Abstract

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.

© 2012 Optical Society of America

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References

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    [CrossRef]
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  7. P. Bhattacharjee, “Exhaustive study of reflection and refraction at spherical surfaces on the basis of the newly discovered generalized vectorial laws of reflection and refraction,” Optik, 381–386 (2011).
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  8. E. Hecht, Optics (Addison-Wesley, 1987).
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    [CrossRef]
  10. The generating MATLAB scripts are available upon request.
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  14. N. Yu, P. Genevet, M. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
    [CrossRef]
  15. C. Perwass, Geometric Algebra with Applications in Engineering (Springer-Verlag, 2009), p. 190.

2012

2011

S. Zhao, K. Wang, F. Chen, D. Wu, and S. Liu, “Lens design of LED searchlight of high brightness and distant spot,” J. Opt. Soc. Am. A 28, 815–820 (2011).
[CrossRef]

E. Tkaczyk, T. Mauring, M. Pajusalu, A. Anijalg, A. Tkaczyk, P. Teesalu, J. Kikas, and K. Mauring, “Cataract diagnosis by measurement of backscattered light,” Opt. Lett. 36, 4707–4709 (2011).
[CrossRef]

P. Bhattacharjee, “Exhaustive study of reflection and refraction at spherical surfaces on the basis of the newly discovered generalized vectorial laws of reflection and refraction,” Optik, 381–386 (2011).
[CrossRef]

N. Yu, P. Genevet, M. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[CrossRef]

2008

2007

2005

P. Bhattacharjee, “The generalized vectorial laws of reflection and refraction,” Eur. J. Phys. 26, 901–911 (2005).
[CrossRef]

2001

1983

1980

1908

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Aieta, F.

N. Yu, P. Genevet, M. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[CrossRef]

Anijalg, A.

Bhattacharjee, P.

P. Bhattacharjee, “Exhaustive study of reflection and refraction at spherical surfaces on the basis of the newly discovered generalized vectorial laws of reflection and refraction,” Optik, 381–386 (2011).
[CrossRef]

P. Bhattacharjee, “The generalized vectorial laws of reflection and refraction,” Eur. J. Phys. 26, 901–911 (2005).
[CrossRef]

Capasso, F.

N. Yu, P. Genevet, M. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[CrossRef]

Chen, F.

Gaburro, Z.

N. Yu, P. Genevet, M. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[CrossRef]

Genevet, P.

N. Yu, P. Genevet, M. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[CrossRef]

Hecht, E.

E. Hecht, Optics (Addison-Wesley, 1987).

Kats, M.

N. Yu, P. Genevet, M. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[CrossRef]

Kikas, J.

Lieb, M.

Liu, S.

Mauring, K.

Mauring, T.

Meixner, A.

Meixner, A. J.

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Oakley, J.

Pajusalu, M.

Perwass, C.

C. Perwass, Geometric Algebra with Applications in Engineering (Springer-Verlag, 2009), p. 190.

Self, S.

Stadler, J.

Stanciu, C.

Stupperich, C.

Teesalu, P.

Tetienne, J.-P.

N. Yu, P. Genevet, M. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[CrossRef]

Tkaczyk, A.

Tkaczyk, E.

Wang, K.

Wiscombe, W. J.

Wu, D.

Yu, N.

N. Yu, P. Genevet, M. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[CrossRef]

Zhao, S.

Ann. Phys.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Appl. Opt.

Eur. J. Phys.

P. Bhattacharjee, “The generalized vectorial laws of reflection and refraction,” Eur. J. Phys. 26, 901–911 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Optik

P. Bhattacharjee, “Exhaustive study of reflection and refraction at spherical surfaces on the basis of the newly discovered generalized vectorial laws of reflection and refraction,” Optik, 381–386 (2011).
[CrossRef]

Science

N. Yu, P. Genevet, M. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[CrossRef]

Other

C. Perwass, Geometric Algebra with Applications in Engineering (Springer-Verlag, 2009), p. 190.

E. Hecht, Optics (Addison-Wesley, 1987).

The generating MATLAB scripts are available upon request.

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Figures (9)

Fig. 1.
Fig. 1.

Geometry for determination of position Q where a reflected light ray q strikes a known plane from (a) a convex ( ρ > 0 ) or (b) a concave ( ρ < 0 ) spherical surface centered at C .

Fig. 2.
Fig. 2.

