giles parameters obtained by Brunet et. al. and what was obtained using the model devised in this project. The extraction was done on the same experimental data measured by Brunet et. al. for the small signal absorption (a) and the small signal gain at complete inversion (b).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77. 6.36 ASE reproduced by the model devised in this project using the Giles parameters extracted by Brunet et. al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77. 6.37 Eective red shift initiated by the various sets of Giles parameters compared to the measured data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79. 6.38 The average small signal absorption calculated over several pump powers. The extraction was done for ytterbium-doped bre with a high doping concentration.. . . . . . . . . . . . . . . .. 6.39 The small signal gain at complete inversion calculated over several pump powers. distinguishable is the peak at around 1073 1069. nm. nm. 80. Clearly. which is very close to the emission line of. in gure 3.6. The extraction was done for ytterbium-doped bre with a high doping. concentration.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81. 6.40 Comparison between literature ionic cross-sections and the ionic cross-sections determined via the Giles parameters using the model devised in this study.. . . . . . . . . . . . . . . . . . . .. 82. 6.41 Fibre laser simulation utilizing the Giles parameters extracted in this project is compared to the measured signal. Simulations utilizing literature ionic cross-sections, [12], are also compared to the measured output for forward ASE.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84. 6.42 Measurements are obtained from Alexander Heidt for a similar bre laser and compared to the laser signal calculated by the simulation.. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85. 1. Laser diode setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90. 2. Diode laser schematics. 91. 3 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Final optical setup used to extract the Giles parameters making use of backward propagating ASE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 92. Program outline. 94. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

(13) List of Tables 5.1. Fibre lengths used to collect backward ASE and residual pump power. . . . . . . . . . . . . .. 48. 5.2. The diode laser currents and output powers used to measure the ASE spectra . . . . . . . . .. 48. 6.1. Parameters used to extract the Giles parameters utilizing forward ASE and residual pump power. The material properties were obtained from Brunet et. al. [13]. . . . . . . . . . . . . .. 6.2. Giles parameters extracted by the numerical model . . . . . . . . . . . . . . . . . . . . . . . . 6.3. 62. Sets of initial value parameters used to test whether these values could have an eect on the 67. Initial guess values for the Giles parameters. Without initial values for the Giles parameters it would not be possible to solve the coupled dierential equations in the amplier model.. .. 68. . . . . . . . . . . . . . . . . .. 69. 6.4. Internal parameters changed in the nonlinear tting algorithm. 6.5. These injected pump powers were used to calculate the Giles parameters to determine the impact of variation in the extrapolated injected power on the Giles parameter extraction. The injected pump power of 687. 6.6. mW. as specied by Brunet et. al. [13] is used as reference.. . . .. with the model devised in this project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7. 73. Parameters used to extract the Giles parameters using forward ASE and residual pump power 80. Parameters used in numerical simulations of bre lasers. The pump and signal wavelengths were measured as described in this project and the Giles parameters were extracted making use of the model devised in this project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. xii. 83.

(14) Chapter 1. Problem statement 1.1 Introduction and problem statement Lasers have become an integrated way of life having well known commercial, industrial, medical and military applications. This fact alone serves as motivation for improving the current techniques utilized for laser construction. Having so many applications no multi-purpose laser can be manufactured for all laser applications, limiting a specic laser design to specic applications. The main diculty with these types of lasers is their emission wavelength bands which limits the laser to a specic spectral region [1]. Depending on the region in which operation is desired an appropriate laser i.e.. gas or solid-state (including bre lasers) as well as. type of active material should be selected. The emission band on its own is not enough to be able to predict the operational wavelength, due to the dependence on laser cavities and other external factors. A benecial procedure to design a laser with a specic operational wavelength would be to simulate laser behaviour by means of a numerical model, modeling the specic type of laser and active material before the assembling of the laser takes place. Simulation of laser behaviour provides the necessary insight into the characteristics of the laser and in doing so, it is possible to see if a system will have useful properties, without the necessity of using trial and error techniques [2], also making it time and cost eective. Considering that bre laser application and technology is a rapidly growing environment with possible high output powers [3, 4] it is of interest for development and manufacturing [5]. These small but ecient lasers have the advantage over other laser systems i.e. high power single mode output resulting in excellent beam quality, heat distribution over much larger areas, laser light focused throughout the whole length of bre, good energy conversion, large emission bands and a great many other benecial aspects [6, 7, 8]. A key reason for investigating bre lasers is to construct lasers with good beam quality. The bre laser is based on the principle that it converts a bad quality pump beam into a good quality beam, hence the name brightness converter was given to it, resulting in a small system featuring good beam quality as well as exibility in the output wavelength.. With this in mind it is clearly understood why bre laser are so. desired above other laser systems. Most bre model formalisms share a common set of input parameters, including bre waveguiding properties, input powers, transition cross-sections and overlap factors. The ionic cross-sections are mostly directly used from literature, but recent research indicates that the techniques used in obtaining these parameters are not always so successful as previously speculated [9, 10]. The reasons for the lack of correct ionic cross-sections is that they depend on various parameters and are typically host and material related.. 1.

