Frontiers in Guided Wave Optics and Optoelectronics Part 15 doc

Polarization Properties of Laser-Diode-Pumped Microchip Nd:YAG Ceramic Lasers 545 be 1.1 μm. An SEM surface image of the micro-grained sample used in this experiment is shown in Fig. 17(b), together with that of large-grain sample. The collimated linearly- polarized LD beam was passed through an anamorphic prism pair and it was focused onto the sample by a microscope objective lens of NA = 0.25, where the focused beam diameter was about 80 μm. The laser exhibited a single-frequency TEM 00 -mode oscillation, which is linearly polarized along the LD pump-beam polarization direction due to the reduced thermal birefringence for mode-matched on-axis pumping condition as mentioned in section 3. By shifting or tilting the laser cavity slightly as depicted by arrows in Fig. 17(a), a variety of MG mode operations were observed, instead of Ince-Gauss (IG) modes, depending on the degree of effective off-axis pumping. Typical far-field patterns, including BG modes, are shown in Fig. 18. For the higher-order BG modes (BG 1 , BG 2 ), an optical vortex having a topological charge of 1 and 2 was formed in the center. Fig. 18. Observed far-field lasing patterns. (a) Mathieu-Gauss laser beam. (b) Bessel-Gauss laser beam. Numerically reproduced intensity patterns corresponding to Fig. 18 and the phase portraits are shown in Fig. 19. Here, the complex amplitude of the m-th order even and odd MG beams propagating along the positive z of an elliptic coordinate system r = (ξ, η, z) is given by (Gutierrez-Vega & Bandres, 2007): ),(),()() 2 exp()( 2 qceqJerGB k zk rMG mm t m e ηξ μ −= , (9) ),(),()() 2 exp()( 2 qseqJorGB k zk rMG mm t m o ηξ μ −= . (10) Here, Je m (·) and Jo m (·) are the m-th order even and odd radial Mathieu functions, ce m (·) and se m (·) are the m-th order even and odd angular Mathieu functions, GB(r) = μ -1 exp(-r 2 /μw 0 2 ) is Frontiers in Guided Wave Optics and Optoelectronics 546 the fundamental Gaussian beam, μ(z) = 1 + iz/(kw 0 2 ), w 0 is the Gaussian width at the waist plane z = 0, and k = 2π/λ is the longitudinal wave number. q = k t 2 f 0 2 /4 is the ellipticity parameter, which carries information about the transverse wave number k t and the semiconfocal separation at the waist plane f 0 . Similarly, m-th order BG beams are given by )exp()()() 2 exp(),( 2 φ μμ φ im rk JrGB k zk rBG t m t m −−= . (11) Here, (r,φ) are the polar coordinates and J m (·) is the m-th order Bessel function of the first kind. Fig. 19. Numerically reproduced intensity patterns corresponding to Fig. 18 and their phase portraits. (a) Mathieu-Gauss laser beam. (b) Bessel-Gauss laser beam. λ (wavelength) = 1064 nm, w 0 = 3 mm. Adopted parameter values (k t , q) are (a)-(i): (2800/m, 0.2); (a)-(ii): (6000/m, 0.2); (a)-(iii): (4300/m, 0.5); (a)-(iv): (7500/m, 25); (b)-(i): (4500/m, 0); (b)-(ii): (5500/m, 0); (b)- (iii): (6500/m, 0). Elliptical-polarization BG modes or dual-polarization MG modes appeared for small effective off-axis pumping. An example of polarization-dependent oscillation spectra is shown in Fig. 20(a). With larger off-axis pumping, linearly polarized single or double longitudinal MG mode operations were observed, where the longitudinal mode spacing coincided with 12.88 GHz, which corresponds to the inverse of two round-trip times as expected for BG and MG mode oscillations. An example oscillation spectrum consisting of two longitudinal modes is shown in Fig. 20(b). B. Effect of fluorescence anisotropy on lasing pattern formation We replaced the micro-grained Nd:YAG ceramic by LiNdP 4 O 12 (LNP) and a-cut Nd:GdVO 4 crystals, which exhibit