Distributed-Feedback Lasers

Transcription

1 Distributed-Feedback Lasers Class: Integrated Photonic Devices Tie: Fri. 8:00a ~ 11:00a. Classroo: 資電 06 Lecturer: Prof. 李明昌 (Ming-Chang Lee) Wavelength Dependence of Bragg Reflections Free-Space Bragg Reflection Waveguide Bragg Reflection dsinθ = lλ and l = 1,, 3,... Let θ = 90 Λ= ( / ) and l = 1,, 3,... l λ0 n e Effective index Usually only one ode will lie within the gain bandwidth of the laser

2 Coupling Efficiency of Wave Reflection The fraction of optical power that is reflected by a grating depends on Thickness of the waveguiding layer Depth of the grating teeth Length of the grating region Coupling Efficiency of Wave Reflection The coupling can be characterized by perturbation assuption ( ) 3 n n1 a 3( λ0/ a) 3( λ0 / a) 1 1/ 0 g 1 1 π κ = + + 3lλ n t π( n n ) 4 π ( n n ) where βλ l β is the propagation constant of the particular ode π

3 Coupling Efficiency of Wave Reflection A + A (incident wave, forward) (reflect wave, backward) For the first order (l=1) case, the reflection is essential liited to coupling between the forward and backward traveling wave [ κ z L ] + + cosh ( ) A ( z) = A cosh( κ L) and [ κ z L ] + κ sinh ( ) A ( z) = A κ cosh( κl) L: the length of grating Coupling Efficiency of Wave Reflection A + A (incident wave, forward) (reflect wave, backward) The reflectance and transittance are R eff = A A + and T eff = A A + + ( L) The incident power decreases exponentially with z, as power is reflected into the backward travelling wave

4 Ray Scattering of Higher-Order Bragg Gratings For higher-order gratings with periodicity Λ lλ0 Λ= l > 0 n e The extra path length ust be integral ultiples of the wavelength to have constructive scattering ray l = l λ0 b +Λ= n and b =Λsinθ Thus, 0 sinθ = 1 e e l λ n Λ l = 0,1,,3,... l l = 0,1,,3,... l Ray Scattering of Higher-Order Bragg Gratings For exaple of a second-order Bragg grating λ0 Λ= Then n e θ = l l = 0,1, sin 1 Three diffraction odes l = l = 0 l =1 l =

5 Coupling Efficiency of Wave Reflection A + A (incident wave, forward) (reflect wave, backward) Generally, the strong coupling in the transverse direction by second-order Bragg grating is undesirable A first-order grating is required to yield optiu perforance However, fabrication of the first-order grating is challenging. Lasing with Distributed Feedback E 0 L E r In a gain ediu (gain:g), κexp( jβ0z)sinh [ S( L z) ] ( β ) sinh ( ) cosh ( ) Er ( z) = E0 g j SL S SL where ( β ) S κ g j + The paraeter E 0 is the aplitude of a single ode incident on the grating (stiulus)

6 Lasing with Distributed Feedback E 0 L E r The phase isatching ter, β = β β 0 Where β 0 is the propagation constant at the Bragg wavelength The oscillation condition for the DFB laser corresponds to the case for E r E0 That is, ( g j β )sinh( SL) = Scosh( SL) Lasing with Distributed Feedback In general, nuerical ethod should be used for exactly solving g and Δβ siultaneously. A special case of solution of lasing frequency 1 πc ω = ω0 ( + ) nl Supposing that g >> κ, β g = 0, ± 1, ±,... ω 0 : Bragg Frequency β β β 0 ( ω ω ) c 0 n g It is interesting to note that no oscillation can occur at exactly the Bragg frequency ω 0. The ode spacing πc ω nl g However, only the lasing odes close to the Bragg frequency have the sallest threshold gain. Therefore, in a noral operating condition, the spectral feature of DFB laser often consists of the two longitudinal odes

7 Separate Confineent Heterostructure Lasers Lattice daage is usually created during the grating fabrication. It is better to separate the active layer out of the grating layer Distributed Bragg Reflection Lasers L 1 L L a Active Region Laser Eission Two Bragg gratings are eployed at both ends of the laser and outside of the electrically-puped active region To achieve a single longitudinal ode, one distributed reflector ust have narrow bandwidth, high reflectivity at the lasing wavelength

8 Distributed Bragg Reflection Lasers L 1 L L a Active Region Laser Eission For the passive grating region, The Transittivity: The Reflectivity: T R = γ exp j( β β) L = ( α + j β) sinh ( γl) + γ cosh ( γl) jκsinh ( γl) ( α + j β) sinh ( γl) + γ cosh ( γl) where (coplex) ( j ) γ κ + α + β Loss of grating Distributed Bragg Reflection Lasers R R1 L eff L 1 L L a Active Region Laser Eission Since the reflectivity is coplex nuber, we can consider an effective cavity length L eff L = L a 1+ + La L 1 L ( α ) 1 + a ( α + κ + α ) 1

9 Distributed Bragg Reflection Lasers R R1 L eff L 1 L L a Active Region Laser Eission The longitudinal ode spacing between the -th and (±1)-th lasing ode is approxiately π β β ± 1 = L eff Wavelength Selectability Copared with Fabry-Perot lasers, DFB or DBR laser is easy to achieve single-longitudinal-ode operation because the spacing between the -th and the (±1)-th ode is generally large and the reflectivity is ode-dependent