(15) CHAPTER 1.. PROBLEM STATEMENT. 2. The ionic cross-sections are normally obtained using absorption spectroscopy for the absorption crosssections and models such as the Fuchtbauer Ladenberg relation, the McCumber relations or uorescence spectroscopy for the emission cross-sections.. The inaccuracy of these parameters in specied wavelength. domains makes it dicult to simulate laser and amplier behaviour in these regions. The overlap factors needed are calculated by rst calculating the various modes inside the bre and successively calculating the overlap factors of the modes. Using the Giles parameters, rather than the ionic cross-sections, reduces the number of parameters used in amplier models. This is because the ionic crosssections as well as the overlap factors are encapsulated within the Giles parameters. The task at hand is to investigate a method involving experimental measurements, on the chosen bre, and a numerical model to extract the Giles parameters, so that they can be used to accurately simulate lasers and ampliers using this bre, without utilizing the tabulated values of the ionic cross-sections or techniques shown in literature [11, 12].. 1.2 Aim This project is aimed at extracting the wide-band emission and absorption cross-sections of bre indirectly by means of the Giles parameters. The wide-band spectral characteristics of the Giles parameters are indispensable in the design of bre lasers and ampliers, as they form the key parameters for laser models. For single mode bre the Giles parameters can be measured in a straight forward manner, but it is more complicated for multi mode bre. One of the parameters used in this formalism is the small-signal gain at complete inversion. Complete inversion in double cladding bre is very dicult to attain, due to the pump intensity being spread over the whole length of the bre for double cladding bre. Currently there exist two models capable of numerically extracting the Giles parameters from experimental data for double cladding bres. The photon balance model used by Brunet et. al. [13] and the newly developed amplier based model (devised in this project). Rather than using already developed models to extract the Giles parameters, a numerical model is devised which is capable of extracting the Giles parameters using bre waveguide properties and experimental measurements of the amplied spontaneous emission (ASE) and the residual pump power. The Giles parameters are extracted with a tting algorithm that adjusts the relevant numerical values to minimize the least square dierence between the numerical data obtained from the amplier model and the measured data. The minimization process makes use of an interior trust region approach to minimization [14]. Collaborating with Brunet et. al., the model devised at Stellenbosch is used to extract the Giles parameters on data measured by Brunet et. al. and compared to the Giles parameters extracted by Brunet et. al.'s model on the same data [13]. Brunet et. al.'s model is constructed using the photon balance principle, where the number of photons entering a bre segment has to equal the number of photons exiting the bre segment. An existing problem with the photon balance model is that it does not suciently accommodate for possible feedback [13].. Any form of feedback drastically inuences the model and results in inaccurate extraction. of the Giles parameters. The model devised at Stellenboch rather makes use of an amplier model where feedback does not have any signicant impact on the model. As this is a numerical model which is combined with experimental data, physical limitations in the.

(16) CHAPTER 1.. PROBLEM STATEMENT. 3. experiment inuence the capability of the model to extract the Giles parameters. It is therefore important to analyze the model and to establish model limitations as far as possible, as it is of no use to work outside the model capabilities. An experimental setup is devised to measure the backward ASE from bre when pumped below threshold, as well as the absorbed pump power for several injected pump powers. Using the data collected and the model the Giles parameters are extracted for each of the injected pump powers. Finally a bre laser is built and the output measured. The extracted Giles parameters are used in a bre laser model and the results compared to the measurements taken for a laser built using the bre laser model specications. This will suce as verication that the Giles parameter extracted can be used to model bre lasers eciently..

(17) Chapter 2. Theory of Light-matter interaction To develop a method by which the ionic cross-sections can be extracted, a good theoretical understanding of light-matter interaction is necessary. Making use of the semi-classical approach, i.e. light (electromagnetic wave) is treated classically (Maxwell) and matter treated quantum mechanically, expressions can be formulated describing the microscopic interactions taking place. In this chapter the derivation of the rate equations from rst principles will be shown following the approach of Milonni and Eberly [15] as well as the book by O. Svelto [16] which will suce as the basis for subsequent work.. 2.1 Light: Classical description Electromagnetic waves are best described by making use of Maxwell's equations (See appendix A3). Consider light traveling in a arbitrary dielectric media and the media to be non-magnetic, then the electric current. (J). density. and the volume charged density. (ρ). both may be assumed to be zero.. Making use of vector. calculus (see appendix A2, relation 4 and 5) and the preceding assumptions the non-linear inhomogeneous wave equation results from rewriting Maxwell's equations:. ~ − µ0 0 ∇2 E → − E. and. P~. ~ ∂2E 2 ∂t. = µ0. ∂ 2 P~ ∂t2. (2.1). denotes the electric eld and the electric dipole moment per unit volume respectively, whilst. is the permittivity of free space and. µ0. 0. the permeability of free space. Equation 2.1 represents the relation. between the electric eld and the dipole moment density and is satised by transverse elds. From a classical perspective light is considered as a transverse eld i.e. a monochromatic plane wave traveling in the z-direction and is of the form. ~ t E R, = Here. εˆ indicates. h i ~R ~ = εˆ1 E0 ei(ωt−~k.R~ ) + cc εÊ0 (x, y) cos ωt − k. 2. the orientation of the polarization for the waves,. the directional vector,. ω. the oscillation frequency and. 4. E0. ~ R. (2.2). is the position of center of mass,. ~k. the time and propagated direction independent.