linearly polarized emission resulting from strong fluorescence anisotropy independently of the pump-beam polarization state. Under the same azimuth LD-pumping conditions as for micro-grained ceramic lasers, neither BG nor MG mode oscillations appeared. Instead, single-frequency linearly polarized IG mode operations on Polarization Properties of Laser-Diode-Pumped Microchip Nd:YAG Ceramic Lasers 547 Fig. 20. Far-field lasing patterns and their polarization-dependent optical spectra. (a) Dual-polarization Mathieu-Gauss beam with small off-axis pumping. (b) Linear-polarization multi-longitudinal mode Mathieu-Gauss beam with large off-axis pumping. elliptical coordinates were observed depending on the pump-beam position (Ohtomo et al., 2007), similar to large grain Nd:YAG ceramic lasers with spatially dependent thermal birefringence discussed in the previous subsection 4.1. Examples are shown in Fig. 21. As for large-grain Nd:YAG ceramic lasers, neither BG nor MG mode oscillations appeared with azimuth LD pumping. Fig. 21. Ince-Gauss mode operations with azimuth LD pumping. (a) Nd:GdVO 4 single crystal. (b) Large-grain Nd:YAG ceramic with average grain size of 19.2 μm. C. Discussion Laser oscillations in BG and MG modes are usually obtained in cavities with an axicon-type lens or mirror (Gutierrez-Vega, 2003; Alvarez-Elizondo, 2008) such that interference between conical lasing fields occurs within the laser cavity. In the present experiment, BG and MG mode oscillations were produced just by azimuth LD pumping. Let us offer a plausible explanation for MG mode oscillations in terms of effective off-axis pumping depicted in Fig. 22(a). In the framework of vector lasers (Kravtsov, 2004), the angular amplification inhomogeneity has been shown to depend on the orientation of the polarization plane of laser radiation from that of pump radiation, in the form of D( θ , Ψ ) = 2A 0 cos 2 (θ - Ψ ) as depicted in Fig. 22(b), Frontiers in Guided Wave Optics and Optoelectronics 548 and the polarization state is almost completely determined by the polarization of the pump radiation for an isotropic cavity with micro-grained Nd:YAG ceramic as described in section 3 (Ohtomo, 2007; Otsuka 2008). For azimuth LD pumping, the laser emission tends to occur such that its polarization direction follows the LD polarization direction within the pumped area. Let us assume a small reflection loss difference at uncoated surfaces of the thermal lens between polarizations along radial and azimuth directions as depicted in Fig. 22(c). With the two effects combined, the laser polarization state may depend on the pump-beam position and size, i.e., gain area, if the LD polarization direction is fixed. For larger off-axis pumping, MG modes with a linear eigen-polarization are expected as a result of the stronger polarization discrimination effect and beam bending through the thermal lens as shown in Fig. 22(a). For small off-axis pumping, BG modes with orthogonal eigen-polarizations appear presumably because radial polarization components with a smaller reflection loss increase within the gain area. Fig. 22. (a) Conceptual illustration of the optical resonator containing a micro-grained Nd:YAG thermal lens with azimuth LD pumping. (b) Angle-dependent dipole moment induced by a linearly-polarized LD pump light. (c) Polarization-dependent reflection loss at un-coated surfaces. In anisotropic lasers or large-grain Nd:YAG ceramic lasers, the laser polarization state is determined by fluorescence anisotropies or local thermal birefringence independently of the pump polarization, and neither BG nor MG mode oscillations take place. 