Abstract

We obtain simple exact formulas for the refracted wavefronts through plano-convex aspheric lenses with arbitrary aspheric terms by considering an incident plane wavefront propagating along the optical axis. We provide formulas for the zero-distance phase front using the Huygens’ Principle and the Malus-Dupin theorem. Using the fact that they are equivalent, we have in the second method found a way to use an improper integral, instead of the usual evaluated integral, to arrive at these formulas. As expected, when the condition of total internal reflection is satisfied, there is no contribution to the formation of the refracted wavefront.

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References

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  1. A. Epple and H. Wang, “Design to manufacture from the perspective of optical design and fabrication,” Optical Fabrication and Testing, OSA Technical Digest, OFB1. (2008).
  2. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-function and its Ramifications (Wiley-VCH Verlag GmbH & Co.KGaA, 2006), Chap. 12, 179–186.
    [CrossRef]
  3. D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A, 25, 2370–2382 (2008).
    [CrossRef]
  4. J. A. Hoffnagle and D. L. Shealy, “Refracting the k-function: Stavroudis’s solution to the eikonal equation for multielement optical systems,” J. Opt. Soc. Am. A, 28, 1312–1321 (2011).
    [CrossRef]
  5. M. Avendaño-Alejo, “Caustics in a meridional plane produced by plano-convex aspheric lenses,” J. Opt. Soc. Am. A, 30, 501–508 (2013).
    [CrossRef]
  6. G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Exp., 18, 19700–19712 (2010).
    [CrossRef]
  7. G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Exp., 15, 5218–5226 (2007).
    [CrossRef]
  8. M. Avendaño-Alejo and O. N. Stavroudis, “Huygens’s Principle and Rays in Uniaxial Anisotropic Media II. Crystal Axis with Arbitrary Orientation,” J. Opt. Soc. Am. A.19, 1674–1679 (2002).
    [CrossRef]
  9. F. A. Jenkins and H. E. White, Fundamentals of Optics (Mc. Graw-Hill, 1976), Chap. 3, 44–57.
  10. S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, London, 1984), 11–35.
  11. M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” Journal of Microscopy, 197, 219–223 (2000).
    [CrossRef] [PubMed]
  12. M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive index mismatched media,” Journal of Microscopy, 192, 90–98 (1998).
    [CrossRef]
  13. B. Scherger, C. Jordens, and M. Koch, “Variable-focus terahertz lens,” Opt. Exp., 19, 4528–4535 (2011).
    [CrossRef]

2013

2011

2010

G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Exp., 18, 19700–19712 (2010).
[CrossRef]

2008

2007

G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Exp., 15, 5218–5226 (2007).
[CrossRef]

2002

M. Avendaño-Alejo and O. N. Stavroudis, “Huygens’s Principle and Rays in Uniaxial Anisotropic Media II. Crystal Axis with Arbitrary Orientation,” J. Opt. Soc. Am. A.19, 1674–1679 (2002).
[CrossRef]

2000

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” Journal of Microscopy, 197, 219–223 (2000).
[CrossRef] [PubMed]

1998

M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive index mismatched media,” Journal of Microscopy, 192, 90–98 (1998).
[CrossRef]

Avendaño-Alejo, M.

M. Avendaño-Alejo, “Caustics in a meridional plane produced by plano-convex aspheric lenses,” J. Opt. Soc. Am. A, 30, 501–508 (2013).
[CrossRef]

M. Avendaño-Alejo and O. N. Stavroudis, “Huygens’s Principle and Rays in Uniaxial Anisotropic Media II. Crystal Axis with Arbitrary Orientation,” J. Opt. Soc. Am. A.19, 1674–1679 (2002).
[CrossRef]

Booth, M. J.

M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive index mismatched media,” Journal of Microscopy, 192, 90–98 (1998).
[CrossRef]

Cornbleet, S.

S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, London, 1984), 11–35.

Epple, A.

A. Epple and H. Wang, “Design to manufacture from the perspective of optical design and fabrication,” Optical Fabrication and Testing, OSA Technical Digest, OFB1. (2008).

