7.5 Higher-order Hermite–Gauss modes

The complete set of Hermite–Gauss modes is given by an infinite discrete set of modes unm (x, y,z) with the indices n and m as mode numbers. The sum n+m is called the order of the mode. The term higher-order modes usually refers to modes with an order n + m > 0. The general expression for Hermite–Gauss modes can be given as [35]
unm (x, y,z) = un(x,z)um (y,z), (128 )
with
( ) u (x, z) = (2)1∕4 exp(i(2n+1)Ψ(z)) 1∕2 × n π ( √- ) 2nn!w((z) ) (129 ) H -2x- exp − i-kx2--− -x2-- , n w(z) 2RC (z) w2(z)
and Hn(x ) the Hermite polynomials of order n. The first Hermite polynomials, without normalisation, can be written
H0 (x) = 1 H1 (x ) = 2x 2 3 (130 ) H2 (x) = 4x − 2 H3 (x ) = 8x − 12x.
Further orders can be computed recursively since
Hn+1 (x ) = 2xHn (x) − 2nHn −1(x). (131 )
For both transverse directions we can also rewrite the above to
−1∕2 unm (x,y,z) = (2n+m −1n!m!π ) w1(z) exp(i(n + m + 1)Ψ(z)) × ( √2x) ( √2y ) ( k(x2+y2) x2+y2) (132 ) Hn w-(z) Hm w(z) exp − i-2RC(z)- − w2(z) .
The latter form has the advantage of clearly showing the extra phase shift along the z-axis of (n + m + 1)Ψ (z), called the Gouy phase; see Section 7.8.
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