5. Concluding remarks In this Chapter, reviews were given on modal and polarization properties of microchip Nd:YAG ceramic lasers with laser-diode end pumping, featuring such effects as average grain sizes and azimuth pumping. Segregations into multiple local-modes and the associated variety of dynamic instabilities occur in LD-pumped Nd:YAG samples with average grain size over several tens of microns resulting from the field interference effect among local-modes. The following results have been obtained for realizing stable single-frequency, linearly-polarized oscillations in Nd:YAG microchip ceramic lasers: 1. Micro-grained ceramics, whose average grain sizes are below 5 μm, can guarantee stable linearly-polarized TEM 00 mode operations. Polarization Properties of Laser-Diode-Pumped Microchip Nd:YAG Ceramic Lasers 549 2. Large-grain ceramics, whose average grain sizes are larger than several tens of microns, can exhibit stable linearly-polarized oscillations in forced Ince-Gauss modes with azimuth/off-axis pumping. 3. Micro-grain ceramics can produce spontaneous Mathieu-Gauss and Bessel-Gauss lasing modes with azimuth/off-axis pumping. 6. References Alvarez-Elizondo, M. B., Rodrlguez-Masegosa, R. & Gutierrez-Vega, J. C. (2008). Generation of Mathieu-Gauss modes with an axicon-based laser resonator. Opt. Express 16, 23 (2008) 18770-18775, eISSN 1094-4087. Arlt, J., Dholakia, K., Allen, L. & Padgett, M. J. (1998). The production of multiringed Laguerre-Gaussian modes by computer-generated holograms. J. Mod. Opt. 45, 6 (1998) 1231-1237, ISSN 0950-0340. Bandres, M. A. & Gutierrez-Vega, J. C. (2004). Ince–Gaussian modes of the paraxial wave equation and stable resonators. J. Opt. Soc. Am. A 21, 5 (2004) 873-880, ISSN 1084-7529. Bielawski, S., Derozier, D. & Glorieux, P. (1992). Antiphase dynamics and polarization effects in the Nd-doped fiber laser. Phys. Rev. A 46, 5 (1992) 2811-2822, ISSN 1050-2947. Cabrera, E., Calderon, O. G. & Guerra, J. M. (2005). Experimental evidence of antiphase population dynamics in lasers. Phys. Rev. A 72 (2005) 043824, ISSN 1050-2947. Chu, S C. & Otsuka, K. (2007). Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers. Optics Express 15 (2007) 16506-16519, eISSN 1094-4087. Durin, J. (1987). Exact solutions for nondiffracting beams. I. The scalar theory. J. Opt. Soc. Am. A, 4, 4 (1987) 651-654, ISSN 1084-7529. Erneux, T (1990). Laser Bifurcations, Northwestern University Press, Evanston, IL. Gutierrez-Vega, J. C., Rodrlguez-Masegosa, R. & Chaves-Cerda, S. (2003). Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis. J. Opt. Soc. Am. A 20, 11 (2003) 2113-2122, ISSN 1084-7529. Ikesue, A., Furusato, I. & Kamata, K. (1995a). Fabrication of polycrystalline, transparent YAG ceramics by a solid-state reaction method. J. Am. Ceram. Soc. 78, 1 (1995) 225- 228, ISSN 0002-7820. Ikesue, A., Kinoshita, T., Kamata, K. & Yoshida, K. (1995b). Fabrication and optical properties of high-performance polycrystalline Nd:YAG ceramics for solid-state lasers. J. Am. Ceram. Soc. 78, 4 (1995) 1033-1040, ISSN 0002-7820. Kawai, R., Miyasaka, Y., Otsuka, K., Ohtomo, T., Narita, T., Ko, J Y., Shoji, I. & Taira, T. (2004). Oscillation spectra and dynamic effects in a highly-doped microchip Nd:YAG ceramic laser. Opt. Express 12, 10 (2004) 2293-2302, eISSN 1094-4087. Kimura, T. & Otsuka, K. (1971). Thermal effects of a continuously pumped Nd 3+ :YAG laser. IEEE J. Quantum Electron. QE-7, 8 (1971) 403-407, ISSN 00189197. Ko, J Y., Otsuka, K. & Kubota, T. (2001). Quantum-noise-induced order in lasers placed in chaotic oscillation by frequency-shifted feedback. Phys. Rev. Lett. 86, 18 (2001) 4025- 4028, ISSN 0031-9007. Koechner, W. & Rice, D. K. (1970). Effect of birefringence on the performance of linearly polarized YAG:Nd lasers. IEEE J. Quantum Electron. QE-6,9 (1970) 557-566, ISSN 00189197. Kravtsov, N. V., Lariontsev, E. G. & Naumkin, N. I. (2004). Dependence of polarisation of radiation of a linear Nd:YAG laser on the pump radiation polarization. Quantum Electron. 