2.3 Possible solutions One possibility is the Gaussian, which is equal to its Fourier transform as we mentioned previousl U(a,y) where w is a characteristic width There are also other higher order solutions Upg(a, y)=HE H where Hp.g are Hermite polynomials, Hu)=1H1(n)=2Hn(u)=(-1)e2dn(a-n2 Note. With this notation, the Gaussian case can be denoted Uo These are the same Hermite polynomials that appear in the solution to the 1d Shrodinger equation for the quantum mechanical simple har nonic oscillat We are considering the case in which the electromagnetic fields are per pendicular. In this case, the solutions are called transverse electromagnetic (TEM) solutions Each choice of p and q leads to a different irradiance pattern. Some xamples are TEMoo consists of a pure circle TEM10 consists of two symmetric spots TEMIl consists of four spots, the total forming a circular shape with the coordinate axes removed from the circle Visualizations of some of these patterns appear in Padrotti, figures 22-17 and 22-18 All TEM modes beyond the 00 mode produce an undesirable spread in the irradiance of the output laser beam. We would like to eliminate all of these higher order modes to create single mode laser with maximum point like irradiance 52.3 Possible solutions One possibility is the Gaussian, which is equal to its Fourier transform as we mentioned previously, U(x, y) = e − (x 2+y 2) w2 where w is a characteristic width. There are also other higher order solutions, Upq(x, y) = Hp √ 2x w ! Hq √ 2y w ! e − (x 2+y 2) w2 where Hp,q are Hermite polynomials, H0(u) = 1 H1(u) = 2u Hn(u) = (−1)n e u 2 d n dun  e −u 2  Note. • With this notation, the Gaussian case can be denoted U00. • These are the same Hermite polynomials that appear in the solution to the 1D Shr¨odinger equation for the quantum mechanical simple har￾monic oscillator. We are considering the case in which the electromagnetic fields are per￾pendicular. In this case, the solutions are called transverse electromagnetic (TEM) solutions. Each choice of p and q leads to a different irradiance pattern. Some examples are: TEM00 consists of a pure circle TEM10 consists of two symmetric spots TEM11 consists of four spots, the total forming a circular shape with the coordinate axes removed from the circle. Visualizations of some of these patterns appear in Padrotti, figures 22-17 and 22-18. All TEM modes beyond the 00 mode produce an undesirable spread in the irradiance of the output laser beam. We would like to eliminate all of these higher order modes to create single mode laser with maximum point￾like irradiance. 5