(18) CHAPTER 2.. THEORY OF LIGHT-MATTER INTERACTION. 5. amplitude. In the quantum mechanical approach these electromagnetic waves consists of photons. The energy contained in each photon is:. Ep with. h. Planck's constant,. c. =. hc |k| hc = 2π λ. the speed of light in vacuum and. λ. (2.3). the photon wavelength. This interpretation. of light, i.e. the quantum mechanical approach, is adequate to understand the quantum phenomena which results from light-matter interaction without scrutinizing in depth the nature of light.. 2.2 Matter: Quantum mechanical description To be able to predict when atoms will absorb or emit light it is imperative to understand how matter responds to external electromagnetic elds.. The consideration of relatively weak external electromagnetic. elds permits us to neglect non-linear processes and a quantum mechanical description suce in describing possible interactions. In this section we give a brief introduction to the Lorentz model and make use of the time-dependent Schrodinger equation in obtaining the probability amplitudes from which upon an in-depth theory can be based to describe light-matter interactions.. 2.2.1 Time-dependent Schro¨dinger equation Consider an electron conned to a time-independent potential caused by the attraction between the opposite charged nucleus and electron, rather than the classical spring force. For this type of energy well one nds the potential to be of the form. V (~r) Z. with. the total number of particles,. e. =. −. Ze2 4π0 r. (2.4). the electron charge and. r. the distance between the nucleus and. charged electron. Making use of quantum mechanics one obtain the Schro ¨dinger equation as:. . with. ~=. h 2π ,. ~2 2 − ∇ + V (~r) ψ (~r, t) 2m. =. m the electron mass and ψ (~r, t) the wave function.. i~. ∂ψ (~r, t) ∂t. To simplify one denes. (2.5). ˆ a = − ~2 ∇2 + V (~r) H 2m. as the unperturbed particle Hamiltonian (operator that represents the potential and kinetic energy of the particle).. Having a time independent potential the solution to the Schrodinger equation would be of the. form:. ψn (~r, t). =. φn (~r) e−iEn t/~. (2.6).

(19) CHAPTER 2.. THEORY OF LIGHT-MATTER INTERACTION. 6. It is clear that the wave equation is separated into time independent functions called stationary states. φn (~r). and a time dependent component, with. En. representing an allowed energy for the electron. For light. (electromagnetic wave) interacting with media the time-independent potential is found to be insucient and a time dependent Hamiltonian is created. ˆ H with. Vext (~r, t). ˆ a + Vext (~r, t) = H. (2.7). the perturbation due to the eect of a wave like source acting in on the particle. The new. Schro ¨dinger equation looks as follows:. −. ~2 2 − ∇ + V (~r) + Vext (~r, t) ψ (→ r , t) 2m. =. ι~. ∂ψ (~r, t) ∂t. Finding a solution to equation 2.8 is somewhat more cumbersome, but due to the fact that. (2.8). φn (~r). is. a complete orthogonal set, a solution satisfying the Schr¨ odinger equation may be expressed as a linear combination of the stationary states, such that. ψ (~r, t). =. X. an (t) φn (~r). (2.9). n Substituting equation 2.9 into equation 2.8 and making use of some mathematical manipulation it can be shown that. X an [En + Vext (~r, t)] φn (~r). =. i~. n. X ∂an n. ∂t. φn (~r). (2.10). The probability that energy exchange will take place can be derived making use of the bra- ket- notation and is found to be.

(20) +

(21) X

(22) φm (~r)

(23) an [En + Vext (~r, t)] φn (~r) =

(24) n. *. Making use of the orthogonality of. φn. and. φm.

(25) +

(26) X ∂a

(27) n φm (~r)

(28) i~ φn (~r)

(29) n ∂t. *. (2.11). (see appendix A4) it can be shown that equation 2.11 is. reduced to the time-dependent Schr¨ odinger equation of the form. Em am +. X an Vmn. =. i~a˙ m. (2.12). n with. Vmn (t). expressed as in appendix A5, equation 19. Equation 2.12 describes the change in the state of. the system over time considered. m. number of possibilities and. Vmn (t). is considered the drive force initiating.

(30) CHAPTER 2.. THEORY OF LIGHT-MATTER INTERACTION. 7. the change. As expected, equation 2.12 is reduced to the unperturbed system if no external electric eld is applied.. 2.2.2 Two-state quantum system with a monochromatic plane wave as source Active media are considered having two, three or four main levels, without considering the ne and hyperne structures (sub-levels).. Ytterbium-doped bre can be quasi-three or -four level systems depending on the. bre's length (quasi-three for short bres and quasi-four for long bres). If for instance one considers a four level energy system then one can approximate it with a two level system, due to the quick decay of atoms from the fourth to the third level and from the second to the rst level. Similarly for a three level system. Reducing the set of equations 2.12 to only two levels will result in. with the terms. V11. E1 a1 (t) + V12 a2 (t). = i~a˙ 1 (t). (2.13). E2 a2 (t) + V21 a1 (t). = i~a˙ 2 (t). (2.14). andV22 zero due to parity consideration. The solution to the above mentioned Schr¨ odinger. equations is of the form. ψ (~r, t). = a1 (t) φ1 (~r) + a2 (t) φ2 (~r). (2.15). which is once again a linear combination of the stationary states. Another advantage of having it reduced to a two level system is that the electron is conned indenitely in one of the two levels and the sum total of the probabilities should therefore add up to one.. |a1 (t) |2 + |a2 (t) |2. =. 1. (2.16). which is equivalent to saying. N01 + N02 with. N01. the number of atoms in state one and. total number of doped atoms. N0 .. N02. = N0. (2.17). the number of atoms in state two, which adds up to the. Considering an external electric eld disturbing an electron one nds the. force exerted on the electron to be of the form. −. dV dx. =. ~ t eE R,. (2.18). Extending equation 2.18 to 3-D space and integrating one obtain the potential enforced by such external elds:.