34, 9 (2004) 839-842, ISSN 1063-7818. Frontiers in Guided Wave Optics and Optoelectronics 550 Lu, J., Prabhu, M., Xu, J., Ueda, K., Yagi, H., Yanagitani, T. & Kaminskii, A. (2000). Highly efficient 2% Nd:yttrium aluminum garnet ceramic laser. Appl. Phys. Lett. 77, 23 (2000) 3707-3709, ISSN 0003-6951. Narita, T., Miyasaka, Y. & Otsuka, K. (2005). Self-Induced instabilities in Nd:Y 3 Al 5 O 12 ceramic lasers. Jpn. J. Appl. Phys. 37 (2005) L1168-L1170, ISSN 0021-4922. Ohtomo, T., Kamikariya, K. & Otsuka, K. (2007). Effect of grain size on modal structure and polarization properties of laser-diode-pumped miniature ceramic lasers. Jpn. J. Appl. Phys. 46 (2007) L1043-L1045, ISSN 0021-4922. Ohtomo, T., Kamikariya, K., Otsuka, K. & Chu, S C. (2007). Single-frequency Ince-Gaussian mode operations of laser-diode-pumped microchip solid-state lasers. Opt. Express 15, 17 (2007) 10705-10717, eISSN 1094-4087. Ohtomo, T. & Otsuka, K. (2009). Yb:Y 3 Al 5 O 12 laser for self-mixing laser metrology with enhanced optical sensitivity. Jpn. J. Appl. Phys. 48 (2009) 070212, ISSN 0021-4922. Otsuka, K. (1999). Nonlinear Dynamics in Optical Complex Systems. Kluwer Academic Publishers. Dordrecht/London/Boston (1999), Chapter 2, ISBN 07923-6132-6. Otsuka, K., Kawai, R., Hwong, S L., Ko, J Y. & Chern, J L. (2000). Synchronization of mutually coupled self-mixing modulated lasers. Phys. Rev. Lett. 84, 14 (2000) 3049- 3052, ISSN 0031-9007. Otsuka, K., Ko, J Y., Lim, T S., and Makino, H. (2002). Modal interference and dynamical instability in a solid-state slice laser with asymmetric end-pumping. Phys. Rev. Lett. 87 (2002) 083903, ISSN 0003-6951. Otsuka, K., Narita, T., Miyasaka, Y., Ching, C C., Ko, J Y. & Chu, S C. (2006). Nonlinear dynamics in thin-slice Nd:YAG ceramic lasers: Coupled local-mode model. Appl. Phys. Lett. 89, 8 (2006) 081117, ISSN 0003-6951. Otsuka, K., Nemoto, K., Kamikariya, K., Miyasaka, Y., Ko, J Y. & Lin, C C. (2007). Chaos synchronization among orthogonally polarized emissions in a dual-polarization laser. Phys. Rev. E 76, 2 (2007) 026204, ISSN 1063-651X. Otsuka, K., Nemoto, K., Kamikariya, K., Miyasaka, Y. & Chu, S C. (2007). Linearly polarized single-frequency oscillations of laser-diode-pumped microchip ceramic Nd:YAG lasers with forced Ince–Gaussian mode operations. Jpn. J. Appl. Phys. 46 (2007) 5865- 5867, ISSN 0021-4922. Otsuka, K. & Ohtomo, T. (2008). Polarization properties of laser-diode-pumped micro-grained Nd:YAG ceramic lasers. Laser Phys. Lett. 5, 9 (2008) 659-663, ISSN 1612-2011. Schwarz, U. T., Bandres, M. A. & Gutierrez-Vega, J. C. (2004). Observation of Ince–Gaussian modes in stable resonators. Opt. Lett. 29, 16 (2004) 1870-1872, ISSN 0146-9592. Shoji, I., Kurimura, S., Sato, Y., Taira, T., Ikesue, A. & Yoshida, K. (2000). Optical properties and laser characteristics of highly Nd3 + -doped Y3Al5O12 ceramics. Appl. Phys. Lett. 77, 7 (2000) 939-941, ISSN 0003-6951. Shoji, I., Sato, Y., Kurimura, S., Lupei, V., Taira, T., Ikesue, A. & Yoshida, K. (2002). Thermal- birefringence-induced depolarization in Nd:YAG ceramics. Opt. Lett. 27, 4 (2002) 234-236, ISSN 0146-9592. Sudo, S., Miyasaka, Y., Kamikariya, K., Nemoto, K. & Otsuka, K. (2006). Microanalysis of Brownian particles and real-time nanometer vibrometry with a laser-diode- pumped self-mixing thin-slice solid-state laser. Jpn. J. Appl. Phys. 45 (2006) L926- L928, ISSN 0021-4922. Tokunaga, K., Chu, S C., Hsiao, H Y., Ohtomo, T. & Otsuka, K. (2009). Spontaneous Mathieu-Gauss mode oscillation in micro-grained Nd:YAG ceramic lasers with azimuth laser-diode pumping. Laser Phys. Lett. 6, 9 (2009) 635-638, ISSN 1612-2011. 