7.9 Laguerre–Gauss modes

Laguerre–Gauss modes are another complete set of functions, which solve the paraxial wave equation. They are defined in cylindrical coordinates and can have advantages over Hermite–Gauss modes in the presence of cylindrical symmetry. More recently, Laguerre–Gauss modes are being investigated in a different context: using a pure higher-order Laguerre–Gauss mode instead of the fundamental Gaussian beam can significantly reduce the impact of mirror thermal noise on the sensitivity of gravitational wave detectors [5412]. Laguerre–Gauss modes are commonly given as [50Jump To The Next Citation Point]
∘ ------- up,l(r,ϕ,z) = --1- --2p!--exp (i(2p + |l| + 1)Ψ (z )) w( (z√) ) π(|l||l|+p)!( ) ( ) (146 ) × -2r- L(pl) -2r22- exp − ik-r2-+ ilϕ , w(z) w(z) 2q(z)
with r, ϕ and z as the cylindrical coordinates around the optical axis. The letter p is the radial mode index, l the azimuthal mode index8 and (l) Lp (x) are the associated Laguerre polynomials:
p ( ) (l) 1-∑ p! l + p j L p (x ) = p! j! p − j (− x ). (147 ) j=0
All other parameters (w (z),q(z),...) are defined as above for the Hermite–Gauss modes.

The dependence of the Laguerre modes on ϕ as given in Equation (146View Equation) results in a spiraling phase front, while the intensity pattern will always show unbroken concentric rings; see Figure 45View Image. These modes are also called helical Laguerre–Gauss modes because of the their special phase structure.

The reader might be more familiar with a slightly different type of Laguerre modes (compare Figure 46View Image and Figure 47View Image) that features dark radial lines as well as dark concentric rings. Mathematically, these can be described simply by replacing the phase factor exp(ilϕ) in Equation (146View Equation) by a sine or cosine function. For example, an alternative set of Laguerre–Gauss modes is given by [55Jump To The Next Citation Point]

∘ ------------ alt -2-- -----p!---- u p,l(r,ϕ, z) = w((z) )(1+δ0lπ(|(l|+p)! e)xp(i(2(p + |l| + 1))Ψ (z)) -√2r |l| (l) 2r2-- -r2-- (148 ) × w (z) L p w(z)2 exp − ik 2q(z) cos(lϕ).
This type of mode has a spherical phase front, just as the Hermite–Gauss modes. We will refer to this set as sinusoidal Laguerre–Gauss modes throughout this document.

For the purposes of simulation it can be sometimes useful to decompose Laguerre–Gauss modes into Hermite–Gauss modes. The mathematical conversion for helical modes is given as [71]

∑N uLnG,m(x, y,z) = ikb(n,m, k)uHGN−k,k(x, y,z), (149 ) k=0
with real coefficients
∘ ----------- (N − k)!k! 1 b(n,m, k) = ---N---------(∂t)k[(1 − t)n(1 + t)m ]t=0, (150 ) 2 n!m! k!
if N = n + m. This relates to the common definition of Laguerre modes as upl as follows: p = min (n,m ) and l = n − m. The coefficients b(n,m, k) can be computed numerically by using Jacobi polynomials. Jacobi polynomials can be written in various forms:
(− 1 )n P αn,β (x ) = --n---(1 − x)−α(1 + x)−β(∂x)n(1 − x)α+n (1 + x )β+n, (151 ) 2 n!
n ( ) ( ) α,β 1--∑ n + α n + β n−j j P n (x ) = 2n j n − j (x − 1) (x + 1) , (152 ) j=0
which leads to
∘ ----------- (N − k )!k! b(n,m, k) = -----------(− 2)kPnk− k,m− k(0 ). (153 ) 2N n!m!
View Image

Figure 46: Intensity profiles for helical Laguerre–Gauss modes upl. The u00 mode is identical to the Hermite–Gauss mode of order 0. Higher-order modes show a widening of the intensity and decreasing peak intensity. The number of concentric dark rings is given by the radial mode index p.
View Image

Figure 47: Intensity profiles for sinusoidal Laguerre–Gauss modes uapllt. The up0 modes are identical to the helical modes. However, for azimuthal mode indices l > 0 the pattern shows l dark radial lines in addition to the p dark concentric rings.

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