High-order mode selection in Yb:YAG ceramic laser

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(1)High-order mode selection in Yb:YAG ceramic laser A thesis submitted by. Manasadevi Priyamvadha Thirugnanasambandam in partial satisfaction to the requirements for the degree of. Doctor of Philosophy in Physics. Supervisor: Prof. UEDA Ken-ichi Institute for Laser Science Department of Applied Physics and Chemistry Graduate school of Electro-Communications The University of Electro-Communications Tokyo, Japan. March, 2011.

(2) High-order mode selection in Yb:YAG ceramic laser. Manasadevi Priyamvadha Thirugnanasambandam. Approved by: Chair of Committee: Prof. Ken-ichi Ueda, Advisor Department of Applied Physics and Chemistry. ————————————–. University of Electro-Communications. Committee Members: Prof. Mitsuo Takeda Department of Information and Communication Engineering. ————————————–. University of Electro-Communications. Prof. Kazuko Shimizu Department of Applied Physics and Chemistry. ————————————–. University of Electro-Communications. Prof. Masayoshi Watanabe Department of Engineering Science. ————————————–. University of Electro-Communications. Assoc. Prof. Masayuki Katsuragawa Department of Applied Physics and Chemistry University of Electro-Communications. ————————————–.

(3) Declarations I, Manasadevi Priyamvadha Thirugnanasambandam, hereby declare that the thesis entitled High-order mode selection in Yb:YAG ceramic laser, is a bonafide record of my original research work and that it has not been submitted anywhere for award of higher degree or diploma. The work was done under the guidance of Prof. Ken-ichi Ueda, at the University of Electro-Communications. Wherever contributions of others are involved, every effort is made to indicate this clearly, with due reference to the literature, and acknowledgement of collaborative research and discussions.. Date. Signature of candidate.

(4) c 2011 Manasadevi Priyamvadha Thirugnanasambandam Copyright All rights reserved..

(5) Dedicated to My parents K. P. Thirugnanasambandam & S. Parvady This work, as with all of my work, belongs to you. I will love you, always and forever..

(6) “ With the Gift of Awareness, comes the realization of the Infinite Knowledge awaiting our grasp.” – Rhonda Lipstein.

(7) Acknowledgements Prof. Ueda Ken-ichi: For having given me this opportunity to work here at ILS. I sincerely appreciate all your brilliant ideas and guidance through the doctoral course. I am grateful for all your efforts; it has made my stay here pleasant. Prof. Senatsky Yuri: For your clarity, patience and perseverance while working together both during your stay here in ILS and in Moscow. Assoc. Prof. Shirakawa Akira: For all your motivating discussions, critical suggestions and offering a helping hand whenever in need. Thank you for translation help with the thesis abstract. Assoc. Prof. Musha Mitsuru: For all the helpful discussions and for always being there to advise on difficulties during experiments. Prof. Hakuta Kohzo, Prof. Takeda Mitsuo, Prof. Nakagawa Ken’ichi: For your kindness, generosity and selfless support. Dr. Li Jianlang : Thank you for all helpful discussions during your stay at ILS. All fellow lab mates (2007-2011): Enjoyed working with you all! Review committee [Prof. Shimizu Kazuko, Prof. Takeda Mitsuo, Prof. Watanabe Masayoshi, Assoc. Prof. Katsuragawa Masayuki]: For all your critical and valuable comments on the thesis. Nivethida, Mirunalini: Wonderful siblings; for having selflessly given me all their support in every way. Raju, Kali, Dinesh, Gopi, Manoj, Chandra, Srinu, Sanjay and all friends: For being supportive, comforting and encouraging through all good and bad times Okudaira, Ito, Ishii: For all the administrative support that helped realize a pleasant stay and research. This journey would not have been possible without the help and support of the kind and precious people (teachers and friends) around me in every walk of life so far, to only some of whom it was possible to give particular mention here. A heartfelt thanks to one and all..

(8) Abstract This dissertation demonstrates an economical and less complicated laser scheme to generate high-order modes from an Yb:YAG ceramic laser cavity with unique properties and hence varied applications. The intra-cavity mode selecting element is a simple plano-convex glass lens and it has been shown to offer excellent mode selection in scalar Laguerre-Gauss modes, modes with near-diffraction free characteristics, and radial and azimuthally polarized modes from within the Yb:YAG cavity. The current study bridges two different regimes that have not been brought together so far; and the end-product is a quasi-three level laser system to realize circular mode selection. In case of a laser resonator with intra-cavity lens, the stability conditions of the cavity in as defined by ray transfer matrix are enclosed between the focusing and imaging conditions of the lens at the resonator mirrors. In the presence of strong spherical aberration, the focal length of the lens varies continuously from its axis to the periphery and is no more a defined point but a line. When such a lens is placed at focal length distance from the cavity mirror and shifted along the resonator axis to span across the line focus, the cavity conditions provide continuous selection in modes of increasing mode diameter at the output coupler because of the stability condition shifting to rings of increasing radius along the refracting surface of the lens, generating pure hollow scalar LaguerreGauss modes of increasing azimuthal orders as high as 28 (order limited only by the diameter of the cavity optics). Generation of the whole lineage of LaguerreGauss modes from a single scheme has been reported by the proposed scheme for the first time. The laser scheme with aberrated intra-cavity lens can also be set up such that the cavity conditions offer stability conditions for both focusing and imaging conditions simultaneously. Coupling of the cavity modes, enhanced by aperture guiding at the gain element and long cavity length, generates beams which exhibit.

(9) near-diffraction free properties in their central parts (maximum or minimum). The coupling of these two modes in the cavity has also been confirmed experimentally. Near-diffraction free properties associated with central parts of doughnut-like and multi-ring modes are reported. The near-diffraction free properties of these beams (divergence of their central parts 10 times smaller than the diffraction limit that is associated with the mode radius at the output and constant power confined within the center) exist for over 40 m of propagation distance. When the intra-cavity lens in combination with a birefringent crystal (positive or negative birefringence) inside the Yb:YAG ceramic laser cavity, the stability region of the cavity between the imaging and focusing conditions of the lens is split into two regions; one corresponding to the extraordinary ray and the other to the ordinary ray. The splitting of the stability region allows mode selection in azimuthal and radially polarized modes. The transition point from azimuthal to radial polarization as the lens is shifted along the resonator axis was clearly distinguishable when the width of the stability region (enclosed between focusing and imaging conditions) matched the birefringence shift. Azimuthal and radial polarized doughnut modes were generated with high polarization purity. Polarization extinction ratio as high as 100:1 in certain cases have been reported. Propagation factor M2 was estimated to vary between 2-2.5 (theoretical value of M2 for doughnut LG∗01 = 2). Apart from these doughnut modes, new polarization distribution patterns wherein azimuthal and radial polarizations existed together in a single output beam were generated for the first time. When the intra-cavity lens had pronounced spherical aberrations, the cavity offered mode selection in high-order azimuthal and radial polarized modes. The experimental work has brought into limelight the realization of certain high-order modes which were only theoretically investigated till date. High-order modes have wide applications related to angular momentum transfer, material processing and many more. Near-diffraction free beams generated from the scheme can be seen as a valuable replacement to Bessel beams which has only a very lim-.

(10) ited propagation distance. Radial and azimuthal polarized modes have extensive applications in the optical toolbox for micromanipulation and some of the new polarization distributions are added tools for realizing new phenomena..

(11) Contents. Acknowledgements. vi. Abstract. viii. List of Figures. xiii. List of Tables. xviii. 1 Introduction. 1. 1.1. Motivation to the research . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2. Ytterbium solid-state lasers . . . . . . . . . . . . . . . . . . . . . .. 3. 1.2.1. Ceramic lasers . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 1.2.2. Yb 3+ doped Yittrium Aluminium Garnet ceramics . . . . .. 7. Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 1.3. 2 Axi-symmetric Circular Modes. 10. 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. 2.2. Laguerre-Gaussian scalar modes . . . . . . . . . . . . . . . . . . . . 12. 2.3. 2.4. 2.2.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. 2.2.2. Properties of LG modes . . . . . . . . . . . . . . . . . . . . 12. 2.2.3. Generation of Scalar Laguerre-Gaussian modes . . . . . . . . 13. Diffraction-free beams . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14. 2.3.2. Properties of Bessel beams . . . . . . . . . . . . . . . . . . . 14. 2.3.3. Generation of Bessel beams . . . . . . . . . . . . . . . . . . 16. Cylindrical vector modes . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. 2.4.2. Properties of Cylindrical Vector Beams . . . . . . . . . . . . 19. 2.4.3. Generation of Cylindrical Vector Beams . . . . . . . . . . . 20. xi.