25 Surface-Emitting Circular Bragg Lasers – A Promising Next-Generation On-Chip Light Source for Optical Communications Xiankai Sun and Amnon Yariv Department of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA 1. Introduction Surface-emitting lasers have been attracting people’s interest over the past two decades because of their salient features such as low-threshold current, single-mode operation, and wafer-scale integration (Iga, 2000). Their low-divergence surface-normal emission also facilitates output coupling and packaging. Although Vertical Cavity Surface Emitting Lasers (VCSELs) have already been commercially available, their single-modedness and good emission pattern are guaranteed only for devices with a small mode area (diameter of ~ μ m). Attempts of further increase in the emission aperture have failed mostly because of the contradictory requirements of large-area emitting aperture and single modedness, which casts a shadow over the usefulness of VCSELs in high-power applications. A highly desirable semiconductor laser will consist of a large aperture (say, diameter larger than 20 μ m) emitting vertically (i.e., perpendicularly to the plane of the laser). It should possess the high efficiency typical of current-pumped, edge-emitting semiconductor lasers and, crucially, be single-moded. Taking a clue from the traditional edge-emitting distributed feedback (DFB) semiconductor laser, we proposed employing transverse circular Bragg confinement mechanism to achieve the goals and those lasers are accordingly referred to as “circular Bragg lasers.” There have been intensive research activities in planar circular grating lasers since early 1990s. Erdogan and Hall were the first to analyze their modal behavior with a coupled- mode theory (Erdogan & Hall, 1990, 1992). Wu et al. were the first to experimentally realize such lasers in semiconductors (Wu et al., 1991; Wu et al., 1992). With a more rigorous theoretical framework, Shams-Zadeh-Amiri et al. analyzed their above-threshold properties and radiation fields (Shams-Zadeh-Amiri et al., 2000, 2003). More recently, organic polymers are also used as the gain medium for these lasers due to their low fabrication cost (Jebali et al., 2004; Turnbull et al., 2005; Chen et al., 2007). The circular gratings in the above-referenced work are designed radially periodic. In 2003 we proposed using Hankel-phased, i.e., radially chirped, gratings to achieve optimal interaction with the optical fields (Scheuer & Yariv, 2003), since the eigenmodes of the wave equation in cylindrical coordinates are Hankel functions. With their grating designed to follow the phases of Hankel functions, these circular Bragg lasers usually take three Frontiers in Guided Wave Optics and Optoelectronics 552 configurations as shown in Fig. 1: (a) circular DFB laser, in which the grating extends from the center to the exterior boundary x b ; (b) disk Bragg laser, in which a center disk is surrounded by a radial Bragg grating extending from x 0 to x b ; (c) ring Bragg laser, in which an annular defect is surrounded by both inner and outer gratings extending respectively from the center to x L and from x R to x b . Including a second-order Fourier component, the gratings are able to provide in-plane feedback as well as couple laser emission out of the