Forbes, G. W.

G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Exp., 18, 19700–19712 (2010).
[CrossRef]

G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Exp., 15, 5218–5226 (2007).
[CrossRef]

Hoffnagle, J. A.

Jenkins, F. A.

F. A. Jenkins and H. E. White, Fundamentals of Optics (Mc. Graw-Hill, 1976), Chap. 3, 44–57.

Jordens, C.

B. Scherger, C. Jordens, and M. Koch, “Variable-focus terahertz lens,” Opt. Exp., 19, 4528–4535 (2011).
[CrossRef]

Juskaitis, R.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” Journal of Microscopy, 197, 219–223 (2000).
[CrossRef] [PubMed]

Koch, M.

B. Scherger, C. Jordens, and M. Koch, “Variable-focus terahertz lens,” Opt. Exp., 19, 4528–4535 (2011).
[CrossRef]

Neil, M. A. A.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” Journal of Microscopy, 197, 219–223 (2000).
[CrossRef] [PubMed]

M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive index mismatched media,” Journal of Microscopy, 192, 90–98 (1998).
[CrossRef]

Scherger, B.

B. Scherger, C. Jordens, and M. Koch, “Variable-focus terahertz lens,” Opt. Exp., 19, 4528–4535 (2011).
[CrossRef]

Shealy, D. L.

Stavroudis, O. N.

M. Avendaño-Alejo and O. N. Stavroudis, “Huygens’s Principle and Rays in Uniaxial Anisotropic Media II. Crystal Axis with Arbitrary Orientation,” J. Opt. Soc. Am. A.19, 1674–1679 (2002).
[CrossRef]

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-function and its Ramifications (Wiley-VCH Verlag GmbH & Co.KGaA, 2006), Chap. 12, 179–186.
[CrossRef]

Wang, H.

A. Epple and H. Wang, “Design to manufacture from the perspective of optical design and fabrication,” Optical Fabrication and Testing, OSA Technical Digest, OFB1. (2008).

White, H. E.

F. A. Jenkins and H. E. White, Fundamentals of Optics (Mc. Graw-Hill, 1976), Chap. 3, 44–57.

Wilson, T.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” Journal of Microscopy, 197, 219–223 (2000).
[CrossRef] [PubMed]

M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive index mismatched media,” Journal of Microscopy, 192, 90–98 (1998).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A.

M. Avendaño-Alejo and O. N. Stavroudis, “Huygens’s Principle and Rays in Uniaxial Anisotropic Media II. Crystal Axis with Arbitrary Orientation,” J. Opt. Soc. Am. A.19, 1674–1679 (2002).
[CrossRef]

Journal of Microscopy

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” Journal of Microscopy, 197, 219–223 (2000).
[CrossRef] [PubMed]

M. J. Booth, M. A. A. Neil, and T. Wilson, “Aberration correction for confocal imaging in refractive index mismatched media,” Journal of Microscopy, 192, 90–98 (1998).
[CrossRef]

Opt. Exp.

B. Scherger, C. Jordens, and M. Koch, “Variable-focus terahertz lens,” Opt. Exp., 19, 4528–4535 (2011).
[CrossRef]

G. W. Forbes, “Robust, efficient computational methods for axially symmetric optical aspheres,” Opt. Exp., 18, 19700–19712 (2010).
[CrossRef]

G. W. Forbes, “Shape specification for axially symmetric optical surfaces,” Opt. Exp., 15, 5218–5226 (2007).
[CrossRef]

Other

F. A. Jenkins and H. E. White, Fundamentals of Optics (Mc. Graw-Hill, 1976), Chap. 3, 44–57.

S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, London, 1984), 11–35.

A. Epple and H. Wang, “Design to manufacture from the perspective of optical design and fabrication,” Optical Fabrication and Testing, OSA Technical Digest, OFB1. (2008).

