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Complex Aperture Theory – Volume Apertures – General Results
Published in Lawrence J. Ziomek, An Introduction to Sonar Systems Engineering, 2017
for t ≥ r/c, where the expression t − (r/c) is known as the retarded time. Retarded time is a measure of the amount of time that has elapsed since the acoustic field transmitted by a source first appears at a receiver located r meters from the source at time t = r/c seconds. Expand the exponent of next.
Controllable ultrashort pulse generation in a graphene mode-locked fibre laser with dispersion-decreasing fibres
Published in Journal of Modern Optics, 2019
Sheida Mahmoodi, Alireza Bananej, Rahman Nouroozi, Daryoush Abdollahpour
A mode-locked fibre ring laser, as shown in Figure 1, can be modelled by the generalized nonlinear Schrödinger equation (GNLSE) that is widely used for the investigation of optical pulse propagation (26). For such a ring cavity, the pulse propagation equation can be written as where T is the retarded time in a reference frame moving with the pulse, is the third-order dispersion coefficient, α and g are the loss and gain parameters, respectively, and , where c is the speed of light, k is the wave number and is the gain bandwidth. Equation (4) can be numerically solved by using the split-step Fourier method, in which linear and nonlinear effects of the propagation are dealt with separately, the former in the frequency domain and the latter in the time domain (17). Moreover, the gain is modelled by (26) where is the small signal gain; and are the average power and the saturation power of the laser, respectively. In addition, pulse propagation in a passive fibre can also be modelled by Equation (4) with neglected gain term. In order to implement a DDF in the model , with a predefined profile function, is used instead of a constant in Equation (4). Finally, the action of the saturable absorber is included in the model by a transfer function for the absorber. In each round trip, calculated pulse envelope is multiplied by a transfer function in the form of where is obtained from the rate equation of the saturable absorber. Since graphene is a fast saturable absorber, can be written as (27) where and are the saturable and non-saturable absorption coefficients, respectively; P is the instantaneous power and is the power at which absorption saturation occurs for graphene.