(12) 3 Resonator theory. 23. 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. 3.2. Ray Transfer Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 23. 3.3. Resonator stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 26. 3.4. Lens Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4.1. Resonator scheme . . . . . . . . . . . . . . . . . . . . . . . . 28. 3.4.2. Ray Transfer Matrix Analysis . . . . . . . . . . . . . . . . . 29. 3.4.3. Nature of mode propagation . . . . . . . . . . . . . . . . . . 30. 4 Scalar Laguerre-Gauss modes generation. 32. 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32. 4.2. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2.1. Diode pumping . . . . . . . . . . . . . . . . . . . . . . . . . 33. 4.2.2. Gain medium . . . . . . . . . . . . . . . . . . . . . . . . . . 33. 4.2.3. Laser scheme for generation of scalar Laguerre-Gaussian modes 34. 4.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35. 4.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38. 4.5. 4.4.1. Resonator with intra-cavity aberrated lens . . . . . . . . . . 38. 4.4.2. Resonator stability and mode selection mechanism . . . . . . 39. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. 5 Generation of beams with near-diffraction free characteristics. 44. 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44. 5.2. Experiment and Results . . . . . . . . . . . . . . . . . . . . . . . . 46. 5.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.3.1. 5.4. Coupling of cavity modes for NDF propagation . . . . . . . 50. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52. 6 Radial and azimuthal polarized modes selection 6.1. 53. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.

(13) 6.2. Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54. 6.3. Observation and results . . . . . . . . . . . . . . . . . . . . . . . . . 55 6.3.1. Radial and azimuthal polarized doughnut modes . . . . . . . 55. 6.3.2. Novel mode compositions . . . . . . . . . . . . . . . . . . . . 59. 6.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61. 6.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66. 7 Conclusion. 68. 7.1. Scalar Laguerre-Gaussian beams . . . . . . . . . . . . . . . . . . . . 69. 7.2. Beams with near-diffraction free characteristics. 7.3. Cylindrical Vector Beams. 7.4. Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.4.1. . . . . . . . . . . . 70. . . . . . . . . . . . . . . . . . . . . . . . 71. Into the world of applications . . . . . . . . . . . . . . . . . 73. Bibliography. 74. Curriculum Vitae. 83.

(14) List of Figures. 1.1. (a) Room temperature spectra of 1 at.% Yb 3+ :YAG (b) Energy level diagram of Yb 3+ :YAG . . . . . . . . . . . . . . . . . . . . . .. 4. 1.2. Transparent ceramics manufactured by vacuum sintering technique. 7. 2.1. A basic configuration for high-order mode selection in a laser resonator with an intra-cavity phase element . . . . . . . . . . . . . . 13. 2.2. Intensity profile of a typical (a) zeroth-order (b) first order Bessel beam Courtesy: Z.Bouchal laboratory, Palacky University, Czech Republic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15. 2.3. Zeroth-order quasi-Bessel beam (a)obstructed by opaque obstacle (b,c)restoring beyond obstacle (d) restored fully beyond an obstacle. Courtesy: Am. J. Phys. 67, 912 (1999) . . . . . . . . . . . . . . . . 17. 2.4. Experimental arrangement for the creation of a J0 beam using an annular slit of mean diameter d and width δd and a positive lens of focal length f and radius R. Zmax indicates the propagation distance of the Bessel beam created. Courtesy: Phy. Rev. Lett., 58, 15, 1499 (1987) . . . . . . . . . . . . . . . . . . . . . . . . . . . 18. 2.5. Plane waves propagating on a cone beyond an axicon . . . . . . . . 18. 2.6. LG01 mode showing (a)radial and (b)azimuthal polarization distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19. 2.7. Intra-cavity polarization selective photonic crystal grating mirror used for mode selection. Coutesy: Opt. Lett., 16, 10841 (2008). . . 21. 2.8. The pie-shaped segments of a half-wave plate were assembled with their slow axes oriented as shown by the arrows (left). The fabricated disk was antireflection-coated at 1.06 m (right). Courtesy: Opt. Lett. 32, 11, 1468 (2007). . . . . . . . . . . . . . . . . . . . . 22. xiv.

(15) 3.1. Transformation of rays by optical systems having one vanishing element in their ray transfer matrix. The ray transfer matrix describes the ray propagation from left to right between the planes indicated by vertical lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25. 3.2. Mode parameters of interest for an arbitrary resonator . . . . . . . 27. 3.3. Stability diagram of passive laser resonator . . . . . . . . . . . . . . 28. 3.4. Resonator scheme with intra-cavity thin lens . . . . . . . . . . . . . 29. 4.1. CCD camera image and intensity profile of the focused pump beam. 4.2. Schematic of the experimental setup. 4.3. CCD camera images of various LGpl modes (indices specified) in. 33. . . . . . . . . . . . . . . . . . 34. near-field. (a) p = 1, l = 2; (b) p = 6, l = 12; (c) p = 10, l = 20; (d) p = 11, l = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4. Dependence of the azimuthal mode order, l of hollow LGpl modes on (a) lens apex position (threshold LD currents are indicated), L = 114 cm; (b) length of the cavity . . . . . . . . . . . . . . . . . . . 36. 4.5. CCD camera images of (a) near and (b) far field patterns of LG10,25 mode and (c) far field of LG7,14 mode (d) with the corresponding intensity profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37. 4.6. Ray tracings in a plane-parallel resonator containing a plano-convex lens with spherical aberration . . . . . . . . . . . . . . . . . . . . . 38. 4.7. Ray tracings showing the variation in focal length with y-coordinate in a lens with strong spherical aberration . . . . . . . . . . . . . . . 40. 4.8. Graphical plots showing the boundaries of the stability region defined by dependences of the lens foci f1 and f2 on lens position d1 in the thin lens resonator of lengths (a) L = 114 cm and (b) L = 20 cm 40. 4.9. Graphical plots showing the dependence of mode radii at the cavity mirrors on the position of lens in the cavity. . . . . . . . . . . . . . 41.

(16) 5.1. Radially polarized ring and arc beams generation in neodymium laser with an intra-cavity axicon . . . . . . . . . . . . . . . . . . . . 45. 5.2. CCD camera images of output in near-field as the lens is shifted from d = fax towards the HR surface (a) - (f) . . . . . . . . . . . . 46. 5.3. CCD camera images of (a) Bessel-like multi-ring beam with central peak (b) Bessel-like multi-ring beam with central hole (c) Doughnutlike beam. Each group contains images captured in near-field, farfield (at the focus of an extra-cavity lens ; f = 50cm) and far-field (at ≈ 10m from the output coupler) . . . . . . . . . . . . . . . . . . 47. 5.4. CCD camera images captured in near-field (a) with 1.7 mm obstacle (b) 1.2 mm obstacle (c) no obstacle along the resonator axis. . . . . 51. 5.5. Ray tracing of conical (type-1) and parallel (type-2) beams in the Yb:YAG plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52. 6.1. Schematic of the laser resonator . . . . . . . . . . . . . . . . . . . . 54. 6.2. CCD camera images in near-field of AP and RP doughnut modes generated from Yb:YAG laser cavity with birefringent (a,b) c-cut YVO4 or (c,d) α-BBO crystals and lens f = 10 cm placed at distance d < fax or d > fax from HR surface. The transmission axis of linear polarizer placed before the camera is indicated by arrows in the corresponding frames wherever used. . . . . . . . . . . . . . . . . . 57. 6.3. Mode profiles and polarization change observed on shifting the lens f = 5cm in the Yb:YAG laser cavity between HR and OC regions. . 59. 6.4. Sequence of modes registered at the output of laser (L = 114 cm) by shifting intra-cavity lens (a) near field images with YVO4 and lens f = 10 cm (b) near field images and far field intensity profiles with α-BBO and lens f = 7.5 cm (c) far field images with YVO4 and lens f = 3.5 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . 60.

(17) 6.5. Schematic of resonator with intra-cavity lens showing ‘imaging’ (Type-1 trajectory) and ‘focusing’ (Type-2 trajectory) positions of the lens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62. 6.6. Schemes of mode selection in the laser with intra-cavity c-cut YVO4 crystal and lens: (a) AP mode at ‘focusing’ (d = (fe )ax ) and (b) RP mode at ‘imaging’ (d = io ) lens positions . . . . . . . . . . . . . 64. 6.7. Dependence of width of the resonator stability region, (I-F) on the cavity length, L for intra-cavity lenses with different foci (dotted curves, calculations; rhombs- experimental data). . . . . . . . . . . 65.

