Order Gaussian Beam Modes

Appendix Higher Order Gaussian Beam Modes 

In this Appendix, we will develop the mathematical background necessary to study higher order Gaussian beam modes. We also will outline how certain integrals that arise in beam mode analysis may be evaluated. Because this material is a compilation from several sources, the original works should be consulted for further details. 

The vector Helmholtz equation may be rewritten in a form suitable for evaluation in curvilinear coordinates as follows, using well-known vector identities  

It is possible to show that the vector Helmholtz equation for a transverse vector function reduces to the following equation for the scalar function, (p, z) in cylindrical coordinates  

We now see that our choice of index on / i is more than a convenient label it characterizes the radial and longitudinal dependence of the scalar part of transverse solutions of the vector Helmholtz equation. 

---

Appendix Higher Order Gaussian Beam Modes References... 

We may now derive the electromagnetic field of higher order transverse Gaussian beam modes. In order to do so, we will use a technique developed for Cartesian coordinates described in Marcuse (1975), but adapted to cylindrical symmetry. For a system with cylindrical symmetry, we may take a trial solution of the form... 

In order to proceed, we will accept that the transverse components of the electromagnetic field are the only ones that are relevant in the problem on the basis of the exact calculation that we have performed for the fundamental Gaussian beam. Instead, we will use trial functions for u that will lead to self-consistent expressions for the transverse components of Gaussian beams of arbitrary order when substituted into the vector Helmholtz equation. The derivation is clearest for the fundamental. We will redrive the transverse field components of the fundamental Gaussian beam here. The deviation of higher order modes is outlined in the Appendix. 

Fig. 4. Scan of cavity modes in the presence of a sample and sample holder. The most intense resonances are the longitudinal resonances, which are the resonances of the fundamental Gaussian beam. Radial and azimuthal modes are also present and appear as shoulders on the longitudinal resonances. Note that the higher order radial and azimuthal modes are slightly dispersive.

If the beam is not a pure Gaussian beam but contains admixtures of higher order modes, the beam quality can be defined by the parameter... 

Figure 4-49 Ray traces indicating the three significant modes of scattaing—reflection and first- and second-order refraction—for a water droplet (top) light scattering of a Gaussian beam from a water droplet, simulated using the Fouiia- Lorenz-Mie theory (bottom courtesy of C. Tropea and N. Damaschke, Technische Universitat Darmstadt,...

In this case the Gaussian beam is incident normally on the endface, but the center of the beam is shifted a distance along the x-axis in Fig. 20-2(e). Thus the fiber is illuminated asymmetrically, so less power enters the fundamental mode, and higher-order modes will be excited. In this situation, it is clear from Eqs. (20-7a) and (20-9) that on the... 

An efficient optical coupling to the WGMs is instrumental in order to harvest the full potential of the high-2 droplet resonators. In most reported experiments, the droplet resonators are probed by free-space excitation, where, e.g., a Gaussian laser beam excites resonator modes and scattered light or fluorescence is detected. This approach... 

This is the rate per atom and so to obtain a value for the signal expected in an experiment we must consider the laser beam geometry and the number density n of hydrogen atoms. Here I will assume that the laser beam can be described by the lowest order TEMoo Gaussian mode. The power per unit area of each beam is then given by... 

The efficiency with which beams excite the fundamental modes of circular fibers, with the fields of Eq. (13-9), is of particular interest when the fiber is single moded. In order to account for weakly guiding fibers of otherwise arbitrary profile, when analytical solutions of the scalar wave equation for Fo (r) are not available, we use the Gaussian approximation of Chapter 15. The radial dependence of the fundamental-mode transverse fields is approximated in Eq. (15-2) by setting... 

---