On Spontaneous Emission Into Guided Modes with Curved Wavefronts

The problem of spontaneous emission into guided modes with curved wavefronts is examined quantum mechanically. A classical result due to Petermann, which shows an increased emission rate relative to modes with planar phase fronts, is corroborated. T HE amount of power which is emitted spontaneously into a laser mode plays an important role in determining the spectral and dynamic features of the laser radiation. It figures prominently in semiconductor lasers in questions such as the resonance peak (“spiking resonance”) in the modulation response as well as in the number of longitudinal modes which oscillate [ 21. An important result due to Petermann [3] is that the spontaneous emission power into modes with curved wavefronts (such as result from gain guiding or a combination of gain and (anti) real-index guiding) is enhanced relative to that emitted into modes in pure index guiding. At first glance, this result seems to contradict [4] the basic quantum mechanical relationship which states that the ratio of stimulated to spontaneous emission rates into a (any) mode is equal to the number of quanta in the mode. This last statement would suggest that, all other factors remaining the same, the spontaneous emission rates into curved wavefront modes and those with planar wavefronts are equal. This apparent contradiction is not resolved by a study of Petermann’s paper [3], which treats the electron in a semiconductor as a localized classical dipole. This classical approach which, in our opinion, still needs to be justified, does not make any contact with the conventional quantum mechanical derivation of the spontaneous emission [ 51 rate. To resolve this problem, we undertook a quantum mechanical derivation of the spontaneous emission problem into a ‘ mode with a field dependence E ( x , z) = E, exp 7 (1 t ix) ipz [zxw’ 1 (1) corresponding to a Gaussian beam height w [2] and a radius of curvature of the wavefront R = 2rrnw2/xh. In a real-index guided mode, the astigmatism factor x is zero, while in the case of pure gain guiding, x = 1. Higher values of x obtain in a combination of anti-index guiding and gain guiding [ 3 ] . The key idea is that when x # 0, the field (1) does not qualifv as a quantum mechanical mode. A ‘‘pyper” mode must be, an eigenfunction of a field Hamiltonian Hfield with the total H of the form