Geometry for calculation of intersection point (small circle) of the half-maximum ray of a Gaussian beam with a paraboloid. The paraboloid illustrated here is the closest second-order Taylor approximation to a sphere of radius ρ , with axis parallel to the laser beam’s axis. The paraboloid has axis parallel to and displaced from the laser axis by ρ / 2 . The Gaussian beam here has the following properties: wavelength λ = ρ / 10 , and the fwhm at beam focus ( fwhm 0 ) is ρ / 10 .

Fig. 3.
Fig. 3.

Intensity profile of a Gaussian laser beam [ λ = 650 nm , fwhm at waist ( fwhm 0 ) is 1 mm], calculated in the detector plane (coincident with beam waist position z = 0 ) after reflection by a sphere ( ρ = 8 mm , center z C = 48 mm ) displaced from the laser central axis by varying amounts ( y C = 0 , 1.5, or 3 mm). (a) Cross-sectional vectorial calculation and (b) lens-law approximation.

Fig. 4.
Fig. 4.

(a) Reflected beam position in the photodetector plane, per lens approximation (circles) or exact vectorial calculation (solid curve). The position of the beam’s center ( y Q for the ray of maximum intensity) is shown for different displacements of the center of the convex reflecting sphere from the laser beam axis ( y C ). The photodetector plane is at position z P = 0 . Remaining calculation parameters are the same as in Fig. 3: λ = 650 nm , ρ = 8 mm (convex sphere), z C = 48 mm . Results are independent of beam waist size fwhm 0 . (b) Reflection by the convex paraboloid most closely approximating the convex sphere of (a). (c) Reflection by the concave sphere (opposite sign of ρ relative to plot (a). (d) Refraction by the convex sphere. The photodetector plane is at position z P = z C ρ / 2 = 44 mm .

Fig. 5.
Fig. 5.

(a) Ratio of the exact to the lens-approximated reflected beam magnification in the photodetector plane for different convex reflecting sphere center displacements y C . The ratio of the fwhm measurements is shown in both the y (the direction of the sphere center displacement from the axis) (no crosses) and the x directions (crosses). The plots are for beam waist size fwhm 0 = 0.5 mm (solid curve) or fwhm 0 = 2.0 mm (dotted curve). The remaining calculation parameters are the same as in Fig. 4. Note that, by symmetry, the sign of ρ (convex or concave surface) does not affect the calculation. (b) Reflection by the paraboloid most closely approximating the sphere. (c) Refraction by a sphere. (d) Reflection by concave sphere.

Fig. 6.
Fig. 6.

(a) Geometry of the position Q where a reflected light ray q strikes a known plane from a convex ( ρ > 0 ) paraboloid of revolution with focal point F , axis u ^ , and focal length ρ / 2 . (b) Focusing property for reflection at point J on a concave ( ρ < 0 ) paraboloid of revolution.

Fig. 7.
Fig. 7.

Reflection of three parallel rays from spherical and paraboloidal surfaces. Surfaces and their normals are shown in black; wave vectors in red. This calculation accurately represents the axis and half-maximum edges of a Gaussian laser beam (in red) with λ ρ . (a) A concave spherical reflecting surface does not reflect all three rays to the same point. (b) The sphere’s nearest approximating paraboloid reflects all three rays to its focal point. The generating MATLAB script is available [10].

Fig. 8.
Fig. 8.

Geometry of light emanating from a point K , striking a reflecting (a) convex or (b) concave spherical or paraboloidal surface at J , and reflecting to a point Q on a photodetector a distance d PD from the apex of the surface.

Fig. 9.
Fig. 9.

Geometry of light emanating from a point K , striking a refracting (a) convex or (b) concave spherical or paraboloidal surface at J , and refracting to a point Q on a photodetector a distance d PD from the apex of the surface.

Tables (2)

Tables Icon

Table 1. Absolute Magnification of fwhm of Beam by Reflection (Refraction in Final Row) to Photodetector from Various Surfaces, in the Direction of Displacement ( y Magnification) (italics) and the Perpendicular Direction ( x Magnification), by Exact Vectorial Calculation (exact) or Approximation with the Lens Law or Cross-Sectional Vectorial Calculation (cross)a

Tables Icon

Table 2. For Each Geometry, the Value of y C (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 y Direction (top) or x Direction (bottom) from the Lens-Law Approximationa

Equations (40)