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-function and its Ramifications (Wiley-VCH Verlag GmbH & Co.KGaA, 2006), Chap. 12, 179–186.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(a) Process of refraction produced by a plano-convex aspheric lens, and its associated parameters, considering that the point source is located at infinity. (b) Wavefronts produced by the wavelets.

Fig. 2
Fig. 2

Wavelets and wavefronts produced by a plano-convex aspheric lens, considering a plane wavefront incident on the lens. (a) Without TIR. (b) Showing TIR.

Fig. 3
Fig. 3

(a) Process to construct Parallel Curves. (b) Wavefronts propagating along the optical axis for different lengths .

Fig. 4
Fig. 4

(a) Plane wavefront refracted by a convex-plano aspheric lens, and its associated parameters. (b) Wavelets and their wavefronts associates after refraction.

Fig. 5
Fig. 5

(a) Plane wavefront refracted by a convex-plano aspheric lens, and its associated parameters. (b) Wavelets and their wavefronts associates after translation.

Fig. 6
Fig. 6

(a) Plane wavefront refracted by a convex-plano aspheric lens, and its associated parameters. (b) Wavelets and their wavefronts associates after refraction.

Fig. 7
Fig. 7

(a) Wavefronts propagating along the optical axis for different lengths . (b) Zoom of the wavefronts propagating near to effective focal length.

Tables (1)

Tables Icon

Table 1 Aspheric Coefficients

Equations (25)