(18) List of Tables. 1.1. Spectroscopic laser parameter values for Nd:YAG and Yb:YAG . . .. 5. 1.2. Thermo-mechanical parameters of YAG ceramics and crystals . . .. 6. xviii.

(19) 1 Introduction. Lasers came forth as an invention in search of an application in 1960. In just 50 years, it is now the foundation for the modern communications society. Installed in machine tools, it has taken manufacturing by storm. In the realm of science it delivers important findings in research efforts. Brought to a remarkable state of sophistication through decades of intensive research and development, today everyone uses the incredible capabilities of laser light. Considering what is going on in the world’s applications laboratories gives us the feeling that we are still at the beginning of the revolution, one triggered by the laser, which still keeps laser physicists occupied over the globe. Adding a drop to the ocean, this thesis is also an effort to bring lasers to the forefront for application and technology.. 1.1 Motivation to the research Yb-doped gain media have been investigated for decades, they have recently gained importance and seen as a vital option in the construction of both high-power and ultra-short pulse lasers. Yb-doped gain medium used in developing diode-pumped terawatt facilities has shown the importance of further studies. Very high efficiencies, diffraction-limited beam quality, and output powers of more than 1 kW have been achieved with ytterbium-doped double-clad fiber lasers and amplifiers. Thin-disk lasers, which most often work with Yb:YAG crystals, can 1.

(20) 1.1. MOTIVATION TO THE RESEARCH. 2. also generate well above 1 kW diffraction-limited output with high beam quality, or even higher powers with non-diffraction-limited beam quality. Various Yb-doped gain media have been used in mode-locked lasers for the generation of femtosecond pulses; the by far highest average output powers of 80 W have been obtained with a passively mode-locked thin-disk Yb:YAG laser [1, 2]. For passive mode locking, problems can arise in the form of Q-switching instabilities. This tendency is a consequence of the relatively small laser cross sections of ytterbium-doped media. Therefore, some of the broadband ytterbium-doped gain media are not very suitable for passively mode-locked lasers, particularly at high power levels, but can still be very useful in regenerative amplifiers. Relatively large cross sections are found for tungstate crystals. Some ytterbium-doped crystals have a fairly broad amplification bandwidth, but the emission curve is not very smooth; it exhibits several maxima. In such cases, wide wavelength tunability may still be achieved, but the realization of very short pulses with mode locking is difficult.Ytterbium doping is also often used together with erbium doping. Typically, ytterbium ions absorb the pump radiation and transfer the excitation energy to erbium ions. Even though the erbium ions could directly absorb radiation e.g. at 980 nm, ytterbium codoping can be useful because of the higher ytterbium absorption cross sections and the higher possible ytterbium doping density in typical laser glasses, so that a much shorter pump absorption length and a higher gain can be achieved. On the other hand, in recent years, considerable interest has been shown in the generation of laser beams containing phase singularities. These beams are best described in terms of Laguerre-Gaussian modes which have an azimuthal phase term and possess a well defined orbital angular momentum per photon. The identification of light beams with a well defined orbital angular momentum, which is distinct from the spin component, has led to a number of exciting studies. These include the transfer of the orbital angular momentum to macroscopic objects [3, 4], the mode transformation upon nonlinear frequency doubling [5], and the interaction of these beams with atomic systems [6, 7]. The methods to generate.

(21) 1.2. YTTERBIUM SOLID-STATE LASERS. 3. such beams have been widely studied in detail and these have involved either using phase elements in Nd-doped lasers or extra-cavity diffraction plates. While Yb- has been proving itself as a potential gain medium, there are not many works that have tried to bring them into the scene as a valuable source for generating high-order circular modes. With high-order modes gaining importance for their wide range of applications especially in the optical toolbox for trapping and tweezing; it is important that we study the role Yb-lasers play in generating such modes. Using an intra-cavity lens inside an Yb:YAG ceramic laser cavity, we have succeeded in generating the highest mode orders in Laguerre-Gauss beams experimentally demonstrated so far. We also generate beams in doughnut-like and Bessel-like profiles that exhibit near-diffraction free characteristics of their central lobes for over tens of meters. Radial and azimuthal polarized doughnuts of high polarization purity and high-order modes were also generated from Yb:YAG ceramic laser cavity.. 1.2 Ytterbium solid-state lasers The realization of narrow-band laser diode pump sources brought major advances in the design and performance of solid-state lasers. Broad band lamps in transversepumped systems found an efficient replacement and end-pumped diode-pumped solid state lasers became more prevalent because of the higher degree of mode matching of the pump and laser beams; hence a high conversion efficiency. The development of semi conducting laser materials, brought in the scope of using different laser ions to produce high-energy, high peak power output pulses for various applications. Trivalent ytterbium ion, Yb 3+ was an obvious candidate with the narrow pump band of the Yb ion lying in the 900-980nm spectral region which matched the emission at InGaAs quantum wells. Figure 1.1 shows room-temperature absorption and emission spectrum of Yb 3+ :YAG [8]. Laser action occurs in the strong emission transition at 1030nm between the lowest Stark level of the 2 F5/2 manifold at 10327cm−1 and the Stark level of the.

(22) 1.2. YTTERBIUM SOLID-STATE LASERS. 4 10679 10624. 2. F5/2. 20x10-21. 10327 Absorption Emission. 10000. 941nm. Energy (1/cm). Cross section (cm^2). 15x10-21. 10x10-21. 5x10-21. 1030nm. 785 2. 0 850. 900. 950. 1000. Wavelength (nm). (a). 1050. 612. F7/2. 1100. 565 0. 0. (b). Figure 1.1: (a) Room temperature spectra of 1 at.% Yb 3+ :YAG (b) Energy level diagram of Yb 3+ :YAG. ground 4 F7/2 manifold at 612cm−1 . At room temperature, the latter Stark level contains a thermal population equal to 4.2% of Yb ions in the 2 F7/2 manifold. Thus, near room temperature, the Yb laser has a quasi-three-level character; when manifold partition functions are taken into account, at least 5.5% of all Yb ions must be pumped into the initial Stark level to achieve gain material transparency. Pumping of Yb:YAG lasers usually is performed at 941nm at the peak of the rather strong and relatively broad feature to the blue of the strong but relatively narrow zero-phonon transition at 976nm. When Yb:YAG is pumped at 941nm the terminal Stark level of the pump transition will have some thermal population at room temperature since it lies only few hundred wave numbers above the lowest lying metastable Stark level. To understand in detail the spectroscopic and lasing properties of Yb:YAG we shall compare its spectroscopic parameters along side with the widely popular Nd:YAG gain medium. Table 1.1 gives a summary of such a comparison. Yb:YAG is characterized by a pump-transition cross section which is about an order of magnitude smaller than that of Nd:YAG. Taking into account the approximately fourfold longer energy-storage lifetime of Yb:YAG, one finds that.

(23) 1.2. YTTERBIUM SOLID-STATE LASERS. 5. Table 1.1: Spectroscopic laser parameter values for Nd:YAG and Yb:YAG Parameter (units) Nd:YAG Yb:YAG Lifetime, μs 230 950 2.8×10−19 at 1064 nm 2.1×10−20 at 1030 nm Emission cross section, cm2 0 at 1064 nm 0.1×10−20 at 1030 nm Absorption cross section, cm2 Emission bandwidth, nm 0.6 6.3-8.5 Quantum defect 24% 9% Quantum efficiency 0.6 1 Opt-to-opt efficiency 45% 90% Transparency intensity, kWcm−2 0 1.4. the pump saturation intensity for Yb:YAG is about 2.3 times larger than that of Nd:YAG. Their laser transition cross sections differ by a factor of 14 and the relatively low transition cross section of Yb:YAG results in a rather large saturation fluence twice smaller than the optical damage fluence of YAG for 10 ns pulses. Also its quantum defect is smaller than that of Nd:YAG which results in lower amount of waste heat deposited in the gain medium for each excited ion produced by pumping. Moreover, Yb- doped laser materials when used as gain medium of a disk laser absorb pump radiation form a soft amplification/loss stop (or gain aperture) in connection with the pumped part. The soft aperture arises from the lower laser level absorption in the unpumped regions and the gain in the pumped regions of the gain element. A soft aperture allows gradual attenuation of a light beam dependent upon its location away from the center of a diffractive optical element. Such a soft aperture is particularly useful in aiding circularizing of an elliptical light beam; which is an added advantage with Yd- doped materials when it comes to generation of circular high-order modes. Unlike gain elements like Nd-doped YAG, the cavity mode stabilization is not governed by thermal guiding in quasi-three level systems; especially in cases of a flat-flat resonator and when the laser is operating nearthreshold. This is because of the small quantum defect and hence a low heat dissipation. ‘Aperture guiding’ becomes the dominant mechanism in determining the cavity mode radius and stabilizing the cavity modes. This strongly reduces diffraction losses that are encountered in hard apertures unlike in Nd-doped gain.