resonator plane in vertical direction. Fig. 1. Surface-emitting circular Bragg lasers: (a) circular DFB laser; (b) disk Bragg laser; (c) ring Bragg laser. Laser emission is coupled out of the resonator plane in vertical direction via the Bragg gratings This chapter will present a comprehensive and systematic study on the surface-emitting Hankel-phased circular Bragg lasers. It is structured in the following manner: Sec. 2 focuses on every aspect in solving the modes of the lasers – analytical method, numerical method, and mode-solving accuracy check. Sec. 3 gives near-threshold modal properties of the lasers; comparison of different types of lasers demonstrates the advantages of disk and ring Bragg lasers in high-efficiency surface laser emission. Sec. 4 discusses above-threshold modal behavior, nonuniform pumping effect, and optimal design for different types of lasers. Sec. 5 concludes this chapter and suggests directions for future research. 2. Mode solving techniques Taking into account the resonant vertical laser radiation, Appendix A presents a derivation of a comprehensive coupled-mode theory for the Hankel-phased circular grating structures in active media. The effect of vertical radiation is incorporated into the coupled in-plane wave equations by a numerical Green’s function method. The in-plane (vertically confined) electric field is expressed as (1) (2) () () () () (), mm E xAxHxBxHx=+ (1) where (1) () m H x and (2) () m H x are the mth-order Hankel functions which represent respectively the in-plane outward and inward propagating cylindrical waves. A set of evolution equations for the amplitudes A(x) and B(x) is obtained: 2 d() () () () () , d ix Ax ux Ax vx Bx e x δ ⋅ =⋅−⋅⋅ (2) 2 d() () () () () , d ix Bx ux Bx vx Ax e x δ − ⋅ =− ⋅ + ⋅ ⋅ (3) where Surface-Emitting Circular Bragg Lasers – A Promising Next-Generation On-Chip Light Source for Optical Communications 553 x = βρ : normalized radial coordinate with β being the in-plane propagation constant; δ = ( β design – β )/ β : frequency detuning factor, representing a relative frequency shift of a resonant mode from the designed value; − ⎧ = ⎨ ⎩ 1 () ,if is within a g ratin g re g ion () (), if is within a no- g ratin g re g ion; A A gx h x ux gx x 12 , if is within a g ratin g re g ion () 0, if is within a no- g ratin g re g ion; hih x vx x + ⎧ = ⎨ ⎩ h 1 = h 1r + ih 1i : grating’s radiation coupling coefficient, representing the effect of vertical laser radiation on the in-plane modes; h 2 : grating’s feedback coupling coefficient, which can always be chosen real; g A (x) = g(x) – α : space-dependent net gain coefficient, the minimum value of which required to achieve laser emission will be solved analytically or numerically; α : nonsaturable internal loss, including absorption and nonradiative scattering losses; g(x) = g 0 (x)/[1 + I(x)/I sat ]: intensity-dependent saturated gain profile; g 0 (x): unsaturated gain profile; and I(x)/I sat : field intensity distribution in units of saturation intensity. It should be noted that, although Eqs. (2) and (3) appear to be a set of coupled equations for in-plane waves only, they implicitly include the effect of vertical radiation due to h 1 . As it will become clearer in Sec. 2.3, the vertical radiation can simply be treated as a loss term during the process of solving the in-plane laser modes. 