(31) CHAPTER 2.. THEORY OF LIGHT-MATTER INTERACTION. ~ mn ~r, R, ~ t ∇V ~ t Vmn ~r, R, ~ t Vmn ~r, R, with. ~rmn ,. 8. ~ = −eE ~ t = −e~rmn .E R, h i 1 ~ ~ = − e~rmn . εÊ0 ei(k.R−wt) + cc 2. (2.19). as indicated in [A5, equation 19], the expectation value of the displacement vector. For further. simplication we dene the dierence in energy between states as. e(~ rab .ˆ ε)E0 and also choose ~. E1 = 0. ω21 =. 4E ~ , the Rabi-frequency as. χab =. as the ground state energy. Then equations 2.13 and 2.14 can be rewritten. as. ia˙ 1 (t) ia˙ 2 (t). 1 χ12 e−iωt + χ∗21 eiωt a2 (t) 2 1 = ω21 a2 − χ21 e−iωt + χ∗12 eiωt a1 (t) 2. = −. (2.20) (2.21). At this stage it is important to note that the energy dierence between levels has been dened as the energy of a photon as mentioned in equation 2.3. This draws a direct relation between the energy of a photon and the energy dierence between atomic levels. Taking into consideration that light-matter interaction will most likely occur if one work at energies representing the transitions between levels, one makes use of the trial solutions. The terms that because. e. a1 (t) = c1 (t). e±i2ωt ±i2ωt. and. a2 (t) = c2 (t) e−iωt .. With this in mind equations 2.20 and 2.21 becomes. 1 χ12 e−2iωt + χ∗21 c2 (t) 2 1 (ω21 − ω) c2 (t) − χ21 + χ∗12 e2iωt c1 (t) . 2. ic˙1 (t). = −. (2.22). ic˙2 (t). =. (2.23). are taken to be zero if one makes use of the rotating wave approximation (which states. oscillates so slow on a time scale, in comparison to the other terms, it can be neglected).. The preceding Schro ¨dinger equations reduce to. ic˙1 (t) ic˙2 (t) with. 4 = (ω21 − ω). and. χ = χ21 .. condition) we may assume. 1 − χ∗ c2 (t) 2 1 = 4c2 − χc1 (t) 2. =. (2.24) (2.25). By considering the atom in the ground state at time zero (an initial. c1 (0) = 1. and. c2 (0) = 0.. Also by taking. Ω = χ2 + 42. Rabi-frequency the coupled equations 2.24 and 2.25 can be solved to obtain:. 21. as the generalized.

(32) CHAPTER 2.. THEORY OF LIGHT-MATTER INTERACTION. c1 (t). =. c2 (t). =. 9. i4 cos (Ωt/2) + sin (Ωt/2) e−i4t/2 Ω iχ sin (Ωt/2) e−i4t/2 Ω. (2.26). (2.27). which brings us to the density matrix.. 2.2.3 The Density matrix and decay processes Let's dene the following for the elements of the density matrix. ρ12. = c1 c∗2. (2.28). ρ21. =. c2 c∗1. (2.29). ρ11. = c1 c∗1 = |c1 |. ρ22 with. ρ21. and. ρ12. various states.. 2. =. c2 c∗2. (2.30). 2. = |c2 |. the complex electron displacement vectors,. ρ11. (2.31). and. ρ22. to indicate the population of the. Rewriting Schr¨ odinger equations 2.24 and 2.25 making use of the density matrix elements. 2.28 to 2.31 we obtain the equations of motion:. χ∗ (ρ22 − ρ11 ) 2 χ = −i4ρ12 − i (ρ22 − ρ11 ) 2 i = − (χρ12 − χ∗ ρ21 ) 2 i = (χρ12 − χ∗ ρ21 ) 2. ρ12 ˙. = i4ρ12 + i. ρ21 ˙ ρ11 ˙ ρ˙ 22. (2.32) (2.33) (2.34) (2.35). These four equations posses the capacity to describe the interactions qualitatively, even though they do not consider any relaxation processes.. If the relaxation process should be accounted for, one should look. individually at each of the components contributing to relaxation i.e. elastic and inelastic collision as well as spontaneous emission. As one would expect due to the physical size of the atom, spontaneous emission is found by far the dominant process such that it is the only process to be considered. Including spontaneous emission equations 2.34 and 2.35 become:. ρ11 ˙. =. ρ˙ 22. =. i (χρ12 − χ∗ ρ21 ) 2 i −A21 ρ22 + (χρ12 − χ∗ ρ21 ) 2. +A21 ρ22 −. (2.36) (2.37).

(33) CHAPTER 2.. with. A21. THEORY OF LIGHT-MATTER INTERACTION. the Einstein. A. coecient and. A21 ρ22. 10. describing the spontaneous emission which results in the. expulsion of a photon. For spontaneous emission to occur there need to be atoms in the excited regime and likewise the electron displacement vectors are also inuenced by these relaxation processes, to obtain:. with. β ≈ 12 A21 .. ρ12 ˙. =. ρ21 ˙. =. χ∗ (ρ22 − ρ11 ) 2 χ − (β + i4) ρ21 − i (ρ22 − ρ11 ) 2. − (β − i4) ρ12 + i. The density matrix has therefore been changed to also accommodate for collisions. For. (2.38) (2.39). β large. the density matrix elements reduce to the adiabatic approximation: the change of the electron displacement vectors over time is zero. The occupational probabilities can therefore be rewritten as follows:. 2. |χ| β2 (ρ22 − ρ11 ) 42 + β 2. ρ11 ˙. = A21 ρ22 +. ρ22 ˙. = −A21 ρ22 −. (2.40). 2. |χ| β2 (ρ22 − ρ11 ) 42 + β 2. (2.41). Using the results obtained for the electron displacement vectors equations 2.38 and 2.39 as well as the occupation probabilities equations 2.40 and 2.41 an amplier model will be developed in subsequent sections.. 2.2.4 Ionic cross-sections and Einstein A & B coecients The ionic cross-sections play an important role in quantifying absorption and emission processes and therefore their connection to the Einstein coecients are of great importance. The steps in this section mostly follow reference [17].. Other than having dierent transition lineshape functions caused by the population. distribution; the absorption and emission cross-sections are considered equal but describe complimentary processes. From the denition of the absorption cross-section. σ with the ux density as. =. φ (~r) =. energy absorption rate per atom in level 1 incident radiant energy ux. I(~ r ,ν) hν. =. (2.42). I(~ r ,λ)λ and inspection of equations 2.40 and 2.41 one can conclude hc. the absorption cross-section to be. 2. σ (λ). =. |χ| β2 42 + β 2. !. hc I (~r, λ) λ. (2.43). For now the absorption and stimulated emission cross-sections are considered equal in magnitude, but will change at a later stage when considering level degeneracies caused by the population distributions. By substituting. χ with its previous denition χab =. e(~ rab .ˆ ε)E0 ~. =. µE0 ~ and the light intensity with its electromagnetic.