(24) 1.2. YTTERBIUM SOLID-STATE LASERS. 6. Table 1.2: Thermo-mechanical parameters of YAG ceramics and crystals Parameters(units) YAG ceramics YAG crystal Melting temperature, celsius 1960 1960 Thermal conductivity, W/mK 14 14 Young’s Modulus, TPascal – 0.31 Thermal shock parameter W/m 4000 790. media. 1.2.1. Ceramic lasers. While crystalline solids have had a long history of 50 years, ceramics have earned their recognition only recently eventhough the first demonstration of lasing with transparent ceramics dates back to 1964 [9]. Ceramic lasers were not a point of interest till 1995 when the first Nd 3+ doped Yittrium Aluminium Garnet (Y3 Al5 O12 ) was demonstrated by Ikesue et al. [10]. Ceramics is a sintered polycrystalline medium possessing properties of both single crystal and polycrystalline material. Ceramics have several advantages over single crystals. To mention a few; they are associated with low fabrication cost, high optical homogeneity and better mechanical properties as compared to single crystals which make it more useful for laser engineering. While a single crystal grown cannot be ideal, ceramics made of single crystals are a potential alternative to overcome inhomogeneity of dopant and defects. While they posses crystalline thermal properties with glass-like size scalability (they show almost the same spectroscopic properties and comparable thermal conductivities as those of single crystals) they have better mechanical properties and fracture toughness. Figure 1.2 shows photograph of transparent ceramics manufactured by vacuum sintering technique at Konoshima Co. Ltd. . Table 1.2 will give a quantitative proof of the advantages of using ceramics as the gain medium as opposed to single crystals..

(25) 1.2. YTTERBIUM SOLID-STATE LASERS. 7. Figure 1.2: Transparent ceramics manufactured by vacuum sintering technique. 1.2.2. Yb 3+ doped Yittrium Aluminium Garnet ceramics. Yb:YAG has been discussed earlier as a promising candidate for high-power laserdiode-pumped solid-state lasers. However, there are defects such as cores, lightscattering particles, and concentration gradient along the growth axis in Yb:YAG crystals grown from the melt by Czochralski and other methods. Nanocrystalline ceramic technology will be a more promising method to obtain large-size, highoptical-quality Yb:YAG ceramic laser materials. Laser performance of Yb:YAG ceramics and single crystals doped with different Yb concentrations investigated using two-pass pumping miniature laser configuration [14] showed highly efficient laser performance in ceramics. For the low doping concentration, the laser performance of ceramics is lower than those of their single crystal counterpart. However, better laser performance was observed for heavy-doped Yb:YAG ceramics (20 at.%) than single-crystal..

(26) 1.3. OVERVIEW OF THE THESIS. 8. 1.3 Overview of the thesis The thesis is composed in seven chapters. Since the current work boasts an economical and convenient option for the generation of high-order circular modes, it is important to provide an overview on their significance and uniques properties that the Gaussian beams do not posses. Chapter 2 discusses the properties of axisymmetric circular modes; scalar Laguerre-Gauss modes, diffraction-free modes and cylindrical vector beams in particular. A number of intra-cavity (active) and extra-cavity (passive) methods have been proposed since decades towards realizing these modes and they are briefly discussed including their merits and demerits. In designing a resonator, it is important to know it’s stability conditions and limits. In the present work, this is derived using the ABCD matrix. The meaning and significance of the matrix elements are discussed in chapter 3, with the case of a lens resonator discussed in detail, which forms a foundation to the laser scheme in this work. The information on stability limits and mode radii at the cavity mirrors predict the cavity modes and their propagation characteristics even while diffraction effects are not considered by this analysis. Following the discussions on the necessary theory, we work towards a cavity configuration that generates hollow scalar Laguerre-Gaussian modes of all azimuthal orders (mode order limited only by the diameter of the cavity optics) in chapter 4. The cavity element of importance that offers mode selection is a simple glass lens with profound spherical aberrations. While spherical aberrations have always been viewed as a demerit in a laser cavity, the current scheme exploits the same towards generating hollow Laguerre-Gaussian modes. A model for mode selection is proposed based on the ray matrix analysis and the mode selection mechanism is studied. A detailed study on the mechanism of mode selection in the earlier chapter, provided a way to manipulate the stability conditions of the resonator and set the resonator to oscillate in coupled modes. The mechanism helped realize near-.

(27) 1.3. OVERVIEW OF THE THESIS. 9. diffraction free propagation properties for over tens of meters in multi-ring Bessellike beams and doughnut-like modes where the divergence of their central parts was recorded 10 times smaller than the diffraction limit associated with their near-field diameters. Mechanism of coupling in cavity modes is investigated to be the reason behind generation of such modes and is discusses in chapter 5 with supportive experimental evidence. While the circular modes generated so far did have interesting properties, these modes becomes more worthwhile, if they have a spatial polarization dependence associated with them. A birefringent crystal introduced in the lens resonator discriminates either the ordinary or extra-ordinary rays at the resonator stability limits generating radial and azimuthal polarized modes; which can be switched at the output of the resonator by a horizontal shift of the intra-cavity lens. In addition to radial and azimuthal polarized doughnut modes, selection in output having a combination of azimuthal and radial polarization can also be realized from the resonator. High-order radial and azimuthal polarized modes can be generated when the intra-cavity lens has strong spherical aberrations. Detailed discussions and experimental observation are provided in chapter 6. To conclude the thesis, the last chapter summarizes the important points and results of the work. Having come up with a novel laser scheme for generating axi-symmetric circular modes, it is important to discuss on the future prospects which are compiled in the same chapter..

(28) 2 Axi-symmetric Circular Modes. 2.1 Introduction Theoretical and experimental studies on the spatial structure and designing of transverse modes have been the of core theme of numerous studies since the invention of lasers. The different spatial distributions of modes arise from the solutions of the scalar and vector Helmholtz equation within the paraxial approximation (∇2 + k 2 )E = 0. 2.1 . where k = 2π/λ is the wavenumber. Applying the slowly varying envelope approximation to the paraxial solution of the above equation in cartesian co-ordinates, E(x, y, z) = u(x, y, z) exp[ı(kz − ωt). 2.2 . The solutions define a family of modes; the Hermite-Gaussian modes which have rectangular symmetry along the propagation axis. The solutions bear a mathe-. 10.

(29) 2.1. INTRODUCTION. 11. matical representation as below . √ y w0 x u(x, y, z) = E0 Hm 2 2 Hn w(z) w(z) w(z) k 2 r exp (−ıϕmn (z)) exp ı 2q(z) √. 2.3 . where Hm (x) denotes the Hermite polynomials, E0 is a constant electric field amplitude, w(z) is the beam size, w0 is the size of the beam at the beam waist, z0 = πw02 /λ is the Rayleigh range, q(z) = (z −ız0 ) is the complex beam parameter, and ϕmn (z) = (m + n + 1) tan−1 (z/z0 ) is the Guoy phase shift. Applying the same approximation to the paraxial solution in cylindrical coordinates, the solutions yield the Laguerre-Gaussian family of modes with cylindrical symmetry along the propagation axis given by √ r l r 2 w 0 2 Llp 2 2 u(r, φ, z) = E0 ω ω w(z) k 2 exp(−ıϕpl (z)) exp ı r exp(ılϕ) 2q(z). 2.4 . where Llp (x) is the associated Laguerre polynomial and ϕpl (z) = (2p+l+1) tan−1 (z/z0 ) is the Guoy phase shift. Apart from the above solutions, a class of diffraction-free solutions to the Helmholtz equation also exist. These non-spreading beam solutions are given by E(r, φ, t) = E0 exp(ıkz z)Jn (kn r) exp(±ınφ). 2.5 . where Jn is the nth-order Bessel function, kz and kr are longitudinal and radial wave vectors, with k = kz2 + kr2 = 2π/λ. While the scalar Laguerre-Gauss beams are solutions to the scalar Helmholtz equation that define an electric field which is uniform polarization over the whole beam, the corresponding solutions from the vector Helmholtz equation is defined.

