Transfer Matrix Simulation of External Cavity Laser based on Vernier Effect. Moto Kinoshita

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1 Transfer Matrix Simulation of xternal Cavity Laser based on Vernier ffect Moto Kinoshita Apr. 24 Introduction In optical transmission networks, Wavelength Division Multiplexing (WDM) which let us have large transmission capacities is very important system for the next generation communication. Widely tunable lasers are needed in order to enhance the efficiency of this system. In this paper, we propose a new external cavity laser which can be step-tuned based on the Vernier effect between a Fabry-Pérot etalon and the longitudinal mode of an external cavity and simulate its lasing spectra by using transfer matrix method. xternal cavity laser Figure shows an external cavity laser with two-section Fabry-Pérot laser chip. This external cavity laser consists of a two-section semiconductor Fabry-Pérot laser chip with igh eflection () and Anti eflection (A) coating, a collimator lens, an etalon filter, and an external mirror. The semiconductor Fabry-Pérot laser chip has a gain section which provides laser gain and a phase section which control longitudinal mode by injected current. Lasing occurs at the frequency that corresponds to both resonant conditions of etalon and external cavity. Therefore, we can control the laser frequency by step-tuning based on Vernier effect between an etalon and the longitudinal mode of an external cavity. coating A coating Laser Gain section Phase section Lens talon Fig. xternal cavity two-section Fabry-Pérot laser xternal mirror

2 Transfer Matrix Simulation Transfer matrix is the method for describing the relation between input and output electric fields at both side regions of an optical part. When the electric fields which are defined in Figure 2 are related r+ r T = T 2 T T 2 22 f + f, () T T2 T = is called the transfer matrix of the optical part. We take an optical T2 T22 reflector and space which has optical path length for example. Input and output electric fields at both side regions of the reflector and space are given by Figure 3 and Figure 4. Thus, the transfer matrix of the reflector M and space P is respectively described by r M = t t, (2) r t t exp( ikl) P =. (3) exp( ikl) Where r is amplitude reflectance, t 2 = r is amplitude transmittance, L is the length of space, and k is wave number. f+ r+ t f+ r r f = r f+ + t Optical part r Fig.2 Transfer matrix of a reflector

3 f+ r+ t f+ r r r t f = r f+ + t r M Fig.3 Transfer matrix of a reflector L f+ r+ t f+ exp( ikl) f = r exp( ikl) r P Fig.4 Transfer matrix of space Lasing spectra are calculated by applying these transfer matrixes to the components of the external cavity laser: coated facet, gain section, phase section, the space between A coated facet and etalon filter (space ), etalon filter, the space between etalon filter and external mirror (space 2), and external mirror. The transfer matrixes and the parameters of each optical component are summarized by Table and Table 2, respectively.

4 Table Transfer matrixes of each optical component Optical component coated facet gain section phase section space etalon filter space 2 t = r t exp G = exp P = P exp = r t t ( gl ikn L ) g ( ikn L ) p p Transfer matrix g ( ikn L ) p tt2 exp r r2 exp = P 2 exp = p g ( ) exp glg + ikng Lg ( ) exp ikn p L p ( iknele ) ( 2ikn L ) ( ikn L ) p2 p2 ( ) exp ikn pl p e e r r2 exp t t exp 2 ( ) exp ikn p2l p2 ( 2ikn ) ( ) ele ikn L e e external mirror tm M = rm tm rm tm t M

5 Table 2 Parameters of each optical component Optical component Parameter coated facet reflectance r =. 99 t 2 r = gain g = 2L g f ln r r ( δ ) M * gain section phase section space etalon filter space 2 reducing factor f ( ) δ δ = π FS laser linewidth δ = Mz Free Spectral ange of external cavity FS = 8 Gz refractive index n g = 3.5 length L g = 3 µm refractive index n p = variable value about 3.5 length L p = 3 µm refractive index n p = length L p = 3 mm reflectance of both end facet r r. 539 (finesse = 5) refractive index n e =.5 Thickness L e = mm refractive index n p2 = length L p2 = mm, 2 = external mirror reflectance r =. 9 t M M 2 rm = * The gain g is slightly smaller than one at the threshold by a reducing factor depending on experimentally expected linewidth δ.

6 We simulated the round trip process which is a circulation of the original light born in the gain section. f+ r+ r f r Fig.5 ound trip process We consider transfer matrix equation f + f r+ = (4) r where round trip matrix is defined by 2 = M P P P G. (5) ere, f+, f, r+, and r represent the original light, feedback light, output light, and exterior noise, respectively (see Fig.5). We can recognize from eq. (4) and Figure 5 that the returned light through a cycle is described by 2, (6) = r f + 22 where 2 and 22 are matrix elements of. Therefore, the superposed light through n cycles n is given by n n = 2 r n = 22 n f +. (7)

7 When n, the geometrical series of eq. (7) can be calculated as = f r 22, (8) 2 since slightly r <. Using, output light out is described by out = (9) 22 from eq. (4). Fig.6 shows out 2 as the lasing spectrum ( out ( ) ) Fig.6 Lasing spectrum 96 2 aving obtained the lasing spectra, I applied various parameters. Figure 7 and Figure 8 show the simulated spectrum for various parameters of refractive index n p and finesse of the etalon filter.

8 n p = ( out ( ) ) n p = ( out ( ) ) n p = ( out ( ) ) Fig.7 Changing the refractive index n p

9 finesse = ( out ( ) ) finesse = ( out ( ) ) ( out ( ) ) 2 finesse = Fig.8 Changing finesse of the etalon 96 2

10 Fig.7 suggests that we can achieve the step-tuning induced by the variation of refractive index n p. As variation of refractive index n p about n p.7 4, we can shift laser frequency to the next channel which is Gz off. And we found from Fig.8 that Side Mode Suppression atio (SMS) is improved with using higher finesse etalon filter. Summary The lasing spectrum of the step-tunable external cavity laser based on Vernier effect between a Fabry-Pérot etalon and the longitudinal mode of an external cavity was simulated by transfer matrix method. These results are likely reasonable because of the validity of each parameter. Therefore, we can design various types of lasers using this method with changing each parameter.

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