(34) CHAPTER 2.. counterpart,. THEORY OF LIGHT-MATTER INTERACTION. I=. cε0 2 2 |E0 | , the absorption cross-section can be expressed as:. β |µE0 |2 h2 2 2 2 2λ0 ~ |E0 | β + 42 = Bf (λ). σ (λ). with. B=. 11. µ2 0 2~2 the well know Einstein. B. =. coecient and. f (λ). (2.44). some arbitrary line-shape function dependent. on the incident light and the population distribution. The average power absorbed at a specied wavelength is dependent on the absorption cross-sections as well as the incident light intensity.. P (λ). =. σ (λ) I (~r, λ). (2.45). Taking the energy absorbed during interactions into consideration the rate of absorption at a specied wavelength can be expressed as. R. = σ (λ) φ (λ). (2.46). Furthermore the relationship between the spontaneous emission rate (or better known as the Einstein coecients) and the Einstein. B. A. coecients can be found as:. A. =. B2~ (2πν) πc3. 3 (2.47). The full derivation can be found in reference [16]. Having related these fundamental aspects the following section is devoted to develop the population rate equations and will specically draw distinction between absorption and stimulated emission cross-sections.. 2.2.5 The population rate equations Multiplying the occupational probabilities. ρ11. and. ρ22. N1. =. N ρ11. (2.48). N2. =. N ρ22. (2.49). with the total doping concentration. N,. the density of. atoms in each level is calculated:. with. N1. the atom density in level 1 and. N2. the atom density in level 2. These relations also need to satisfy. equation 2.17 which states that the the total number of active atoms should stay xed. Substituting equations 2.48, 2.49 and 2.46 into equations 2.40 and 2.41 the population rate equations results:.

(35) CHAPTER 2.. with. THEORY OF LIGHT-MATTER INTERACTION. 12. N˙ 1. =. +A21 N2 + σφ (N2 − N1 ). (2.50). N˙ 2. =. −A21 N2 − σφ (N2 − N1 ). (2.51). σ dened as expressed in equation 2.43 representing the absorption or stimulated emission cross-sections.. Adding these equations will result in zero which is expected considering the rate of change in level one needs to be countered by the rate in change in level two. This is mainly due to the fact that collisions have been ignored. Considering collisions will result in a negative counterpart:. N˙ 1 + N˙ 2. = −Γ1 N1 − Γ2 N2. (2.52). which states that inelastic collisions will cause all the atoms to decay from states 1 and 2 into lower states. This is, however, negligible in view of what is to be achieved and will be ignored for all further purposes. What is important though, is to draw distinction between absorption and emission cross-sections by considering sub-level degeneracies ignored previously.. The population changes in levels 1 and 2 are governed by the. transition strengths between the individual sublevels that make up each level. Consider degeneracy in level 1 having. g1. sub-levels denoted. m1. and level 2 having. g2. sub-levels denoted. m2. as depicted in gure 2.1, then. the emission and absorption to and from level 2 will be a summation of terms corresponding to transitions between sub-levels.. Figure 2.1: Possible transitions between two energy levels. Making use of references [15, 18] the new rate equations are obtained:. N˙ 1. =. 1 X 1 X [A (m1 , m2 ) + R (m1 , m2 )] N2 − R (m1 , m2 ) N1 g2 g1. =. A21 N2 + φ (σ21 N2 − σ12 N1 ). m1 ,m2. m1 ,m2. (2.53).