(30) 2.2. LAGUERRE-GAUSSIAN SCALAR MODES. by ∇ × ∇ × E − k2 E = 0. 12. 2.6 . impose a cylindrical symmetry to the electric field polarization. The vector solutions in cylindrical co-ordinates are grouped as cylindrical vector beams. Their axially symmetric polarization which is responsible for high numerical aperture focusing properties are of great importance in optical manipulation and other fields of research. The rest of this chapter will consider in detail the properties, methods of generation adopted to generate Bessel, scalar Laguerre-Gauss and cylindrical vector beams.. 2.2 Laguerre-Gaussian scalar modes 2.2.1. Overview. Laser beams propagating in Laguerre-Gaussian (LG) modes are of considerable interest due to their characteristic phase singularity, orbital angular momentum, and multiply connected topology. Laguerre-Gaussian beams (LGpl ) described by equation 2.4 are a set of propagation modes where the equation for the radial electric field is proportional to the product of a Gaussian and an associated Laguerre polynomial LGpl . They have proved their importance in the field of optical trapping. Optical tweezers with more sophisticated features can be realized with higher-order LG modes. Also known as ‘optical vortices’, they can transfer their angular momentum to trapped particles. 2.2.2. Properties of LG modes. LG modes possess rotational symmetry along propagation axis. They carry an intrinsic angular momentum of l¯h per photon [39, 40] which makes the trapped particle along the propagation axis experience a torque. When l is greater than zero, the electric field has an azimuthal phase change of 2πl, which results in a.

(31) 2.2. LAGUERRE-GAUSSIAN SCALAR MODES. 13. phase singularity in the field and a node in the intensity at the center of the beam. 2.2.3. Generation of Scalar Laguerre-Gaussian modes. Since the interest in spatial structure of laser modes developed as early as the 1960’s, several active and passive methods have been proposed for efficient generation of LG modes. Most mode selection methods involve discrimination and selection of a specific “natural” Laguerre-Gaussian mode by introducing relatively high losses to all modes except the specific desired mode. While doing so, the desired mode will maintain the same transverse field distribution at all locations within the resonator. One of the earliest methods was presented by Rigrod [43], HR Concave mirror. Phase element Aperture Gain medium z Output coupler. Figure 2.1: A basic configuration for high-order mode selection in a laser resonator with an intra-cavity phase element. who isolated Laguerre-Gaussian modes in a laser resonator, by inserting wires into the resonator and adjusting the intra-cavity aperture diameter. Laser resonators with diffractive mirrors [44] that operate with single high-order modes were also used for mode selection followed by pseudorandom phase plates [45]. Intracavity phase plates were introduced and showed much efficient selection [46]. A typical basic experimental scheme for generating higher order modes using phase elements is shown in figure 2.1. This triggered the use of phase plates as one of the resonator mirrors [47]. The back mirror of the laser resonator was replaced with a diffractive mirror in order to obtain an intensity distribution which is the Fourier transformation of the desired distribution. In order to introduce the appropriate phase to the emerging output beam (which has a uniform phase), another external.

(32) 2.3. DIFFRACTION-FREE BEAMS. 14. phase element was placed adjacent to the output coupler. After this successful method, several works using various diffractive elements, spatial light modulators and holographic plates were used both inside and outside the cavity to generate modes of the desired intensity distribution [48, 49]. Modulating the spatial pump profile was also exploited as a method to generate higher order modes [50, 51].. 2.3 Diffraction-free beams 2.3.1. Overview. First noted by Durnin [19] as diffraction-free solutions to the Helmholtz equation, diffraction-free beams have been investigated as an attractive alternative to Gaussian beams because of their immunity to diffraction. Mathematically defined by the Bessel function, their spatial structure is an infinite set of rings around the diffraction-free central maximum for the zeroth order mode. Ideally such beams spreading over infinite distance can carry infinite energy if they are diffraction-free. An approximation to such a Bessel beam(quasi-Bessel beam) was demonstrated by Durnin et al. immediately after defining them mathematically [20] considering Bessel beams as the optical Fourier transform of a ring. Herman et al. [21] used spherical aberrations and created quasi-bessel beams that possessed the properties of the mathematical entity over a finite distance and detailed on the properties and uses of them. Ever since this demonstration, several researchers have investigated in detail the properties and methods of generation. 2.3.2. Properties of Bessel beams. An ideal Bessel beam given by (equation 2.5) describes an intensity distribution with a bright core for the zeroth order while it is a dark core for higher-order because of the phase singularity arising from the higher order Bessel functions. However, the most alluring property described by this solution of the Helmholtz equation is that the intensity for propagation in the z-axis obeys the following.

(33) 2.3. DIFFRACTION-FREE BEAMS. 15. inequality. 2.7 . I(x, y, z ≥ 0) = I(x, y). which means that the beams are non-diverging and propagation invariant. A. (b). (a). Figure 2.2: Intensity profile of a typical (a) zeroth-order (b) first order Bessel beam Courtesy: Z.Bouchal laboratory, Palacky University, Czech Republic.. typical intensity profile of zeroth-order and first order Bessel beam are shown in the figure 2.2. Bessel beams can be considered as a set of plane waves propagating on a cone whose opening angle is defined by θ = tan−1. kr kz. 2.8 . which in fact defines the size of the central core as r0 =. 2.405 kr. 2.9 . It has been observed from experimental demonstrations [22] that the total energy of the Bessel beam is equally distributed between its rings and larger the number of rings, the larger is the diffraction-free propagation distance. It was also shown that the strong field at the core of the Bessel beams comes at the cost of the power in the central core. Lin et al. [23] showed that there exists a π phase shift between adjacent rings of the Bessel beams arising from the fact that they are made up of.

(34) 2.3. DIFFRACTION-FREE BEAMS. 16. plane waves propagating on a cone. Another most interesting property of Bessel beams that arises from the same fact and has attracted the attention of many is self-healing [24,25].The nondiffracting beam exhibits resistance against amplitude and phase distortions. If the transverse intensity profile of the nondiffracting beam disturbed by an opaque obstacle, it regenerates during free-propagation behind the obstacle. After certain distance, the initial transverse intensity profile is restored. The distance after which the beam is able to restore itself is given by zmin ≈. ak 2kz. 2.10 . The theoretical explanation of the effect based on the application of Babinet’s principle has been proposed and its experimental verification has been realized. The results of the experiment are presented in figure 2.3 [25]. The beam with transverse intensity profile approximately corresponding to the zero-order Bessel function J0 and pseudo non-diffracting is disturbed by the nontransparent rectangular obstacle figure 2.3a. During free-propagation of the beam behind the obstacle its transverse intensity profile regenerates figure 2.3(b-d). As is obvious from figure 2.3d, the initial Bessel-like profile is restored with a very good fidelity beyond the obstacle after a distance given by equation 2.10. 2.3.3. Generation of Bessel beams. Several methods to create Bessel beams have been proposed starting with the work of Durnin et al. [20] who were the first to observe Bessel beams in reality considering the Bessel beam as the Fourier transform of a ring. An annular slit placed at the back focal plane of a lens as shown in figure 2.4 [20] created a Bessel beam whose propagation distance is given by Zmax = (R/ tan θ) where R is the radius of the imaging lens and θ is the angle of the generated cone. Though this method offered Bessel beam with the smallest spot size and longest depth in focus as compared to the Gaussian or Airy beam created using the same lens, it.

(35) 2.3. DIFFRACTION-FREE BEAMS. 17. (a). (b). (c). (d). Figure 2.3: Zeroth-order quasi-Bessel beam (a)obstructed by opaque obstacle (b,c)restoring beyond obstacle (d) restored fully beyond an obstacle. Courtesy: Am. J. Phys. 67, 912 (1999). was rather inefficient as most of the incident power is obstructed by the annular slit [26].. Bessel beams may also be created using axicons [27, 28] or conical. lens element. Figure 2.5 clearly illustrates plane waves propagating on a cone. The propagation distance of the nondiffracting beams thus created depends on the angle of the conical lens and demonstrated to less than a meter. The use of axicon is much efficient than the annular slit method. Several other authors have investigated axicons in detail [29–31] to propose more efficient generation of Bessel beams. Other methods of creating Bessel beams include holographic techniques which change the phase of the incoming Gaussian beam appropriately to create nondiffracting beam. The holograms could be static, etched types [32] or variable designed by computer controlled devices like spatial light modulators [33,34]. Sev-.

(36) 2.3. DIFFRACTION-FREE BEAMS. 18. Δd. d. Zmax R. f. Figure 2.4: Experimental arrangement for the creation of a J0 beam using an annular slit of mean diameter d and width δd and a positive lens of focal length f and radius R. Zmax indicates the propagation distance of the Bessel beam created. Courtesy: Phy. Rev. Lett., 58, 15, 1499 (1987). wave. Axicon. Plane. Plane wave Figure 2.5: Plane waves propagating on a cone beyond an axicon. eral experimental demonstrations show the possibility of generating Bessel beams from cavities containing axicon elements. [35–38].