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( Z [ t + S h ] ) 2 + ( Y h ) 2 = ( n i n a S h ) 2 , h [ H , H ]
S h = c h 2 1 + 1 ( k + 1 ) c 2 h 2 + i = 2 N A 2 i h 2 i ,
( Y h ) = [ ( Z [ t + S h ] ) + ( n i n a ) 2 S h ] S h ,
S h = c h 1 ( k + 1 ) c 2 h 2 + i = 2 N 2 i A 2 i h 2 i 1 .
Z ± = t + S h n a 2 + ( n a 2 n i 2 ) S h 2 [ n a 2 + ( n a 2 n i 2 ) S h 2 ± n i ] n a 2 ( 1 + S h 2 ) , Y ± = h n i S h S h [ n i ± n a 2 + ( n a 2 n i 2 ) S h 2 ] n a 2 ( 1 + S h 2 ) ,
( Z p c , Y p c ) = ( t + ( n a 2 n i 2 ) S h n a 2 + ( n a 2 n i 2 ) S h 2 n a 2 [ n i + n a 2 + ( n a 2 n i 2 ) S h 2 ] , h + n i ( n a 2 n i 2 ) S h S h n a 2 [ n i + n a 2 + ( n a 2 n i 2 ) S h 2 ] ) ,
[ F , G ] = [ f , g ] [ h [ f , g ] ( f / h ) 2 + ( g / h ) 2 ] d h ( f / h ) 2 + ( g / h ) 2 ,
z p c = t + S h + [ n a 2 + ( n a 2 n i 2 ) S h ] [ n a 2 + n i n a 2 + ( n a 2 n i 2 ) S h 2 ] n a 2 ( n a 2 n i 2 ) S h , y p c = h [ n a 2 + ( n a 2 n i 2 ) S h 2 ] S h n a 2 S h ,
𝒲 = ( F ± [ G / h ] ( F / h ) 2 + ( G / h ) 2 , G [ F / h ] ( F / h ) 2 + ( G / h ) 2 ) .
Z p c = Z p c + [ n a 2 + n i n a 2 + ( n a 2 n i 2 ) S h 2 n i + n a 2 + ( n a 2 n i 2 ) S h 2 ] n a , Y p c = Y p c [ [ n a 2 n i 2 ] S h n i + n a 2 + ( n a 2 n i 2 ) S h 2 ] n a ,
( Z S h ) 2 + ( Y h ) 2 = ( n a n i [ S H S h ] ) 2 , h [ H , H ]
Z = S H ( S H S h ) n i 2 + ( n i 2 n a 2 ) S h 2 [ n i 2 + ( n i 2 n a 2 ) S h 2 n a ] n i 2 ( 1 + S h 2 ) , Y = h + n a ( S H S h ) S h [ n a n i 2 + ( n i 2 n a 2 ) S h 2 ] n i 2 ( 1 + S h 2 ) ,
Z in = S H + ( n a 2 n i 2 ) ( S H S h ) n i 2 + ( n i 2 n a 2 ) S h 2 n i 2 ( n a + n i 2 + ( n i 2 n a 2 ) S h 2 ) , Y in = h + n a ( n a 2 n i 2 ) ( S H S h ) S h n i 2 ( n a + n i 2 + ( n i 2 n a 2 ) S h 2 ) .
y i [ h , z ] = h ( n i 2 n a 2 ) ( z S h ) S h n i 2 + n a n i 2 + ( n i 2 n a 2 ) S h 2 ,
H = n i ( t S H ) ( n a + n i 2 + ( n i 2 n a 2 ) S H 2 ) n i 2 + n a n i 2 + ( n i 2 n a 2 ) S H 2 ,
Z in = Z in + [ n i 2 + n a n i 2 + ( n i 2 n a 2 ) S h 2 n a + n i 2 + ( n i 2 n a 2 ) S h 2 ] H n i , Y in = Y in [ ( n i 2 n a 2 ) S h n a + n i 2 + ( n i 2 n a 2 ) S h 2 ] H n i .
Z in [ 0 ] = n a n i 2 t + [ n i 2 t + ( n a 2 n i 2 ) S H ] n i 2 + ( n i 2 n a 2 ) S H 2 n i ( n i 2 + n a n i 2 + ( n i 2 n a 2 ) S H 2 ) , Y in [ 0 ] = 0 .
0 = t Z in [ 0 ] = ( n i n a ) [ n i 2 t + ( [ n a + n i ] S H n i t ) n i 2 + ( n i 2 n a 2 ) S H 2 ] n i ( n i 2 + n a n i 2 + ( n i 2 n a 2 ) S H 2 ) ,
i = ( n i n a ) [ n i 2 t + ( [ n a + n i ] S h n i t ) n i 2 + ( n i 2 n a 2 ) S h 2 ] n i ( n i 2 + n a n i 2 + ( n i 2 n a 2 ) S h 2 ) .
( Z t ) 2 + ( Y y i [ h , t ] ) 2 = ( n i n a i ) 2 , h [ H , H ]
y o [ h , z ] = y i [ h , t ] + ( n i 2 n a 2 ) ( t z ) S h n a 2 ( n a + n i 2 + ( n i 2 n a 2 ) S h 2 ) 2 ( n i 2 n a 2 ) 2 S h 2 ,
a = n a ( n a + n i 2 + ( n i 2 n a 2 ) S h 2 ) ( t z ) n a 2 ( n a + n i 2 + ( n i 2 n a 2 ) S h 2 ) 2 ( n i 2 n a 2 ) 2 S h 2 .
n a 2 ( n a + Δ ) ( z t ) n a 2 ( n a + Δ ) 2 ( n i 2 n a 2 ) 2 S h 2 = ( n i n a ) [ n i 2 t + ( [ n a + n i ] S h n i t ) Δ ] ( n i 2 + n a Δ ) ,
Z c p = t + ( n i n a ) { n i 2 t + ( { n a + n i } S h n i t ) Δ } n a 2 ( n a + Δ ) 2 ( n i 2 n a 2 ) 2 S h 2 n a 2 ( n a + Δ ) ( n i 2 + n a Δ ) , Y c p = h + ( n i 2 n a 2 ) { n i 2 ( n a n i ) t + n a 3 ( S h t ) { ( n a 2 + n a n i n i 2 ) t + ( n i 2 2 n a 2 ) S h } Δ } S h n a 2 ( n a + Δ ) ( n i 2 + n a Δ ) ,
Z c p = Z c p + [ [ Y c p / h ] ( Y c p / h ) 2 + ( Z c p / h ) 2 ] , Y c p = Y c p [ [ Z c p / h ] ( Y c p / h ) 2 + ( Z c p / h ) 2 ] ,

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