Appendix 5 Spontaneous Emission Term and Factors Consider the spontaneous emission term describing the contribution of the spontaneous emission to the laser oscillation, which appears as the final term on the right-hand side of the rate equation for the photon density (Eq. (6.21)). Since this term represents the component of the spontaneous emission belonging to the same mode as the laser oscillation (angular frequency ! m ), it is closely related to the gain for the oscillation mode. Consider a laser of index-guiding type, and let E ¼ E(r) ¼ E(x, y)E(z) be the complex electric field of the oscillation mode. The field is normalized in the similar manner as Eq. (2.6), so that the optical energy in the resonator corresponds to the energy of a photon: Z c " 0 n r n g 2 jEðrÞj 2 dV ¼ hh! m ðA5:1Þ Then, from Eqs (2.50a) and (2.57), the transition probability relevant to the photons of this mode can be written as w abs n ¼ w stm n ¼ w spt ¼ p 2hh EðrÞ E h 1 jerj 2 i     2 ðE 1 þ hh! m À E 2 ÞðA5:2Þ where n is the number of the photons in the resonator. Using the direct- transition model, the net number of the stimulated emission transition per unit volume of the active region per unit time can be calculated by integrating w stm multiplied by (1/2p 3 )( f 2 À f 1 )dk. The average value of jE(r)j 2 in the active region is given by hjEj 2 i a ¼ Gð2hh! m =" 0 n r n g Þ V a ðA5:3Þ Copyright © 2004 Marcel Dekker, Inc. G ¼ R a jEðrÞj 2 dV R c jEðrÞj 2 dV ðA5:4Þ where V a is the volume of the active region, À is the confinement factor, and use has been made of Eq. (A5.1). Replacing E in Eq. (A5.2) by the average given by Eq. (A5.3), and calculating the relative time variation in the mode photon number in the resonator of volume V a , we obtain an expression for the mode gain: GGð! m Þ¼G pe 2 n r n g " 0 m 2 ! m jMj 2 ðf 2 À f 1 Þ r ðhh! m ÞðA5:5Þ In the derivation of the above expression, use has been made of Eqs (3.14)– (3.17). This result is consistent with that obtained by rewriting the material gain g given by Eq. (3.16) in G ¼ v g g, and then in the mode gain ÀG. Let R sp (! m ) be the number of photons spontaneously emitted per unit time in the active region of a volume V a . Then R sp can be calculated by integrating w stm given by Eq. (A5.2) multiplied by V a (1/2p 3 ) f 2 (1 À f 1 )dk in a similar manner as above, to yield R sp ð! m Þ¼G pe 2 n r n g " 0 m 2 ! m jMj 2 f 2 ð1 À f 1 Þ r ðhh! m ÞðA5:6Þ Accordingly, from Eqs (A5.5) and (A5.6), the spontaneous emission term and the mode gain are correlated by R sp ð! m Þ¼n sp GGð! m ÞðA5:7Þ n sp ¼ f 2 ð1 À f 1 Þ ð f 2 À f 1 Þ ¼ 1 À exp hh! m À ÁF k B T  ! À1 ðA5:8Þ where ÁF ¼ F c À F v is the difference between the quasi-Fermi levels. The parameter n sp in Eq. (A5.8) is referred to as the population inversion factor. The optical waves in an ordinary laser structure include many radiation modes with the propagation vector not parallel to the waveguide axis. The majority of the spontaneous emissions belong to such radiation modes, and therefore the guided mode component is negligibly small. Therefore, the expression given by Eq. (3.20) for a homogeneous semiconductor can also be used to describe approximately the spontaneous emission spectrum in a laser structure. By integrating it, the total spontaneous emission R sp can be calculated. Approximating the spontaneous emission spectrum by a Lorentzian distribution with a half-width Á! at half-maximum and using 300 Appendix 5 Copyright © 2004 Marcel Dekker, Inc. an approximation that spontaneous emission peak frequency % oscillation frequency, we obtain an expression for R sp : R sp ¼ n r e 2 ! pm 2 c 3 " 0 jMj 2 f 2 ð1 À f 1 Þ r ðhh! m Þ Â Z ðÁ!=2Þ 2 ð! À ! m Þ 2 þðÁ!=2Þ 2 d! ¼ n r e 2 ! Á! 2m 2 c 3 " 0 jMj 2 f 2 ð1 À f 1 Þ r ðhh! m ÞðA5:9Þ The spontaneous emission term C s N/ s at the end of the right-hand side of the rate equation given by Eq. (6.21) was introduced, by representing the total spontaneous emission per unit volume in the active region approximately as R sp ¼ N/ s , and by representing the component belonging to the oscillation mode per unit volume in the active region R sp (! m )/V a as C s N/ s ¼ C s R sp . Therefore, we see from Eqs (A5.6) and (A5.9) that the spontaneous emission factor C s is given by C s ¼ G 2pc 3 n 2 r n g V a ! 2 Á! ¼ G 4 4p 2 n 2 r n g V a Á ðA5:10Þ It should be noted that the above result does not apply for lasers of gain- guiding type, where the guided mode cannot be described independently to the carrier injection. Since the guided mode in gain-guiding lasers has curved wavefronts, the coupling of the spontaneous emission to the guided mode is stronger, and the value of C s is several times the value given by Eq. (A5.10). Spontaneous Emission Term and Factors 301 Copyright © 2004 Marcel Dekker, Inc. . Appendix 5 Spontaneous Emission Term and Factors Consider the spontaneous emission term describing the contribution of the spontaneous emission. ðhh! m ÞðA5:6Þ Accordingly, from Eqs (A5 .5) and (A5.6), the spontaneous emission term and the mode gain are correlated by R sp ð! m Þ¼n sp GGð! m ÞðA5:7Þ n