(37) 2.4. CYLINDRICAL VECTOR MODES. 19. 2.4 Cylindrical vector modes 2.4.1. Overview. Apart from the spatial structure of laser modes, the spatial distribution of polarization states which lead to new effects and phenomenon have been the purpose of many ongoing researches as they increase the functionality and capability of optical systems in applications. Mathematically represented as axi-symmetric solutions to the vector Helmholtz equation in cylindrical co-ordinates, they are called ‘Cylindrical Vector Beams’ (CVB) . These CVB are associated with their unique and interesting properties that normal gaussian beams do not possess. 2.4.2. Properties of Cylindrical Vector Beams. Extensive interest in CVB are due to their unique focusing properties. When there is a radial spatial distribution in polarization (figure 2.6), because of the strong localized longitudinal component created at the focus, it offers a tighter spot as compared to beams with homogeneous polarization [52, 53] The longitudinal. (a). (b). Figure 2.6: LG01 mode showing (a)radial and (b)azimuthal polarization distribution. component also experiences an apodization effect different from the transverse component and is spatially separated from the transverse focal field which is an.

(38) 2.4. CYLINDRICAL VECTOR MODES. 20. added advantage giving possibilities of 3D tailoring of the focus shape [54]. 2.4.3. Generation of Cylindrical Vector Beams. Since the last three decades, numerous methods for generating CVB actively or passively have been reported. Active methods involve an amplifying media and intra-cavity elements to discriminate the fundamental mode. Passive methods convert beams with homogeneous polarization into CVB with desired polarization distribution outside the cavity. To summarize on the various methods of generating CVB , we broadly classify them as intra-cavity and extra-cavity methods and discuss briefly. Intra-cavity Methods. Polarization selection by intra-cavity methods typically involved polarization selective elements inside the laser resonator. One of the earliest intra-cavity methods for generating CVB was proposed by Mushiake et al. in 1972 [55] using a conical mode selector (a conical glass substrate coated with a dielectric thin film on the conical surface) as a part of the laser resonator. The transmission of the dielectric thin film for radially polarized component was larger than for the azimuthally polarized component generating an output that is predominantly a radially polarized mode. More refined schemes of the conical mode selector involved conical prisms which took advantage of the Brewster angle effect to generate CVB. A more popular method that has been refined with time is the use of corrugated mirrors or grating mirrors inside the resonator. The image of one such contemporary commercial grating mirror is shown in figure 2.7 [58]. Varying the depth, period and refractive index of the gratings/mirrors, reflection of either the s- or p-polarized mode is high. Thus the polarization state of interest will be amplified accordingly. Several works have been published in order to obtain polarization selective resonators oscillating at high powers and efficiency [56, 58]. Another intra-cavity scheme proposed by Pohl [59] which has taken a comeback recently is.

(39) 2.4. CYLINDRICAL VECTOR MODES. (a). 21. (b). Figure 2.7: Intra-cavity polarization selective photonic crystal grating mirror used for mode selection. Coutesy: Opt. Lett., 16, 10841 (2008).. using birefringent crystals for mode selection. The stability conditions set for the extra-ordinary and ordinary rays by the birefringent c-cut calcite helped discriminate the radial polarized T EM01 at the output of the laser cavity. Extra-cavity Methods. The earliest extra-cavity methods proposed for mode conversion involved coherent combining of T EM01 and T EM10 with mutually orthogonal polarizations (since the doughnut beam is a linear combination of T EM01 and T EM10 beams and is ∗ beam) in interferometers. The phases of the beams often identified as the T EM01. are controlled by the interferometer and their combination generated CVB [60,61]. Apart from the interferometric methods, gratings similar to those used in intracavity generation methods are widely used. The typical principle most of them work is based on the principle of mode filtering or polarization selection, where sub-wavelength gratings employed in extra-cavity schemes exhibit birefringence and act as uniaxial birefringent materials [62]. Light polarized in the direction parallel and perpendicular to the gratings experiences different refractive indices. As a result, the gratings are capable of changing the polarization of an incident.

(40) 2.4. CYLINDRICAL VECTOR MODES. 22. light and generation of CVBs becomes possible with proper design of the orientation of the gratings. Birefringent crystals used as polarization converters when designed as a combination of them with different axis orientations. Figure 2.8 shows one such 8-vector spatially varying retarder involving a combination of half-wave plates, each with a different slow axis orientation [63]. Another popular. Figure 2.8: The pie-shaped segments of a half-wave plate were assembled with their slow axes oriented as shown by the arrows (left). The fabricated disk was antireflection-coated at 1.06 m (right). Courtesy: Opt. Lett. 32, 11, 1468 (2007).. mode conversion method involves liquid crystals; where the applied electric field affects the orientation of the liquid crystals and hence allowing mode conversion accordingly..

(41) 3 Resonator theory. 3.1 Introduction In designing an optical resonator and understanding its stability conditions and the modes that it can support, analysis based on geometrical optics plays a vital role. The technique has been discussed in detail in numerous texts [15–17]. Though diffraction effects are not taken care of in such an analysis, it is still seen as a source of valuable information to fully understand a laser resonator. In this chapter, we will discuss briefly the Ray Transfer Matrix analysis and its role in understanding the stability of a laser resonator. We will use the understanding we develop in the first two sections of the chapter to discuss in detail the behavior and stability of a plane-plane resonator with intra-cavity thin lens.. 3.2 Ray Transfer Matrix The Ray Transfer Matrix (RTM) or ABCD matrix technique has been exploited for optical designing and is of high significance when it comes to lasers because they act as tools to analyse the stability conditions of the resonator. Most optical resonator designing softwares have the ABCD matrix analysis as one of the key elements. The technique involves the construction of a matrix for the optical system whose elements give information about the propagation of light through 23.

(42) 3.2. RAY TRANSFER MATRIX. 24. the system. In order to get a better understanding of RTMs, we will now discuss the meaning of the matrix elements as far as ray transformation is concerned. This approach is very helpful for the initial layout of an optical system since each matrix element represents a characteristic property of the beam transformation. The most efficient way to visualize the meaning of the matrix elements is to set them to zero and analyze the propagation changes generated. If the ray starts in a plane perpendicular to the z-axis, it is completely determined by its starting point x and the inclination angle α in x-direction. If (x1 , α1 ) indicates the starting point of the ray, (x2 , α2 ) the end point then, they are related by the RTM ⎛ M =⎝. ⎞ A B. ⎠. C D. then, x2 = Ax1 + Bα1 α2 = Cx1 + Dα1. 3.1 3.2 . (a) When A=0 The relationship between input and output ray parameters then reads: x2 = Bα1 α2 = Cx1 + Dα1 The position x2 of the ray does not depend on its initial position x1 . All rays going through the system under an angle α1 end up at the same coordinate x2 . This means that a parallel beam will be focused (Figure 3.1a). (b) When B=0.

(43) 3.2. RAY TRANSFER MATRIX. 25. (x2,α2). (x1,α1) A=0. (a) Focusing. B=0. (b) Imaging. C=0. D=0. (c) Magnifying. (d) Collimating. Figure 3.1: Transformation of rays by optical systems having one vanishing element in their ray transfer matrix. The ray transfer matrix describes the ray propagation from left to right between the planes indicated by vertical lines.

(44) 3.3. RESONATOR STABILITY. 26. The output ray parameters in the case conform as: x2 = Ax1 α2 = Cx1 + Dα1 All rays starting at point x1 , under arbitrary angles will be reunified in point x2 . Thus setting B equal to zero creates an imaging system (Figure 3.1b), with the lateral magnification A. (c) When C=0 At the output plane the ray parameters read: x2 = Ax1 + Bα1 α2 = Dα1 Parallel rays will still be parallel after passage through the optical system, but their inclination angle has changed by a factor of D. If we look through this optical system at a distant object, the size of the object is magnified by D (Figure 3.1c). (d) D=0 At the output plane the ray parameters read: x2 = Ax1 + Bα1 α2 = Cx1 We see that optical systems having D=0 will generate a collimated beam from a divergent ray pattern (Figure 3.1d).. 3.3 Resonator stability Consider an optical resonator formed by mirrors 1 and 2 with radius of curvatures R1 and R2 separated by a distance ‘L’ (Figure 3.2. The RTM for the round trip.