(36) CHAPTER 2.. THEORY OF LIGHT-MATTER INTERACTION. N˙ 2. = −. 1 X 1 X [A (m1 , m2 ) + R (m1 , m2 )] N2 + R (m1 , m2 ) N1 g2 g1 m1 ,m2. = Here. 13. m1 ,m2. −A21 N2 − φ (σ21 N2 − σ12 N1 ). (2.54). A (m1 , m2 ) represents the spontaneous transition rate and R (m1 , m2 ) the absorption and stimulated. transition rates between sublevels. m1. stimulated emission cross-section and. and. σ12. m2 . A21. presents the total spontaneous emission,. σ21. the total. the total absorption cross-section. The total spontaneous emission. rate, also known as the inverse radiative lifetime, can also be expressed in terms of the ionic cross-sections. A21 where. σe. 1 8π = 2 τ λ. =. Z. σ e (λ) dλ. (2.55). presents the stimulated emission. Notice that the absorption was taken as continuous and therefore. the summation was extended to integration. Equations 2.53 and 2.54 describe the populations which are in essence inecient due to the lack of applied electromagnetic elds. Specifying electromagnetic elds which will force atoms from the level 1 to level 2, considering level degeneracies and including spontaneous emission (equation 2.55) it can be shown that equations 2.53 and 2.54 can be written as:. dN1 dt. dN2 dt with. η. processes,. " =. # " # η η σpe Ip P σje Ij σap Ip P σja Ij 1 + + τ N2 − + N1 hνp hνj hνp hνj j j. ". # " # η a η a P σ I σpe Ip P σje Ij I σ p j p j = − + + τ1 N2 + + N1 hνp hνj hνp hνj j j. indicating the summation of positive and negative propagating direction,. a. (2.56). indicating absorption processes,. p. relating to the pump and. j. (2.57). e. indicating emission. relating to the ASE. Inspecting. the population rate equations, equation 2.56 and 2.57, one nds them dependent on intensity parameters of which the values vary with distance along the bre length. The subsequent section is to get a handle on these quantities and to develop an understanding for those parameters which will eventually lead on to the Giles parameters.. 2.2.6 Light amplication and Gain Examining the population rate equations they are found dependent on light intensities which change with positions inside the active media. Knowing that absorption takes place it should be evident that this results in light intensity uctuations and hence the light intensities, as a function of position, are of great importance. To approach this problem consider monochromatic light being transmitted through a thin layer of nongain media propagating in the z-direction with frequency. ν.. The rate at which electromagnetic energy is.

(37) CHAPTER 2.. THEORY OF LIGHT-MATTER INTERACTION. transmitted through an area. A. and a layer thickness of. 4z. [Iν (z + 4z) − Iν (z)] A Taking a very thin layer,. 4z ,. uν =. is:. ∂ Iν A4z. ∂z. =. (2.58). equation 2.58 is reduced to:. − with. 14. ∂ uν A4z ∂t. =. ∂ Iν A4z ∂z. Iν c . Considering the physical properties of the media i.e.. A and 4z. (2.59). as xed, equation 2.59 reduces. to one of Pointing's theorems :. 1 ∂Iν ∂Iν + c ∂t ∂z. =. 0. (2.60). This theory states that the temporal and spatial dependence of the intensity is directly related and cannot be considered separately. Substituting the non-gain media with gain media will result in interactions between the active media and the light intensities. The dominant processes taking place, i.e. absorption and stimulated emission, will result in the revised Pointing's theorem :. ∂Iν 1 ∂Iν + v ∂t ∂z. =. Iν (σ21 (ν) N2 − σ12 (ν) N1 ) ,. (2.61). but normally the gain would be substituted with the following:. g (ν). =. (σ21 (ν) N2 − σ12 (ν) N1 ). (2.62). which is not really of signicance other then a simplied writing method. What one should notice is that spontaneous emission is not present and will be dealt with at a later stage. This technique can now be applied to the whole active media to assume N adjacent media increments (gure 2.2) such that the input of the second increment would be the output of the rst increment and so forth.. Figure 2.2: N adjacent bre increments representing the bre as a whole. Having obtained a continuity equation at one wavelength for the intensity it can easily be obtained for.

(38) CHAPTER 2.. THEORY OF LIGHT-MATTER INTERACTION. 15. each of the wavelengths in the media. Not having dealt with bre criteria yet, the modications due to the use of bre will be introduced in section 4.1.2 after a basis for bre characteristics has been developed..

(39) Chapter 3. Fundamental Properties of Optical bres This chapter aims to give a brief but descriptive insight into bre as gain media and suce as description to the spatial characteristics of bre whilst dealing with practical problems that may arise.. Furthermore. the most important aspects of bre will be dealt with aiding us in the construction of numerical models in consecutive chapters imitating bre laser behaviour.. 3.1 Fibres Modern bre lasers have advantages over other laser systems which makes them benecial for wide spread use, but not without consequences. In single mode bre the small face-front within the. µm. range and high. photon densities result in bre damage. The objective of having high output powers is quickly demolished due to thermal fracture caused by the small dimensions of the single mode bre.. The problem is easily. overcome making use of double cladding bre whereupon the inner core is doped and the light is injected into a multi-mode inner cladding. Higher output powers are obtained due to separation of the pump and signals, limiting thermal eects caused by high photon densities . This can be seen visually in gure 3.1. Notice the active media is only conned inside the core and not in the inner cladding.. Figure 3.1: Dierent sections of a double cladding optical waveguide.. 16.

(40) CHAPTER 3.. FUNDAMENTAL PROPERTIES OF OPTICAL FIBRES. 17. The desired bre face-front is that of an asymmetrical prole and hence the inner cladding geometries are asymmetrical in nature. This is primarily because certain symmetric proles will allow light to propagate in a spiral way, by means of internal reection, without actually crossing the bre core. Breaking the circular symmetry will force the injected light to cross the bre core and excite the active media [19, 20]. The type of bre used in this experiment can be seen visually in gure 3.2.. Figure 3.2: Fibre face-front of the optical waveguide used at Stellenbosch with courtesy of FIBERCORE [21]. One can observe quite a dierence between cross-sectional areas of the core and the inner cladding. This is expected since the number of allowed modes relates to the cross-sectional area of each section.. Even though the bre face-front has a symmetric prole the spiraling eect is broken by the ower (quasi polygonal, sinusoidal) formation of the inner cladding.. 3.1.1 Optical coupling in waveguides The power injected into the gain bre is of great signicance in this project and has been studied in great detail to establish a qualitative understanding of measurements taken. Due to unavoidable conditions only about 70-90 % of the optical power delivered by the diode laser is injected into the gain bre. Cleaving the. ◦. diode laser bre at 8. Fresnel reection is at an minimum and there will be minimal feedback back into the. diode laser caused by reections from the gain bre face-front.. At the same time light emitted from the. diode laser is shifted in respect to the reference axis. This makes it dicult to inject light into the gain bre situated on the reference axis and light is lost in the process. Another drawback is that bre only accepts light incident on the bre face-front which comes in at an angle smaller than the acceptance angle. This phenomena mainly occurs due to the lack of internal reection and can be described making use of Snell's law and gure 3.3. If light enters the bre with an incident angle larger than the acceptance angle, total internal reection would not occur, the light would not be guided and coupled to the outer cladding. According to Snell's law light being emit from one material to another yields:.

