3 Description of the beam field 3.1 Motivation Our discussion thus far is in terms of U. a scalar function. To return vector fields to the problem, assume all fields are polarized in the x direction Our discussion of Fraunhofer diffraction also assumed plane waves. The waves emitted by a laser have a very limited transverse extent and are cer- tainly not plane waves. To better understand the case of waves comprising the laser beam, we must return to the underlying physics encapsulated in Maxwell's equations 3.2 Calculation We consider the case of an insulator, which will simplify the resulting equa- tions. In this case, the conductivity o is equal to 0, and there is no attenua- tion of the beam within the material. Maxwell's equations become E=0 Going from the vector to the scalar case, assume x polarization E(phys)=xE(phys) where the physical field is, as usual, the real part of a complex field. Assume a monochromatic wave. so that the fast timescale variation of the wave can be expressed as e-t. The complex field contains the slower variations in the field E(phys)=(E(r)e(z-ut) e(r)is dependent on z, although the variation in the complex field low relative to the ntial depender E(r) is depende and y, and in particular, has important dependence on p=va2+ Now, substitute this assumed form of the E field into Maxwells equation 63 Description of the beam field 3.1 Motivation Our discussion thus far is in terms of U, a scalar function. To return vector fields to the problem, assume all fields are polarized in the xˆ direction. Our discussion of Fraunhofer diffraction also assumed plane waves. The waves emitted by a laser have a very limited transverse extent and are cer￾tainly not plane waves. To better understand the case of waves comprising the laser beam, we must return to the underlying physics encapsulated in Maxwell’s equations. 3.2 Calculation We consider the case of an insulator, which will simplify the resulting equa￾tions. In this case, the conductivity σ is equal to 0, and there is no attenua￾tion of the beam within the material. Maxwell’s equations become,  ∇2 − µ ∂ 2 ∂t2  E = 0 Going from the vector to the scalar case, assume xˆ polarization, E(phys) = xˆE(phys) where the physical field is, as usual, the real part of a complex field. Assume a monochromatic wave, so that the fast timescale variation of the wave can be expressed as e −ıωt. The complex field contains the slower variations in the field, E(phys) = < ￾ E(r)e ı(kz−ωt)
E(r) is dependent on z, although the variation in the complex field is slow relative to the exponential dependence on z. E(r) is dependent on x and y, and in particular, has important dependence on ρ = p x 2 + y 2 . Now, substitute this assumed form of the E field into Maxwell’s equation 6