(45) 3.3. RESONATOR STABILITY. 27. 1. 2 R2. Beam contour w1 Waist(w0). w2 R1 L. Figure 3.2: Mode parameters of interest for an arbitrary resonator. inside an arbitrary optical resonator starting at mirror 1 is given by ⎡ Mres = ⎣. ⎤. 2g1 g2 − 1. 2Lg2. [(2g1 g2 − 1)2 − 1]/2Lg2 2g1 g2 − 1. ⎦. 3.3 . where gi = 1 − (l/R1 ); i = 1, 2. The parameters g1 , and g2 , are called the gparameters of the optical resonator. According to 3.3, the imaging properties of two-mirror resonators are fully defined by the g-parameters and the mirror spacing L. We can further simplify the RTM (Mres ) introducing the equivalent G-Parameter [G = 2g1 g2 − l]: ⎤. ⎡ G. Mres = ⎣ (G2 − 1)/2Lg2. 2Lg2 G. ⎦. 3.4 . To show graphically which type of resonator is stable and which is unstable, it is useful to plot a stability diagram (g2 vs g1 ) on which each particular resonator geometry is represented by a point. The stability diagram for the passive laser resonator along is shown in figure 3.3. The stability criterion in terms of g-parameters can be expressed as: 0 < g1 g2 < 1. 3.5 . All cavity configurations are unstable unless they correspond to points located in the area enclosed by a branch of the hyperbola g1 g2 = 1 and the coordinate axes. Geometrical optics analysis also provides information about the mode behavior at.

(46) 3.4. LENS RESONATOR 1-L/R2. 28. g1g2 = 1 High loss. High loss Hemispherical (0,1). Confocal (0,0). (1,1) Plane-parallel. STABLE Hemispherical (1,0). STABLE. 1-L/R1. Concentric (-1,-1). High loss. High loss g1g2 = 1. ht! Figure 3.3: Stability diagram of passive laser resonator. the output in terms of the radii of the modes at the resonator mirrors. According to [18], the mode radii at the mirrors 1 and 2 represented as w1 and w2 are related to the g-parameters as: wi2. λL = π. . gj gi (1 − g1 g2 ). 3.6 . where i, j = 1, 2 ; i =j. 3.4 Lens Resonator 3.4.1. Resonator scheme. In this section, we shall discuss the case of a resonator of length L formed by two mirrors M1 and M2 enclosing a thin lens of focal length ‘f’ placed at distance d from M1 and d2 from M2 . Figure 3.4 shows the schematic of such a resonator.. 1.

(47) 3.4. LENS RESONATOR. 29 L f. M1 M2. d1. Figure 3.4: Resonator scheme with intra-cavity thin lens. 3.4.2. Ray Transfer Matrix Analysis. We recall from equation 3.5 that the stability condition of resonator in terms of the ray matrix elements the stability condition is given by: −1 <. A+D 2. <1. 3.7 . where A and D are the round trip RTM elements. In case of a plane-plane resonator, the expressions turn out simpler in terms of the one-way propagation matrix. A = D = 2A1 D1 − 1. 3.8 . Hence the inequality 3.7 becomes, 0 < A1 D1 < 1. 3.9 . For the resonator with intra-cavity lens defined above, the one-way RTM is given by, ⎤⎡ ⎤ ⎡ ⎤ ⎤⎡ ⎡ 1 0 1 d1 1 − d2 /f d1 + d2 − d1 d2 /f A B 1 d2 ⎦⎣ ⎦=⎣ ⎦ ⎣ ⎦⎣ ⎦=⎣ 0 1 −1/f 1 − d1 /f −1/f 1 0 1 C D ⎡. ⎤. 3.10 .

(48) 3.4. LENS RESONATOR. 30. The stability condition defined in 3.7 becomes, d2 d1 0< 1− 1− <1 f f. 3.11 . which corresponds to the two inequalities as given below, 1 1 − (d1 + d2 ) +1>0 d1 d2 2 f f 1 1 d1 d2 − (d1 + d2 ) <0 2 f f . 3.12a . 3.12b. For d2 > d1 , 3.12a is valid for 1 1 < f d1 1 > d2 and 3.12b is valid for 1 < 0 f 1 1 > + d2 d1 Hence, both inequalities are verified when, 0<. 3.4.3. 1 1 < f d2. ;. 1 1 1 1 < < + d1 f d2 d1. 3.13 . Nature of mode propagation. It is interesting to consider the nature of mode propagation using the above stability condition in cases considered in figure 3.1 and equation 3.6. When 1/f = 0 (C1 = 0), it corresponds to the case of no lens inside the resonator and hence a plane wave inside the resonator. When 1/f = 1/d2 (A1 = 0), then as per the definition of mode radii at the mirrors in section 3.3 w1 → ∞ and w2 → 0. This.

(49) 3.4. LENS RESONATOR. 31. would describe a plane wave focused on mirror M2 . If 1/f = 1/d1 , then the mode radii w1 → 0 and w2 → ∞ define a mode originating from M1 which is collimated by the intra-cavity lens and transformed into a plane wave at M2 . Finally, when 1/f = 1/d1 + 1/d2, w1 → 0 and w2 → 0 which means the lens images a spherical wave originating from M1 onto M2 . It can be seen that although the geometrical optics analysis of a Gaussian beam loses its validity at the stability limits, these can help in predicting the mode radii and hence their behavior at the mirrors as the stability limit is approached..

(50) 4 Scalar Laguerre-Gauss modes generation. 4.1 Introduction Scalar Laguerre-Gaussian (LG) modes have been in the limelight for over two decades. Their properties and methods proposed to generate these modes since then have been discussed in detail in Chapter 2. Methods to generate LG modes have used special optical elements both inside and outside the laser cavity. While cylindrical lens mode transformers, holograms, diffractive optical elements, phase plates and spatial light modulators were used to convert an incoming He-Ne laser beam with Gaussian, Hermite-Gaussian or top-hat profile to LGpl modes, where p is the radial and l is the azimuthal mode order, of the required intensity profile [64–67], they could also be generated directly in a solid-state laser cavity using phase elements [68, 69]. ‘Hollow’ LG0l (l > 0) beams of high azimuthal order were produced in Nd:YAG and Nd:YVO4 lasers by shaping the pump beam intensity profiles [70–73]. Thermo-optical effects in the gain medium were used for powerful LG∗01 mode formation in Nd:GdVO4 bounce oscillator [74]. In the forthcoming sections, we will present a novel cavity scheme and mode selection element that can generate along with many low order modes, the whole lineage of very high Parts of this chapter contain text reproduced from the article, Generation of very-high order Laguerre-Gaussian modes in Yb:YAG ceramic laser, Laser Phys. Lett. 7, No. 9, 637-643 (2010) c Wiley-VCH Verlag GmbH & Co. KGaA.. 32.

(51) 4.2. EXPERIMENTAL SETUP. 33. order pure ‘hollow’ LGpl modes (p = 5 − 10, l = 7 − 28). Though spherical aberrations in laser cavity has always been a disadvantage, we get optimistic and exploit the spherical aberration of an intra-cavity lens to render selection of ‘hollow’ LGpl modes inside an Yb:YAG ceramic laser cavity. The resonator stability conditions and mechanism of mode selection are discussed in detail.. 4.2 Experimental setup 4.2.1. Diode pumping. The laser scheme is end-pumped by a fiber-coupled InGaAs laser diode at 940 nm (NA= 0.22; Core diameter = 100μm). Pump power (1-4W) from the fiber coupled diode is focused to a 100μm spot with the help of coupling optics. The pump profile is bell-shaped and the CCD camera (frame size = 6.4 × 4.8mm2 ) image (recorded with filters because of limitations on the damage threshold of the CCD) shows a 14 times magnification of the spot size as shown in figure 4.1.. P (W/cm^2) 0.1. 0.05. 0 2. -1. 0. 1. y (mm). Figure 4.1: CCD camera image and intensity profile of the focused pump beam. 4.2.2. Gain medium. The laser scheme uses a 9.8 at.% doped Yb:YAG ceramic plate (9 × 11 × 1.5)mm3 as the lasing medium. Optical and spectroscopic properties of Yb:YAG ceramics.

