Материал: Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo

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Ref. p. 131]

3.1 Linear optics

129

Table 3.1.25. Coupling of waveguides .

Given

Solution

–

Source WG1 (laser,

waveguide) which

emits a Hermite-Gaussian beam,

η 00−00 =

π w I w O

receiver WG2 (laser, waveguide) which can

w I w O

–

−

accept Hermite-Gaussian eigenmodes:

w O

w I

R I

R O

−

8 (w 0 I w 0 O ∆ x)2

k2 ψ2

(w20 I + w20 O)3

−

Plane of coupling

w I + w O .

R O

(3.1.137)

w0O

w0I		x
WG1	w I	WG2

R I

Fig. 3.1.45. Coupling of Gaussian beams. w 0 I and w 0 O: beam waist radii for WG1 and WG2, respectively; w I and w O : beam radii in the coupling plane; R I and R O: curvature radii of the beam wavefronts in the coupling plane; k = 2 π/λ ; λ : the wavelength of light; ∆ x : the lateral o set between the waveguides; ψ : the tilt of the axis.

Asked : The e ciency of the excitation of the modes in WG2, here the fundamental mode 00.

η 00−00 = 1 for ∆ x = ψ = 0 and the exact beam radii and curvature ﬁtting w I = w O and R I = R O , otherwise

η 00−00 < 1 .

Equation (3.1.137) contains the approximations:

–Gaussian beams (paraxial optics).

–Right-hand side of (3.1.137): 2nd and 3rd term 1st term.

About coupling coe cients for higher-order modes and without the approximation: see [64Kog]; on couplings with Hermite-Gaussian modes and Laguerre-Gaussian modes: see [94Kri, 80Gra].

3.1.7.5.4 Laser ﬁber coupling

Methods of treatment:

–Launching of fundamental-mode laser radiation into the fundamental mode of a single-mode ﬁber:

–Calculation of the overlap integral (3.1.134) for a Gaussian mode and the mode ﬁeld for di erent ﬁber cross sections: see [88Neu, p. 179], [80Gra].

–Approximation of the exact ﬁber fundamental modes by a Gaussian ﬁeld distribution (see [88Neu, pp. 68]) and the application of the waist transformation from laser via an optical system with the methods of Sects. 3.1.7.2–3.1.7.4 and calculation of the overlap integral equation (3.1.134) or mode-coupling equation (3.1.137) [91Wu].

–Launching of fundamental-mode laser radiation or multimode laser radiation or incoherent light sources into multimode ﬁbers:

–Overlap integral techniques in the framework of partial coherence theory: see [87Hil].

–Geometric optical methods (ray tracing and phase space techniques): see [90Gec, 95Sny, 91Gra, 91Wu, 01I ].

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New Series VIII/1A1

130	3.1.7 Beam propagation in optical systems	[Ref. p. 131

–Inclusion of the aberrations and stops of the optical system used:

–Monomodal and partial coherent case: calculation of the wave aberrations of the optical system by ray-tracing methods and inclusion of these aberrations into the overlap integral: see [82Wag, 95Gae, 89Hil, 99Gue].

–Ray-tracing methods are adequate for stops and aberrations, but not reliable for a few mode waveguides: rough design [01I ]: the spot diagram of the ray tracing in the ﬁber facet should be within the core area and the angles of incidence should be smaller then the aperture angle [88Neu] of the ﬁber.

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References for 3.1	131

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