10 Generation of High Order Modes

Several methods have been proposed to generate LG modes. We discuss here a particular method based on fiber optics. It is known that cylindrical fibers having three layers (core plus two claddings) can exhibit modes having an annular structure more or less analogous to an LG mode. Consider a fiber having a core of radius rC, a cladding confined to the zone rC < r < rG and an extra cladding for r > rG. The latter can be assumed infinite for the guided modes, which have an evanescent behavior in that region, so that the external radius is never reached by the light. The refractive indices are n C in the core, n G in the first cladding and n ext in the external cladding. We assume nC > nG > next. Assuming a wave of the form (p is any integer and k ≡ 2π∕ λ)

ℰ (r,z) = eikneffzEp(r)eipφ, (10.1 )
where ℰ is any component of the optical wave and n eff is a parameter depending on the fiber geometry (radii and indices). The wave equation reduces to
[ 1 p2 ] ∂2r + -∂r − -2-+ k2(n2 − n2eff) Ep(r) = 0. (10.2 ) r r
There exist families of modes depending on the value of ne ff compared to nc,nG, and next. Modes such that nC > neff > nG are called core modes. Modes such that nC > nG > neff > next are called cladding modes. We are interested in cladding modes because the central part of the beam is vanishing in this case (as in an LG , n ⁄= 0 n,n mode). A further (realistic) assumption is that the indices are slightly different. In this case, the weak guidance model holds, leading to linearly polarized modes called LP. Solving Equation (10.2View Equation) leads to a wave of the form
( { AJp (U r∕rC) (r < rC) Ep(r) = BJp (Ugr ∕rG) + BYp (Ugr∕rG ) (rC ≤ r < rG ) (10.3 ) ( DKp (W r∕rC ) (r > rG )
with the following notation:
∘ --------- ∘ --------- ∘ ---------- 2 2 2 2 2 2 U = krC nC − n eff, Ug = krG n G − neff, W = krG neff − next. (10.4 )
J p and Y p are Bessel functions of the first and second kind, respectively. K p is a modified Bessel function of the second kind. The structure of the solution was dictated by the following considerations: The solution must be regular at r = 0, (no Yp contribution in the core), the solution must be evanescent in the external cladding (no Ip contribution there). Now the arbitrary constants A,B, and C can be reduced to one after taking into account the boundary conditions. The boundary conditions require continuity of the components of the field tangential to the cylindrical interfaces at rC, and rG. In the weak guidance model, this is equivalent to requiring a smooth solution at the interfaces and smooth derivatives. Only discrete values of neff make it possible, and these discrete values determine families of modes. The central issue of the guide theory is thus to find these values. If we adopt the following notation (S ≡ rG∕rC):
M (neff) = UgYp+1 (Ug)Kp (W ) − W Kp+1 (W )Yp(Ug ) (10.5 )
Ug N (neff) = UJp+1 (U )Jp(Ug ∕S) − ---Jp+1(Ug∕S )Jp(U ) (10.6 ) S
P (neff) = UgJp+1(Ug )Kp(W ) − W Kp+1 (W )Jp(Ug) (10.7 )
Ug Q (neff) = ---Yp+1(Ug∕S )Jp(U ) − U Jp+1(U )Yp(Ug∕S ), (10.8 ) S
then the solutions neff are determined by the dispersion equation
M N − P Q = 0. (10.9 )
Solutions of Equation (10.9View Equation) may or may not exist, depending on the parameters. Their (finite) number depends also on these parameters. Standard numerical procedures allow one to extract the number of solutions and the effective indices corresponding to each of the modes. For a given mode, neff being known, the quantities U, Ug, and W are known, and the constants B, C, D can be computed from A, which can be used for normalization. Specifically, we have
[ ] B = π- U Jp+1(U)Yp (Ug ∕S ) − Ug-Jp(U )Yp+1(Ug∕S ) (10.10 ) 2 S
π [U ] C = -- --gJp+1(Ug∕S )Jp(U ) − U Jp+1(U )Jp(Ug∕S ) (10.11 ) 2 S
BJ (U ) + CY (U ) D = ---p--g-------p--g-. (10.12 ) Kp (W )
It is possible to find parameters such that an LP mode has a structure comparable to an LG mode. We show a specific (but somewhat arbitrary) example in Figure 67View Image, for which the Hermitian scalar product of the fiber (LP5,5) mode with an LG mode is about 75% in power. The parameters of the LP mode are
rC = 4μm, rG = 58.99 μm, nC = 1.452126, nG = 1.446846, next = 1.4452, neff = 1.4452017 (10.13 )
The LG5,5 mode has w = 4.472 cm. It is surely possible to have a better matching after an optimization study that we are planning. The last step is to couple a TEM0,0 laser beam with such a fiber and then the resulting fundamental LP mode to the LP5,5 via a Bragg coupler.
View Image

Figure 67: Mode LP 5,5 (solid line), Mode LG 5,5 (dashed line)


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