(41) CHAPTER 3.. FUNDAMENTAL PROPERTIES OF OPTICAL FIBRES. 18. Figure 3.3: Acceptance cone of injected pump power. Any light entering the bre with a angle than the acceptance angle. (θc ). ni sin (θi ) with. ni and nf. (θi2 ) greater (θi1 ).. will be coupled to the outer cladding, but not light with an angle of. = nf sin (θf ). (3.1). the refractive index of the initial and nal media respectively. Using Snell's law for both. the face-front and the interface between the inner and outer cladding one nds the maximum incident angle to be. θacc. with. θint. =. −1. sin. q. n21. −. n22. . the critical angle for total internal reection (material dependent) and. (3.2). n2. and. n1. the outer and. inner cladding refractive index respectively. Though this is not quite the case, due to the refractive index being dependent on the incident light wavelength, equation 3.2 models the acceptance angle adequately. Taking the energy loss into consideration the optical output is not the injected power into the bre. The injected power can be resolved making use of a bre cut-back technique, residual pump power measurements and extrapolation, but will be explained in chapter 6.1.3. Another important aspect that should be taken into account is the dierent modes in which light is allowed to propagate inside the bre. The ASE propagates inside the bre core which is single mode and one therefore expects the intensity distribution to be that of a Gaussian distribution, as illustrated in gure 3.4. Respectively, the pump light propagates inside the core as well as the inner cladding which is multi-mode. Taking all the modes into consideration a top hat prole is obtained for the pump propagating inside the bre. Illustrated in gure 3.5 is the expected pump intensity distribution as well as the rst three modes of the inner cladding..

(42) CHAPTER 3.. FUNDAMENTAL PROPERTIES OF OPTICAL FIBRES. 19. Figure 3.4: The ASE propagates inside the bre core and can be seen to exhibit some intensity distribution which is of signicance if one wants to calculate the overlap factors.. Figure 3.5:. As the pump is injected into the inner cladding one can once again expect some intensity. distribution conned to the core and inner cladding. What can be seen is a top hat prole considered as the intensity distribution as well as the rst three modes of the inner cladding.. Considering the impact dierent modes would have on the intensity distribution it should be clear that the power conned in waveguides will be dependent on the overlap between the modes and the physical.

(43) CHAPTER 3.. FUNDAMENTAL PROPERTIES OF OPTICAL FIBRES. 20. dimensions (area) these modes are conned to. The revised optical power in a waveguide will therefore be of the form:. P (z, λi ) with. z, θ, r. = conf inedinwaveguides. the radial coordinate system components,. eective area and the dierent modes.. Aef f. I (z, θ, r, λi ) Aef f Γ (λi , r, θ). the eective area and. It should be clear that. r, extends to outside the radius of the inner cladding,. Rclad .. I (z, θ, r). (3.3). Γ. the overlap between the. will drop o to 0 if the radius,. Also noticing that optical instruments work. with optical power and not the intensity distribution, it would be ill advised to stick with a formalism not representing the physical measurements taken.. Considering that a solution-doping technique was used to. dope the bre, it is appropriate to approximate a uniform distribution throughout the bre core. With the eective area none other than the doped region, it is found that. 2 Aef f = πRcore. with. Rcore. the core radius.. The overlap factors are somewhat more dicult to determine as they are mode dependent which consist of some intensity distribution. The overlap factors are obtained as. 2π Z ∞ Z. Γ (λi , r, θ). =. d (r, θ) φ (λi , r, θ) rdrdθ. (3.4). 0 0 where. d (r, θ) and φ (λi , r, θ) are the normalized dopand and signal intensity distributions, respectively.. There-. fore one calculates the various modes inside the bre and successively calculates the overlap factors. As a crude estimate one can assume the overlap to be the ratio of the cross-sectional area of the doped region (core) to the cross-sectional area of the various guides the modes are conned to (can be the core or the inner cladding depending if it is ASE or pump). This assumption will be used at a later stage when trying to elaborate on some key points. Calculation of the overlap factors is a tedious process and one would prefer an analogy where the overlap factors are not present.. 3.2 Dopand and hosts The ionic cross-sections are a material property which are mainly dependent on the rare-earth ions and secondly dependent on the host material (in which the rare-earth ions are doped). Yb level multiplets al. [22].. 2. F7/2. and. 2. F5/2. separated up to 11 630 cm. −1. 3+. ions have energy. as seen in gure 3.6 obtained from Newell et..

