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128

3.1.7 Beam propagation in optical systems

[Ref. p. 131

 

 

 

3.1.7.5.2 Mode matching for beam coupling into waveguides

The calculation of the excitation coe cient of an eigenmode in a waveguide (output mode) by the incident mode (input mode) at the surface of the waveguide is described in Table 3.1.24.

This task occurs

–if a laser beam is formed by an optical system and coupled afterwards into an optical fiber,

–if a laser beam of a master oscillator is to be coupled into a power amplifier,

–in the case of waveguide-waveguide coupling especially fiber-fiber coupling or coupling between semiconductor lasers.

Solutions are available in commercial optical design programs.

Table 3.1.24. Definitions for waveguide coupling.

Given

Solution

 

 

 

 

 

 

 

–

Incident beam (emitted by a laser

Coupling coe cient (power relation):

 

 

(and) transformed by an optical system):

 

O IO O IO

 

 

 

Einput (x, y) .

η =

.

(3.1.133)

 

 

–

Waveguide with an eigenmode field the

 

N I N O

 

 

coupling to which is asked: Eoutput (x, y) .

Overlap integral :

 

Plane of mode

O IO =

∞

∞

(3.1.134)

x

 

 

d x

d y E I(x, y) E O(x, y) .

matching

 

 

 

 

 

 

 

 

 

−∞

−∞

 

Einput ( x)

Eoutput ( x)

Normalization:

 

 

z

 

∞

∞

 

 

 

 

N I =

 

d x

d y E I(x, y) E I (x, y) .

(3.1.135)

 

Waveguide

 

−∞

−∞

 

 

Fig. 3.1.44. Mode matching.

Normalization:

 

 

Asked: Part of power transmitted into the waveguide (fiber, laser, integrated optical waveguide).

∞

∞

 

 

 

 

N O =

d x d y E O(x, y) E O(x, y) .

(3.1.136)

−∞

−∞

 

E ective antireflection layers are assumed to be on the waveguide.

3.1.7.5.3 Free-space coupling of Gaussian modes

For the case that a Gaussian output waist of a source waveguide and a Gaussian input waist of a receiver waveguide are separated by air, the coupling of both waveguides is generally treated in [64Kog]. Higher-order modes are also included. The approximation of small misalignments (o set and tilt) is given in Table 3.1.25, large o sets and tilts are treated in [64Kog, 91Wu].

Landolt-B¨ornstein

New Series VIII/1A1

Ref. p. 131]

 

 

3.1 Linear optics

 

 

 

 

 

 

 

 

 

 

129

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3.1.25. Coupling of waveguides .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Given

 

 

Solution

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

–

Source WG1 (laser,

waveguide) which

 

 

 

 

 

 

 

 

4

 

 

 

 

 

 

 

 

 

 

 

emits a Hermite-Gaussian beam,

η 00−00 =

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

π w I w O

2

1

 

 

1

 

 

2

 

receiver WG2 (laser, waveguide) which can

 

 

w I w O

 

 

 

 

 

 

 

–

 

 

 

+

 

+

 

 

 

 

 

 

−

 

 

 

 

accept Hermite-Gaussian eigenmodes:

 

 

w O

w I

 

λ

 

 

R I

R O

 

 

 

 

 

−

8 (w 0 I w 0 O ∆ x)2

 

k2 ψ2

 

2

 

2

 

 

 

 

 

 

 

 

 

(w20 I + w20 O)3

−

2

 

 

 

 

 

 

 

 

 

Plane of coupling

 

 

 

 

 

 

 

 

 

 

 

w I + w O .

 

 

 

 

R O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(3.1.137)

 

 

w0O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

w

O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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 ].

Landolt-B¨ornstein

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.

Landolt-B¨ornstein

New Series VIII/1A1

References for 3.1

131

 

 

References for 3.1

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28Koe

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der Physik, Vol. 20, Berlin: Springer-Verlag, 1928, p. 141–262.