(52) 4.2. EXPERIMENTAL SETUP. 34. used are discussed in detail in Chapter 1. The ceramic plate is sandwiched and conductively cooled between copper plates. The surface facing the pump beam is anti-reflection (AR) coated for pump wavelength 940 nm and high-reflection (HR) coated for lasing wavelength 1030 nm. The other surface was AR coated at 1030 nm. 4.2.3. Laser scheme for generation of scalar Laguerre-Gaussian modes. The schematic of the laser resonator is shown in figure 4.2. The resonator was formed by the HR surface of the ceramic plate (acting as a rear HR mirror) and a plane parallel output coupler (OC) of reflectivity 95 or 98%. The length of the resonator was varied from 20-127 cm. A plano-convex uncoated glass lens of 25 mm diameter and 25 mm focal length was placed initially at a distance dap ≈ fax , where dap is the position of the lens apex relative to HR mirror and fax is the axial focal length determined using a reference He-Ne laser. The lens could be moved along the resonator axis on a mechanical stage in steps of 10 μm. The laser output could be monitored using a power meter and a charge-coupled Output coupler R=95-98%. Yb:YAG LD 940nm. M Coupling optics. Lens M. Lext. CCD Filters. Figure 4.2: Schematic of the experimental setup. device (CCD) camera placed either at 1 m from the OC (to record the near-field pattern) or at the 1 m focus of an extra-cavity lens Lext (to record the far-field pattern). Near field images of higher order modes (with reduction ∼ 0.5× ) were also recorded by placing the camera in front of the focus of lens Lext . Over 80% of pump radiation was absorbed by the Yb:YAG plate. In order to prevent the residual pump radiation beyond the OC from reaching the power meter or CCD.

(53) 4.3. RESULTS. 35. camera, external selective mirrors M and filters were used. Intra-cavity obstacles (metal spheres) of 1-2 mm diameter were placed along the resonator axis in certain cases.. 4.3 Results. (a). p=1, l=2. (c). p=10, l=20. (b). (d). p=6, l=12. p=11, l=2. Figure 4.3: CCD camera images of various LGpl modes (indices specified) in near-field. (a) p = 1, l = 2; (b) p = 6, l = 12; (c) p = 10, l = 20; (d) p = 11, l = 2. The Yb:YAG laser operated at 1030 nm in CW regime. The cylindrical symmetry that could be obtained by precision alignment of the cavity supported generation of circular beams 2-13 mm in diameter and 1-30 mW power. The experiments on mode selection were performed mostly with resonators of length L > 100 cm. For short cavities, L = 20-50 cm ring-like incoherent oscillations were usually observed. For L > 100 cm, when dap ≈ fax , the fundamental LG0,0 mode was observed. On shifting the lens (< 1 mm) towards the HR mirror, low order modes.

(54) 4.3. RESULTS. 36. (LG01 , LG∗01 , LG10 , and others) with core intensity alternating between central maximum and central minimum (Figure 4.3a) were observed. This alternation corresponded to successive appearance of modes with p ≥ 0, l > 0 and modes with p ≥ 0, l = 0. Peaked and hollow multi-ring modes with increasing diameters and increasing number of rings dominated on shifting the lens further towards the HR mirror by 12 mm. It was observed that near the threshold of generation, when an intra-cavity obstacle (metal spheres) of diameter 1-2mm as introduced along the resonator axis, multi-ring patterns with a central hole broke down to discrete multi-lobe spatial structures corresponding to profiles of pure hollow very high order LGp,l modes (l = 7-28, p = 5-10), like figure 4.3(b-d). With the intra-cavity obstacle, it was possible to realize selection in modes in increasing azimuthal orders from l = 1 to 28 continuously as the lens is shifted towards the HR surface. Figure 4.4a (a) Azimuthal mode order. Azimuthal mode order. (b). Displacement of lens from reference focus (mm). Length of cavity (cm). Figure 4.4: Dependence of the azimuthal mode order, l of hollow LGpl modes on (a) lens apex position (threshold LD currents are indicated), L = 114 cm; (b) length of the cavity. illustrates the experimental dependence of the azimuthal mode index l, on the lens shift in cavity of L = 114 cm. Figure 4.4a also indicates that a consistent increase in the pump power/LD current was required for excitation of consecutive higher order modes while shifting the lens. Increasing pump power above the threshold of a particular mode, a discrete structure of the LGpl mode transformed back to a hollow multi-ring structure corresponding to the LG∗pl mode. An LG∗pl.

(55) 4.3. RESULTS. 37. mode is in fact the superposition of two alike LGpl modes, one rotated π/2 about the optical axis relative to the other. Due to the limited size of the CCD camera frame, peripheral parts of some of modes in the near field are outside of this frame, figure 4.3(b,c) and exact p values cannot be determined in such cases from mere experimental observation. Radial indices of modes were determined usually using far field images. Figure 4.5 shows the near and far-field images of pure LG10,25 mode (a), (b) and the far field image of the LG7,14 mode with the corresponding intensity profile (c),(d). The angles of divergence for high order modes estimated from far-field data were (2-5)×10−3 rad. To the best of our knowledge, we report the generation of such pure very high order hollow multi-lobe Laguerre-Gaussian modes in lasers for the first time. It was also observed that for a fixed position. (b). (c). (d) Power density (W/cm^2). (a). Horizontal Line profile 0.2 0.1 0 -4. 4 -2 2 0 X-position (mm). Figure 4.5: CCD camera images of (a) near and (b) far field patterns of LG10,25 mode and (c) far field of LG7,14 mode (d) with the corresponding intensity profile. of lens, a transition from one high order mode to the other could be achieved by changing the cavity length, figure 4.4b. For the fixed L and lens position, it was also possible to select LGpl modes using an intra-cavity obstacle or iris diaphragm,.

(56) 4.4. DISCUSSION. 38. or a combination of coaxial diaphragm and obstacle placed near OC. By this way, it was found, for example, that the spatial region comprising the LG8,17 mode near the OC corresponded to a caustic of external diameter ≈ 5.5 mm and wall thickness 1.5-2 mm. Use of an obstacle helped to increase the range of pump above the threshold where pure hollow LGp,l modes of low and high orders appeared. So, the whole lineage of hollow LGpl modes with l = 1-28 and p = 1-10 was generated in our Yb:YAG laser scheme. For L = 80-100 cm, it was possible to obtain other LGpl modes with p ≥ l, like LG11,2 , figure 4.3d. Apparently, it is also the first observation of this type of very high order LGpl modes. Beams from Yb:YAG laser output were usually found unpolarized. Linear polarization (with a contrast ratio about 3:1) was observed for some pure LGpl modes.. 4.4 Discussion 4.4.1. Resonator with intra-cavity aberrated lens. dap < fax Type-2 feedback γ”. γ’. y”. Output coupler. HR surface. y. y’. z. d” = f” d’ > f’. ack. edb. fe pe-1. Ty. Figure 4.6: Ray tracings in a plane-parallel resonator containing a plano-convex lens with spherical aberration. The ray tracings in the plane-parallel cavity (without a gain medium) containing a plano-convex lens exhibiting spherical aberration is shown in figure 4.6. The distance between HR mirror and the lens apex, dap is shown shorter than the.

(57) 4.4. DISCUSSION. 39. length of the paraxial focus, i.e. dap < fax . The paraxial focus, fax is determined using a reference He-Ne laser beam propagating in the −z direction along the laser cavity. Figure 4.6 shows two types of ray trajectories, which provide feedback in the cavity as per the stability conditions discussed earlier in section 3.3. Type-1 trajectory corresponds to a conical beam at the OC and type-2 to a parallel beam. The peripheral refracting zones corresponding to y-coordinates y and y may be considered as separate thin annular lenses of foci f and f (fax > f > f ). One of these zones at a distance d > f from the HR surface, images it on the plane of OC, and forms a conical beam. The other zone has a focus f at the HR surface (d = f ) and forms a parallel beam. For a given position of the lens, the angle of incidence for type-1 trajectory on the HR, γ is always larger than γ for the type-2 trajectory. Shifting the lens towards the HR, the angles γ and γ increase and both trajectories get shifted towards the lens periphery. 4.4.2. Resonator stability and mode selection mechanism. A thick short focus lens has strong spherical aberrations. Figure 4.7 shows the refraction of parallel rays at the spherical surface of such a lens. It can be seen that the focal length of such a lens is not fixed and varies with along the surface depending on the y-coordinate. Hence, an aberrated lens can be considered as a sequence of ring-like thin lenses. A resonator with an ideal (aberration-free) thin lens is considered in section 3.4. Since an aberrated lens is a collection of thin lenses, the ray transfer matrix analysis from section 3.4 may be applied for every resonator configuration with a particular thin annular lens. Such an analysis should permit finding the region of resonator stability and mode radii at the mirrors for every configuration with a thin lens. In order to simplify the analysis of the current laser scheme, we will initially eliminate the thickness of the Yb-doped plate and ignore the influence of thermal lens in the active medium on beam tracings. With these assumptions, we can apply results of analysis of a thin lens resonator (3.4) to our scheme. The stability factor for stable cavity given by.