(44) CHAPTER 3.. FUNDAMENTAL PROPERTIES OF OPTICAL FIBRES. 21. Figure 3.6: The energy level diagram for ytterbium-doped bre obtained from Newell et. al.[22] is shown. Particularly important are the emission lines. One can observe four emission lines i.e. 977 1032. nm. and 1069. nm. nm,. 1020. nm,. for ytterbium-doped bre. The rst emission line corresponds to a quasi-three level. system and the rest to a quasi-four level system.. One should realize the ionic cross-sections will have minor dierences for dierent hosts materials, but will exhibit drastic changes for dierent rare-earth ions used. Mainly due to the close similarity of Germanosilicate (used in literature [12]) and Aluminosilicate (used in this study) as host, there will be no distinction made between them. A nal remark concerning dierent hosts: the reader can read up on the dierent types of hosts and eects contributed by dierent silica on bre in references [10, 23, 24].. 3.2.1 Transitions in active media 3+. Knowing the energy level scheme of Yb. there are basic assumptions which should be made. The active. media consists of well spaced rare-earth ions such that the atoms can decay independently from each other therefore excluding ion-ion interactions. Possible processes that one should account for in bre when working with low intensities include the following:. •. Spontaneous emission at a probability dependent random wavelength within the emission band. •. Multi-photon decay (not likely to occur). •. Stimulated emission and absorption at all wavelengths. For our purpose we ignore multi-photon decay since it is minute in comparison to other processes and will mostly occur at very high intensities.. The energy diagram in gure 2.1 illustrates the possible processes.

(45) CHAPTER 3.. FUNDAMENTAL PROPERTIES OF OPTICAL FIBRES. 22. taking place which corresponds to the absorption or emission of light in the form of a photon with energy. (Em2 − Em1 ).. What is not evident in this diagram is the fact that transitions can occur to and from any one. of the sub-levels, but the one with the highest probability is favoured.. 3.3 Methods for obtaining the ionic cross-sections Currently several methods exist for obtaining the ionic cross-sections. The absorption cross-sections can be obtained using normal absorption spectroscopy. The bre is illuminated with a white light source and the transmitted light is measured. An existing problem with this technique is that one cannot distinguish if light has been re-absorbed after emission or absorbed for the rst time. The general assumption is that light is absorbed for the rst time when working with short bres. Unlike the absorption cross-sections the emission cross-sections are quite dicult to obtain directly.. The emission cross-section can be calculated from the. absorption cross-sections by making use of the Fuchtbauer Ladenberg (FL) relation, the exact McCumber (EMC) relation or the approximate McCumber (AMC) relation [9, 10]. It can also be measured making use of uorescence saturation spectroscopy [9]. The spectroscopy procedures are currently considered the most accurate, but cannot always be applied due to experimental limitations. For now the spectroscopy procedures are considered as correct and used as benchmark to show the shortcomings of the other methods, but in due time will also be evaluated. For the FL relation the assumptions break down mainly because of the use of rare-earth ions and the FL relation overestimates the peak emission cross-sections, by up to 50 %. Another possible reason why the assumptions break down is because silica as host media causes Stark splitting, due to local electric elds, lifting the degeneracy and causing spectral broadening. The accuracy in relating the absorption and emission cross-sections are therefore typically low. The dierence between what is obtained by the FL relation and what is obtained by uorescence spectroscopy can be seen in the study of Digonnet et. al. [10] for Erbium doped bres (conceptually similar to ytterbium-doped bre). The EMC relation is rarely ever used due to the fact that once again Stark sub-energies of the main peaks should be known. Stark sub-energies coincide with the shifting and splitting of spectral lines of atoms due to the presence of external static electric elds, which are host dependent and extremely dicult to measure. The EMC relation fails for broad transitions, particularly towards the tails of spectra and worsens with increasing homogeneity.. Rare earth ions in silica-based hosts are known for their broad transitions. and homogeneous broadening and are therefore subject to EMC failure [9]. The dierence between what is obtained by the EMC relation and what is obtained by uorescence spectroscopy can be seen in the study of Digonnet et. al. [9] for an Erbium-doped Aluminosilicate glass bre. Having generalized the EMC to be independent on the Stark sub-energies we end up with the AMC relation. The simplicity and yet great success in accuracy of this relation has made it very popular in the scientic community. The derivation of the AMC relation can be followed in the paper of McCumber [11]. What is troublesome though is that recent research indicates that the AMC relation is not as trustworthy as speculated [9].. The AMC relation tends to overestimate the peak emission cross-sections.. According. to literature [9] this is partly because the AMC procedure underestimates the energy spread of the higher energy manifold and overestimate the energy spread of the lower energy manifold. In the paper of Digonnet.

(46) CHAPTER 3.. FUNDAMENTAL PROPERTIES OF OPTICAL FIBRES. 23. et. al. [9] the AMC relation in comparison to what was obtained by uorescence spectroscopy for 16 dierent bres was investigated. The best and worst relation between the AMC and uorescence spectroscopy for the 16 bres can be seen in gures 3.7 and 3.8 respectively. In both gures the emission spectra for the AMC and uorescence spectroscopy were normalized to the same peak value. The McCumber Relation tends to underestimate the emission cross-section spectra for. λ>. main peak. For the best case, gure 3.7, this is a. few percent to almost 10 %. For the worst case it tends to be much higher. Simulating Erbium bre lasers at 1590. nm. using the AMC relation for either bre hosts (NA or Al/Ge-SiO2 ) will yield inaccurate results. as there is at least a dierence of 10 % between what is obtained by the AMC relation in comparison to uorescence spectroscopy.. Figure 3.7: Making use of absorption spectroscopy and the McCumber relation the emission cross-sections are obtained for Erbium doped in Al/Ge-SiO2 . This is the best McCumber t for Erbium-doped bre obtained from Digonnet et. al. [9]..

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