Gaussian beam paraxial approximation

Although paraxial approximation becomes imsuitable for higher-WA optics and for non-Gaussian beams, the above insights should remain qualitatively valid in these cases as well. Since only the above-the-threshold intensity part of the spatio-temporal envelope of the beam is important for photomodification, usually this part can be reasonably well approximated by a Gaussian. 

We note that the term quasioptics implies that it is not sufficient to borrow familiar optical concepts, such as point focus, the lensmaker s equation, etc. without modification. In fact, diffraction plays a crucial role in characterizing system behavior. Fortunately, the quasioptics formalism allows us to avoid the time-consuming computation of diffraction integrals that would otherwise be necessary for a complete system analysis. We will concentrate instead on those aspects of quasioptics that are readily amenable to calculation in the paraxial approximation (see subsequent text). In particular, we will study the propagation of Gaussian beams. 

The paraxial approximation is essentially a Taylor series expansion of an exact solution of the wave equation in powers of p/w, terminated at ( p/tv), that allows us to exploit the rapid decay of a Gaussian beam away from the optical axis. We will develop a more precise criterion in the sequel. We will also show that the phase and amplitude modulation of the underlying plane wave structure of the electromagnetic field is a slowly varying function of distance from the point where the beam is launched. 

We have now found a physical meaning for the complex origin shift Zpi it characterizes the 1 /e radius of the dipole radiation at z = 0. The Cornell FIR spectrometer uses Zp = 117 mm, which corresponds to Wp = 6.7 mm. Thus, even at z = 0, where z — izp has its smallest magnitude, the quantity (tVp/Zp) = 3.28 X 10 (which characterizes the paraxial approximation) is a small number. We may use the paraxial approximation with confidence to derive the properties of the fundamental Gaussian beam. 

It is common to call tvp the beam-waist radius. In the literature one often finds the phrase at the beam waist, which refers to that value of z for which the function u has its minimum radial extent. For u defined by (8), this occurs at z = 0. The distance Zp is called the confocal distance. When z < Zp we say that the Gaussian beam is in the near field. When z > Zp, the Gaussian beam is in the far field. The majority of this chapter is concerned with the behavior of u in the range 0 < z < Zp, the near-field region. The phase and amplitude of m is a complicated function of position in the near field. When z Zp and p z or when we are in the far field and the paraxial approximation is valid, it is straightforward to show that the asymptotic behavior of u approaches a diverging spherical wave from a point source at z = 0. 

At this point, we have quantified the domain of validity of the paraxial approximation and established when we may neglect the nontransverse components of the Gaussian beam. We still need to examine our solution in more detail, because we have not yet addressed diffraction effects. This analysis is necessary because the wavelength of FIR radiation is of the... 

Within the paraxial approximation for Gaussian beams, Kogelnik (1965) has shown that the analog of the transfer matrix formalism is the ABCD law for the parameter q = z - iz [cf. Eq. (17)] that is, = Aq + B)/i.Cq + D), where the ABCD coefficients may be taken from the corresponding optical system transfer matrbc in the form... 

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