2.1 Analytical mode solving method When solving the modes at threshold with uniform gain (or pump) distribution across the device, the net gain coefficient g A is x independent. The generic solutions of Eqs. (2) and (3) in no-grating regions are trivial: 0 () , A g x A xAe= (4) 0 () . A g x Bx Be − = (5) In grating regions, by introducing () () ix A xAxe δ − =  and () () ix Bx Bxe δ =  , Eqs. (2) and (3) become: () d() () (), d Ax ui Ax vBx x δ =− −    (6) () d() () (), d Bx ui Bx vAx x δ =− − +    (7) whose generic solutions lead to sinh[ ( )] cosh[ ( )] () (0) , sinh[ ] cosh[ ] ix Sx L Sx L Ax A e SL SL δ − +− = −+ ^ ^ (8) Frontiers in Guided Wave Optics and Optoelectronics 554 (0) [( ) ]sinh[ ( )] [ ( ) ]cosh[ ( )] () , sinh[ ] cosh[ ] ix A e u i S Sx L u i S Sx L Bx vSLSL δ δδ − −− −+ −− − =⋅ −+ ^^ ^ (9) where 22 ()Sui v δ ≡−− , ^ is a constant to be determined by specific boundary conditions, and L is a normalized length parameter (see Fig. 2). The determination of the constant ^ in Eqs. (8) and (9) requires the specific boundary conditions be applied to the grating under investigation. We focus on two typical boundary conditions to obtain ^ and the corresponding field reflectivity in each case. L A(0) B(0) A(L) B(L) r r 1 1 (a) L A(0) B(0) A(L) B(L) r r 1 1 (a) L x 0 A(x 0 ) B(x 0 ) A(L) B(L) = 0 (b) r r 2 2 L x 0 A(x 0 ) B(x 0 ) A(L) B(L) = 0 (b) r r 2 2 Fig. 2. Two types of boundary conditions for calculating reflectivities. (a) A(0) = B(0), r 1 (L) = A(L)/B(L); (b) B(L) = 0, r 2 (x 0 , L) = B(x 0 )/A(x 0 ) Case I: As shown in Fig. 2(a), the grating extends from the center x = 0 to x = L. An inward propagating wave with amplitude B(L) impinges from outside on the grating. The reflectivity is defined as r 1 (L) = A(L)/B(L). The finiteness of E(x) at the center x = 0 requires A(0) = B(0), leading to ( )sinh[ ] cosh[ ] sinh[ ] ( ) cosh[ ] uvi SL S SL SSLuvi SL δ δ − −+ = +−− ^ and to the reflectivity 2 1 ( ) ( )sinh[ ] cosh[ ] () . ( ) ( )sinh[ ] cosh[ ] iL A LuviSLSSL rL e B LuviSLSSL δ δ δ − −+ == −−− + (10) Case II: As shown in Fig. 2(b), the grating extends from x = x 0 to x = L. An outward propagating wave with amplitude A(x 0 ) impinges from inside on the grating. The reflectivity is defined as r 2 (x 0 , L) = B(x 0 )/A(x 0 ). No inward propagating wave comes from outside of the grating, i.e., B(L) = 0. This condition leads to () δ = −^ Sui and to the reflectivity 0 2 00 20 000 () sinh[( )] (,) . ( ) ( )sinh[ ( )] cosh[ ( )] ix Bx v SL x rxL e A x ui SLx S SLx δ δ − − == −−−− (11) It should be noted that, as seen from their definitions, the above reflectivities Eqs. (10) and (11) include the propagation phase. With the obtained reflectivities for the two types of boundary conditions, it is easy to derive the laser threshold condition for each circular Bragg laser configuration. 1. Circular DFB laser: The limiting cases r 1 (x b ) → ∞ or r 2 (0, x b ) = 1 lead to the same result [...]... values are kept continuous at every interface between grating and no-grating regions to satisfy BC(iii) After the integration, we have B(xb) whose absolute value marks a contour map in the 2-D plane of g0 and δ Now each minimum point in this contour map satisfies BC(ii) and thus represents a mode with corresponding g0 and δ Retrieving A(x) and B(x) for this mode and substituting them into Eq (1) give... deeper and deeper and the development of 576 Frontiers in Guided Wave Optics and Optoelectronics highly recognized passive optical network (PON), it is expected that time-division multiplexing PON (TDM-PON) and wavelength-division multiplexing PON (WDM PON) will be the most promising candidates for next-generation access systems A TDM PON, including asynchronous-transfer-mode (ATM) and broadband PON... modal pump level using the obtained g0 Assuming a linear pump–gain relationship above transparency, the unsaturated gain g0(x) follows the profile of pump intensity Ipump(x), and we may define the pump level