Eric R. Tkaczyk, Koit Mauring, and Alan H. Tkaczyk, "Gaussian beam reflection and refraction by a spherical or parabolic surface: comparison of vectorial-law calculation with lens approximation," J. Opt. Soc. Am. A 29, 2144-2153 (2012)
A ray-tracing approach is used to demonstrate efficient application of the
vectorial laws of reflection and
refraction to computational optics
problems. Both the full width at half-maximum (fwhm) and offset of Gaussian
beams resulting from off-center reflection and refraction are calculated for
spherical and paraboloidal surfaces of revolution. It is found that the
magnification and displacement depend nonlinearly on the miscentering. For these
geometries, the limits of accuracy of the lens approximation are examined
quantitatively. In contrast to the ray-tracing solution, this paraxial
approximation would predict a magnification of a beam’s fwhm that is
independent of miscentering, and an offset linearly proportional to the
miscentering. The focusing property of paraboloidal surfaces of revolution is
also derived in setting up the calculation.
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Absolute Magnification of fwhm of Beam by Reflection (Refraction in Final
Row) to Photodetector from Various Surfaces, in the Direction of
Displacement ( Magnification) (italics) and the
Perpendicular Direction
( Magnification), by Exact Vectorial
Calculation (exact) or Approximation with the Lens Law or
Cross-Sectional Vectorial
Calculation (cross)a
Beam and Surface
Lens
Law
Vector
Calculation
No Displacement Magnification
1.5 mm Displacement
Magnification
Magnification
exact
cross
exact
cross
exact
cross
2 mm
Convex sphere
11.000
11.258
11.900
13.388
Concave sphere
Concave paraboloid
Convex sphere (refraction)
0.875
0.876
0.879
0.885
0.5 mm
Convex sphere
11.000
11.016
11.621
12.925
Concave sphere
Concave paraboloid
Convex sphere (refraction)
0.875
0.875
0.878
0.883
The fwhm of the beam at the waist is 0.5 mm for the top half or
2.0 mm for the bottom half of the table. The remaining calculation
parameters are the same as in Fig. 4.
Table 2.
For Each Geometry, the Value of (the Displacement of the Surface Center
from the Laser Axis, in Millimeters) Is Shown, Where the fwhm Magnification
Calculation Gives the Corresponding Deviation
(5%, 10%, or 25%) in the
Direction (top) or
Direction (bottom) from the Lens-Law
Approximationa
Beam and Surface
for
for
for
exact
cross
exact
cross
exact
cross
2 mm
Convex sphere
0.589
0.980
1.605
Concave sphere
0.554
0.936
1.553
Concave paraboloid
0.827
1.272
2.053
Convex sphere (refraction)
3.116
4.088
5.387
0.5 mm
Convex sphere
0.830
1.160
1.756
Concave sphere
0.799
1.117
1.699
Concave paraboloid
1.008
1.417
2.162
Convex
sphere (refraction)
3.207
4.179
5.523
Beam and Surface
for
for
for
exact
cross
exact
cross
exact
cross
2 mm
Convex sphere
1.034
1.701
2.693
Concave sphere
0.972
1.635
2.616
Concave parabola
1.435
2.198
3.434
Convex sphere (refraction)
5.069
6.305
—
—
0.5 mm
Convex sphere
1.413
1.954
2.862
Concave sphere
1.372
1.899
2.789
Concave parabola
1.726
2.398
3.570
Convex sphere (refraction)
5.160
6.350
—
—
Either the exact vector or cross-sectional approach is used. For example,
5% relative error means . The remaining calculation parameters
are the same as in Table 1
and Fig. 4.
Tables (2)
Table 1.
Absolute Magnification of fwhm of Beam by Reflection (Refraction in Final
Row) to Photodetector from Various Surfaces, in the Direction of
Displacement ( Magnification) (italics) and the
Perpendicular Direction
( Magnification), by Exact Vectorial
Calculation (exact) or Approximation with the Lens Law or
Cross-Sectional Vectorial
Calculation (cross)a
Beam and Surface
Lens
Law
Vector
Calculation
No Displacement Magnification
1.5 mm Displacement
Magnification
Magnification
exact
cross
exact
cross
exact
cross
2 mm
Convex sphere
11.000
11.258
11.900
13.388
Concave sphere
Concave paraboloid
Convex sphere (refraction)
0.875
0.876
0.879
0.885
0.5 mm
Convex sphere
11.000
11.016
11.621
12.925
Concave sphere
Concave paraboloid
Convex sphere (refraction)
0.875
0.875
0.878
0.883
The fwhm of the beam at the waist is 0.5 mm for the top half or
2.0 mm for the bottom half of the table. The remaining calculation
parameters are the same as in Fig. 4.
Table 2.
For Each Geometry, the Value of (the Displacement of the Surface Center
from the Laser Axis, in Millimeters) Is Shown, Where the fwhm Magnification
Calculation Gives the Corresponding Deviation
(5%, 10%, or 25%) in the
Direction (top) or
Direction (bottom) from the Lens-Law
Approximationa
Beam and Surface
for
for
for
exact
cross
exact
cross
exact
cross
2 mm
Convex sphere
0.589
0.980
1.605
Concave sphere
0.554
0.936
1.553
Concave paraboloid
0.827
1.272
2.053
Convex sphere (refraction)
3.116
4.088
5.387
0.5 mm
Convex sphere
0.830
1.160
1.756
Concave sphere
0.799
1.117
1.699
Concave paraboloid
1.008
1.417
2.162
Convex
sphere (refraction)
3.207
4.179
5.523
Beam and Surface
for
for
for
exact
cross
exact
cross
exact
cross
2 mm
Convex sphere
1.034
1.701
2.693
Concave sphere
0.972
1.635
2.616
Concave parabola
1.435
2.198
3.434
Convex sphere (refraction)
5.069
6.305
—
—
0.5 mm
Convex sphere
1.413
1.954
2.862
Concave sphere
1.372
1.899
2.789
Concave parabola
1.726
2.398
3.570
Convex sphere (refraction)
5.160
6.350
—
—
Either the exact vector or cross-sectional approach is used. For example,
5% relative error means . The remaining calculation parameters
are the same as in Table 1
and Fig. 4.