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By Louis De Broglie

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22) and E refers to a single transverse polarization component. In this form the most rapid wavelength-scale variation of the field is carried by the e−jkz 18 INTRODUCTION AND REVIEW term; u(x, y, z) is a slowly varying envelope. Substitution into the wave equation, eq. 57) where ∇T2 = ∂2 /∂x2 + ∂2 /∂y2 . Since u is assumed to be slowly varying, ∂2 u/∂z2 ≪ 2k |∂u/∂z|, and therefore ∂2 u/∂z2 can be neglected. 58) The paraxial wave equation has Gaussian beam solutions that provide a good description of laser beams both inside and outside the laser cavity.

However, such lasers do not generate short pulses! To obtain a mode-locked laser generating ultrashort pulses, one must incorporate either active or nonlinear pulse-forming elements (modulators) into the laser cavity. Furthermore, one must also consider new processes affecting the short pulse as it propagates within the laser cavity. The following nonexhaustive list describes some of the processes that can be important in mode-locked lasers: r Gain. Gain is required in mode-locked lasers just as it is in single-mode lasers.

This is especially useful in fiber lasers for optical communications applications, where the round-trip time can be very long (often corresponding to hundreds of nanoseconds), while pulse trains may be desired at multiple gigahertz rates [20]. It should be noted, however, that there is no guarantee that a pulse will be present in every transmission peak in harmonic mode-locking; therefore, if this is important, other stabilizing mechanisms must be incorporated into the laser. r In another form of mode-locking most appropriate for optically pumped systems, the laser is synchronously pumped by the pulses from another mode-locked laser [21,22].

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An introduction to the study of wave mechanics by Louis De Broglie


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