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d o 1 + [ d i + z R 2 / ( d i f ) ] 1 = f 1 .
( w ( z ) / w 0 ) 2 = 1 + ( z / z R ) 2 , R ( z ) / z = 1 + ( z R / z ) 2 , I ( r , z ) = ( w ( z ) / w 0 ) 2 exp ( 2 r 2 / w ( z ) 2 ) .
( A ) | A | , the length of A a ^ A / | A | , the unit vector in the direction of A A v v A A · v e a e ^ · a ^ A E E A ( A E ) v A E · v ( A E ) | A E | .
P O P the vector from O to the point P ( P N ) | P N | = distance from P to N .
y P = P · y = P y = ( O P ) y = O P · y = ( P O ) · y .
Q = J + ( ( J P ) p / q p ) q .
q = k 2 n k n ,
q = k ( sqrt ( k n 2 + ( μ 2 1 ) | k | 2 ) + k n ) n .
J = K + t k ,
n = C J / ρ = ( C K + t k ) / ρ .
( k ) 2 t 2 + 2 ( C K ) k t + ( C K ) 2 ρ 2 = 0.
n = sign ( ρ ) F J / | F J | .
J = J ( u ^ · F J + ρ ) u ^ .
( J J ) = J J · u ^ = ( J K + K J ) · u ^ = ( J K ) u + t k u = ( F K ) u + t k u , ( J J ) 2 = ( k u ) 2 t 2 + 2 ( F K ) u k u t + ( F K ) u 2 ,
( F J ) 2 = | F K + K J | 2 = | F K + t k | 2 = ( F K + t k ) · ( F K + t k ) = ( k ) 2 t 2 + 2 ( F K ) k t + ( F K ) 2 .
[ ( k u ) 2 ( k ) 2 ] t 2 + 2 [ ( F K ) u k u ( F K ) k ] t + [ ( F K ) u 2 ( F K ) 2 ] = 0.
α H J F , β F H J , γ J F H .
y J 2 = ( τ w 0 ) 2 + ( τ w 0 / z R ) 2 z J 2 .
( 0 x C ) 2 + ( y J y C ) 2 + ( z J z C ) 2 = ρ 2 .
2 y C · y J = 2 y C · a 2 · z J 2 + 2 y C · a 1 · z J + 2 y C · a 0 ,
2 · y C · a 2 = 1 + ( τ · w 0 / z R ) 2 , 2 · y C · a 1 = 2 · z C , 2 · y C · a 0 = ( τ · w 0 ) 2 + ( x C 2 + y C 2 + z C 2 ρ 2 ) .
0 = b 4 · z J 4 + b 3 · z J 3 + b 2 · z J 2 + b 1 · z J + b 0 , with b 4 = a 2 2 , b 3 = 2 · a 2 · a 1 , b 2 = 2 · a 2 · a 0 + a 2 2 ( τ · w 0 / z R ) 2 , b 1 = 2 · a 1 · a 0 , b 0 = a 0 2 ( τ · w 0 ) 2 .
( 0 x C ) 2 + ( y J y C ) 2 2 ρ ( z J z C ) = 2 ρ 2 .
2 · y C · a 2 = ( τ · w 0 / z R ) 2 , 2 · y C · a 1 = 2 · ρ , 2 · y C · a 0 = ( τ · w 0 ) 2 + ( x C 2 + y C 2 + 2 · ρ · z C 2 ρ 2 ) .
h o ( C O ) y , h i ( C I ) y , or y I = ( h i y C ) ,
d o ( O J ) z , d i ( I J ) z .
( y Q h i + h o ) / ( d i + d PD ) = ( y J ( h i h o ) ) / ( d i ) .
y J / d o = y K / z K .
( y Q h i + h o ) / ( d i + d PD ) = ( d o y K ( h i h o ) z K ) / ( d i z K ) .
y Q = d o y K ( d i + d PD ) / ( d i z K ) + [ M o 1 ] ( d PD h o / d i ) ,
M PD d o ( d i + d PD ) / ( d i z K ) .
M o = d i / d o = f i / ( f i d o ) .
y Q = M PD y K ( d PD / f i ) h o .
M PD = ( f i d o + f i d PD d o d PD ) / ( f i z K ) , with f i = ρ / 2.
d o ( O J ) z , d i ( I J ) z .
M o = d i / ( d o n i / n o ) = f i / ( f i d o n i / n o ) .
M PD = ( n i f d o + n o f d PD d o d PD ) / ( n i f z K ) , with f = ρ / ( n i n o ) .
M PD = 1 ( d PD / f i ) .
[ y Q + h o y z | Q ] = [ 1 d PD 0 1 ] · [ 1 0 2 ρ 1 ] · [ 1 d o z K 0 1 ] · [ y K + h o y K z K ] ,
[ y Q + h o y z | Q ] = [ 1 d PD 0 1 ] · [ 1 0 n o n i n i ρ n o n i ] · [ 1 d o z K 0 1 ] · [ y K + h o y K z K ] .

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