Ppump ≡ ∫ Ipump(x) · 2πρ · dρ = 556 Frontiers in Guided Wave Optics and Optoelectronics P0 ∫ g0(x) · x · dx, where P0 having a power unit is a proportionality constant determined by specific experimental... standard single mode fiber (SMF-28) through the Georgia Institute of Technology’s (GT) fiber network 578 Frontiers in Guided Wave Optics and Optoelectronics 2 Optical mm -wave up-conversion for downstream An intermediate frequency (IF) or baseband signal can be transmitted over optical fiber to the BS, where the baseband or IF signal are up-converted to mm -wave carriers for broadcasting in the air In. .. radiation coupling induced mode selection mechanism (Sun & Yariv, 2007) Increased gain results in the excitation of higher-order modes 2 All the displayed modes of the disk Bragg laser are confined to the center disk with negligible peripheral power leakage and thus possess very low thresholds and very small modal volumes as will be shown in Sec 3.3 558 3 Frontiers in Guided Wave Optics and Optoelectronics. .. The modes are labeled in accord with those shown in Table 1 The single-mode range for each laser type is marked in pink After (Sun & Yariv, 2008) 560 Frontiers in Guided Wave Optics and Optoelectronics operation is usually preferred in laser designs, in the rest of this subsection we will limit xb to remain within each single-mode range and focus on the fundamental mode only Quality Factor As a measure... fundamental mode, within each single-mode range, for the three laser types As expected, 562 Frontiers in Guided Wave Optics and Optoelectronics all the lasers possess a larger in with a larger device size Comparing devices of identical dimensions, only the disk and ring Bragg lasers achieve high emission efficiencies This is a result of their fundamental modes being located in a band gap while the circular... nonuniform pumping effects, the typical exterior boundary radius xb = 200 is again assumed for all the circular DFB, disk, and ring Bragg lasers In addition, for the disk Bragg laser the inner disk radius is set to be x0 = xb/2, and for the ring Bragg laser the two interfaces separating the grating and no-grating regions are located at xL = xb/2 – π and xR = xb/2 + π Following the calculation procedure in Sec... threshold conditions Eqs (12), (13), and (14) govern the modes of the lasers of each type and will be used to obtain their threshold gains (gA) and corresponding detuning factors (δ) With these values, substituting Eqs (4), (5), (8), and (9) into Eq (1) and then matching them at the interfaces yield the corresponding in- plane modal field patterns Despite their much simpler and more direct forms, these threshold... E(x) and E’(x) be continuous at every interface between the grating and no-grating regions (Sun & Yariv, 2009c) 2.2 Numerical mode solving method When solving the modes at threshold with uniform gain (or pump) distribution across the device, gA is independent of x so that Eqs (2) and (3) can have analytical solutions Eqs (4) and (5), or (8) and (9) In the case of using a nonuniform pump profile and/ or . unsaturated gain g 0 (x) follows the profile of pump intensity I pump (x), and we may define the pump level P pump ≡ ∫ I pump (x) · 2π ρ · d ρ = Frontiers in Guided Wave Optics and Optoelectronics. from that of pump radiation, in the form of D( θ , Ψ ) = 2A 0 cos 2 (θ - Ψ ) as depicted in Fig. 22(b), Frontiers in Guided Wave Optics and Optoelectronics 548 and the polarization state. )] () (0) , sinh[ ] cosh[ ] ix Sx L Sx L Ax A e SL SL δ − +− = −+ ^ ^ (8) Frontiers in Guided Wave Optics and Optoelectronics 554 (0) [( ) ]sinh[ ( )] [ ( ) ]cosh[ ( )] () , sinh[ ] cosh[