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Part III

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Figure 45. Sketch of a transverse amplitude and phase variation of a paraxial wave (after [1]).

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Figure 46. A plane wave traveling at a small angle θ to the optic axis (after [1])

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Figure 47. Spherical waveform coming from a real point source (after [1]).

Figure 48. Gaussian-spherical wave from a complex point source (after [1], p. 658).

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Figure 49. Notation for a lowest-order gaussian beam diverging away from its waist (after [1], p. 664).

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Analogy between the lowest-order and higher order solutions for Paraxial wave equation and Schrodinger equation for Hydrogen atom

Paraxial wave equation Schrödinger equation

for resonator of cylindrical symmetry

for resonator of square symmetry

for 3D Hydrogen atom of spherical symmetry

lowest-order solution lowest-order solution lowest-order solution

higher order solutions higher order solutions higher order solutions

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Figure 50. Gaussian beam in a two-mirror resonator.

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Figure 51. Example of an optical resonator or lensguide containig arbitrary paraxial elements.

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Figure 52. Explanation to connection between the parameters of complex Gaussian beams at the input ( 1z ) and output ( 2z ) of an arbitrary ABCD system.

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Figure 53. Sketches of 2D hard aperture and 2D Gaussian aperture. ( )yxt ,~ is 2D amplitude transmission function; 0Re, 22 >∈ aa (after

Error! Reference source not found.).

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Figure 54. A gaussian aperture with transversely varying amplitude transmission ( 0Re, 22 >∈ aa ).

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Figure 55. Explanation to confined Gaussian eigenbeams of resonators / lensguides.

Figure 56. Propagation of Gaussian beam ( λ= 30 mm) in air (after [5]).

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Figure 57. Fractional power transfer of a cylindrical gaussian beam through a circular aperture.

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Figure 58. Explanation to the Rayleigh range & collimated range.

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Figure 59. A gaussian beam spreads with a constant diffraction angle in the far field.

Figure 60. Illustration to the cone, corresponding to the solid angle

e1Ω .

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Figure 61. Intensity distribution for the first 12 Hermite-Gaussian modes (TEMnm modes for n = 0,…,3 & m = 0,…,3 ) (after [5]).

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Figure 62. Intensity distribution for the first 12 Laguerre-Gaussian modes (after [5]).

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Figure 63. Illustration to analysis of a guided gaussian beam in a two-mirror cavity (Problem III - 1).

Figure 64. Explanation to the Problem III - 2.

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Figure 65. Cavity sketch (Problem III - 3).

Figure 66. Hard aperture inserted into the cavity (Problem III - 3).

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References : [1]. A. Siegman. Lasers.(University Science books 1986) [2]. A. Yariv. Quantum electronics (3rd edition, Wiley, 1989) [3]. T. Gavlin, G.Eden (ECE Illinois) Optical Resonator Modes ECE 455 Optical Electronics.

https://courses.engr.illinois.edu/ece455/Files/Galvinlectures/02_CavityModes.pdf [4]. A. Fox, T. Li, Resonant modes in a maser interferometer, Bell Labs 1961. [5]. http://en.wikipedia.org/wiki/Gaussian_beam (as of the date 18.03.2015) [6]. https://ashtriferous.wordpress.com/2012/06/19/activity-2-scilab-basics/ (as of the date

1.03.2017)