Ref. p. 131] |
3.1 Linear optics |
123 |
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3.1.7.2.1.2 Higher-order Hermite-Gaussian beams in simple astigmatic beams
Treatment of the x- or y-component of Hermite-Gaussian-beams after (3.1.27): The complex q-
parameter transformation is treated as above, the fundamental mode part is given as above, the
√
new beam radius for the Hermite polynom of order m, Hm( 2 x/w Ix) is calculated from the new q-parameter and the phase is derived from it, too [70Col].
For complex Hermite-Gaussian beams: see [86Sie].
3.1.7.2.2 General astigmatic beam
In Table 3.1.19 the Q−1-matrix transfer for general astigmatic beams is given. The matrix Q−1 is the matrix scheme of inverses of q-parameters and no inverted matrix [96Gro].
Table 3.1.19. Q−1-matrix transfer for general astigmatic beams.
Given |
Propagated field |
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– General Gaussian beam in the input plane:
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k |
1 r |
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U I(r) = exp −i |
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r Q−I |
, (3.1.118) |
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2 |
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r (x, y) the transverse position vector
perpendicular to the propagation axis z .
– Q−I 1-matrix:
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1 |
1 |
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Q−1 |
= |
qxx |
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qxy |
(3.1.119) |
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I |
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1 |
1 |
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qxy |
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qyy |
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with qxx, qxy , qyy complex terms describing the general amplitudeand phasedistribution of U I , and
r Q−1 r = |
x2 |
+ 2 |
xy |
+ |
y2 |
. (3.1.120) |
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I |
qxx |
qxy |
qyy |
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–S-matrix of the optical system (see Table 3.1.15) with
A B
S = C D
– Transformation of the Q−I 1-matrix to its output value:
Q−O1 = C + D Q−I |
1 A + B Q−I 1 −1 , |
(3.1.121) |
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see [88Sim, 96Gro, 05Hod]. |
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– Field in the output plane: |
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k |
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U O(r) = exp −i |
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r Q−O1 r . |
(3.1.122) |
after (3.1.99).
Example 3.1.16. Transformation of a simple astigmatic Gaussian beam (no mixing between x and y)
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1 |
qyy " |
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I |
0 |
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with a θ-rotated cylindrical lens to a general astigmatic beam: We start with Q−1 = |
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qxx |
0 |
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1 . |
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The rotation of an x-aligned cylindrical lens, given as Scyl in Table 3.1.15, is performed by multiplying first Scyl with the rotation matrix R of Table 3.1.15, and then the product with the inverse of R is:
Landolt-B¨ornstein
New Series VIII/1A1