28Szi

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41Jon

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57Mue

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58Fed

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58Mac

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58Shu

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61Hoe

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61Ram

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Materie in ihren Aggregatzust¨anden, part 8: Optische Konstanten, Hellwege, K.H., Hell-

 

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62Shu

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64Bro

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64Kog

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64Lun

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1964.

64Wol

Wolf, E., Marchand, E.W.: J. Opt. Soc. Am. 54 (1964) 587.

66Kog1

Kogelnik, H., Li, T.: Appl. Opt. 5 (1966) 1550.

66Kog2

Kogelnik, H., Li, T.: Proc. IEEE 54 (1966) 1312.

66Rub

Rubinowicz, A.: Die Beugungswelle in der Kirchho schen Theorie der Beugung,

 

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67Loh

Lohmann, A.W., Paris, D.P.: Appl. Opt. 6 (1967) 1739.

67Rud

Rudolph, D., Schmahl, G.: Umschau Wiss. Tech. 67 (1967) 225.

67Sok

Sokolov, A.V.: Optical properties of metals, New York: American Elsevier, 1967.

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New Series VIII/1A1

132

References for 3.1

 

 

68Goo

Goodman, J.W.: Introduction to Fourier optics, San Francisco: McGraw-Hill, 1968.

68Pap

Papoulis, A: Systems and transforms with applications in optics, New York: McGraw

 

Hill, 1968.

68Ves

Veselago, V.G.: Sov. Phys. Usp. 10 (1968) 509.

69Arn

Arnaud, J.A., Kogelnik, H.: Appl. Opt. 8 (1969) 1687.

69Cam

Campbell, J.P., DeShazer, L.G.: J. Opt. Soc. Am. 59 (1969) 1427.

69Kog

Kogelnik, H.: Bell Syst. Tech. J. 48 (1969) 2909.

69Str

Stroke, G.W.: Di raction gratings, in: Fl¨ugge, S. (ed.): Handbuch der Physik, Vol. 29,

 

Berlin: Springer-Verlag, 1969, p. 425.

70Abr

Abramowitz, M., Steguhn, I.A.: Handbook of mathematical functions, New York: Dover

 

Publications, 1970.

70Ber

Berek, M.: Grundlagen der praktischen Optik, Berlin: Verlag Walter de Gruyter, 1970.

70Col

Collins, S.A.: J. Opt. Soc. Am. 60 (1970) 1168.

71Col

Collier, R.J., Burckhardt, C.B., Lin, L.H.: Optical holography, New York: Academic

 

Press, 1971.

71Pan

Pankove, J.I.: Optical processes in semiconductors, New York: Dover, 1971.

71Sch

Schell, R.G., Tyras, G.: J. Opt. Soc. Am. 61 (1971) 31.

72Mar

Marcuse, D.: Light transmission optics, New York: Van Nostrand Reinhold Company,

 

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72Sta

Stavroudis, O.N.: The optics of rays, wavefronts and caustics, New York: Academic

 

Press, 1972.

72Woo

Wooten, F.: Optical properties of solids, New York: Academic Press, 1972.

73Men

Menzel, E., Mirande, W., Weing¨artner, I.: Fourier-Optik und Holographie, Wien:

 

Springer-Verlag, 1973.

74Jun

Junghans, J., Keller, M., Weber, H.: Appl. Opt. 13 (1974) 2793.

74Sle

Slevogt, H.: Technische Optik, Berlin: Walter de Gruyter, 1974.

74Wel

Welford, W.T.: Aberrations of the symmetrical optical system, London: Academic Press,

 

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75Ger

Gerrard, A., Burch, J.M.: Introduction to matrix methods in optics, London: John Wiley

 

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75Jac

Jackson, J.D.: Classical electrodynamics, New York: John Wiley & Sons, 1975.

75Lax

Lax, M., Louisell, W.H., McKnight, W.B.: Phys. Rev. A 11 (1975) 1365.

76Arn

Arnaud, J.: Beam and fiber optics, New York: Academic Press, 1976.

76Fed

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77Azz

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