Paraxial approximation

If the angles are small i.e. close to the optical axis, then the paraxial approximation can be made ... 

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. 

The term in curly brackets describes the deviation from the first-order theory. For paraxial rays h = 0 and that term vanishes this is the first-order result. But for other values of h the rays do not cross the axis at the same point as the paraxial rays, and this causes aberration. When parallel rays are incident from infinity as illustrated in Fig. 2.2, 1 /s y = 0. If the aberration is not too big, then in the term in the curly brackets the approximation Si q can be made. With n = nyjni, (2.4) then reduces to... 

The passage of electrons or other particles with charge q and mass m through an electrostatic lens system is governed by their motion under the action of the electric field. In the case considered here, cylindrical symmetry around the optical axis (z-axis) and paraxial rays will be assumed. Of the cylindrical coordinates only the transverse radial coordinate p and the distance coordinate z are of relevance, and the electrostatic potential of the lens is given by q>(p, z). As shown in Section 10.3.1, in the paraxial approximation the potential q>(p, z) is fully determined by the potential symmetry axis. Hence, the equations of motion and the fundamental differential equation of an electrostatic lens depend only on this potential. The fundamental lens equation is given by (see equ. (10.38))... 

Similarly, from the shaded triangles in Fig. 4.31(c) in the paraxial approximation, i.e., for small y and a values one obtains for the angular magnification the equivalent form... 

This equation with sin a in the transverse momentum is more general than equ. (10.39a) with tan a a a derived in the paraxial approximation. 

A final remark in this Section concerns axially symmetric problems. We usually treat these in radial coordinates and apply a numerical Hankel transform instead of the Fourier transform. This is a slow transform with a dense matrix, but due to the relatively small computational domain radially symmetric problems require, this is not a big problem. Alternatively, one could treat such situations by finite differencing in the radial dimension, but it would mean accepting additional (paraxial) approximation, and would introduce artificial numerical dispersion into the algorithm. 

The above derivation shows explicitly what approximations need to be adopted to obtain NLS Approximating K to second order in frequency and transverse wavenumber amounts to the paraxial, and quasi-monochromatic approximations for the linear wave propagation. The approximation in the nonlinear coupling Q also requires a narrow spectrum in order to be able to represent Q by a constant. 

The Nonlinear Envelope Equation [33] is a paraxial equation with some additional approximations related to chromatic dispersion. This equation appears to be extremely close to the paraxial version of UPPE. 

Once again we follow the general procedure and approximate the linear propagator by its paraxial version ... 

This is essentially the second-order (paraxial) Taylor expansion in transverse wavenumbers with only minor additional approximation. Namely, we replaced rib(oj) —> n, (wr) in the denominator of the diffraction term, and thus partly neglected the chromatic dispersion. 

Thus, the additional approximations underlying the NEE are paraxiality both in the free propagator and in the nonlinear coupling, and a small error in the chromatic dispersion introduced when the background index of refraction is replaced by a constant, frequency independent value in both the spatio-temporal correction term and in the nonlinear coupling term. Note that the latter approximations are usually not serious at all. 

This section provides three illustrative applications of the z-UPPE model. The first is the computationally more challenging as it involves a full 3D + time simulation of the propagation of a wide pancake shaped pulse in air.The second provides a nice illustration of the need to go beyond the paraxial approximation for nonlinear X-wave generation in condensed media and the last illustrates the subtle interplay between plasma generation and chromatic dispersion in limiting the extent of the supercontinuum spectrum. 

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. 

Physically, the paraxial approximation limits attention to beams that are not rapidly diverging. We will establish criteria for the validity of the paraxial approximation while discussing typical applications and constraints. In this way, the reader will come to appreciate the advantages and limitations of quasioptics vis-a-vis microwave technology. 

We will lay particular emphasis in this chapter on factors that influence the design and evaluation of high performance EPR spectrometers. This means that we must take into account the vector properties of the electromagnetic field, the effect of diffraction fringes, and the assumption of paraxial beams. We will then discuss approximations to the complete treatment that are specially useful in spectrometer design. 

It will prove useful in the sequel to use the hertz potentials (Born and Wolf, 1980, pp. 76-84) to describe the electromagnetic field. The hertz potentials also satisfy the homogeneous vector Helmholtz equation in free space. The advantage of the hertz potentials is that they display much higher symmetry than the conventional vector and scalar potentials. Furthermore, they may be written in a form that displays the paraxial approximation of quasioptics directly. 

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... 

In the integral over x, or equivalently po, from 0 to we have assumed that the Gaussian function in the integrand has decayed sufficiently rapidly so that there is negligible error in taking the limit of integration to x = , where the paraxial approximation would otherwise be inapplicable. 

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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