Abstract

Snell’s law describes the relationship between the incidence angle and reflection (or refraction) angle of a light ray impinging on the interface between two different isotropic media. In this paper, Snell’s law is used to derive the unit normal vectors of an aspherical surface given a knowledge of the unit directional vectors of the incoming and outgoing rays. The proposed method has important applications in the design and fabrication of aspherical surfaces since the surface normal vectors determine not only the optical performance of the surface but also the cutting tool angles required to machine the surfaces.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Takaoka, N. Kawano, Y. Awatsuji, and T. Kubota, “Design of a reflective aspherical surface of a compact beam-shaping device,” Opt. Rev. 13, 77–86 (2006).
    [CrossRef]
  2. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4, 1400–1403 (1965).
    [CrossRef]
  3. J. J. M. Braat and P. F. Greve, “Aplanatic optical system containing two aspheric surfaces,” Appl. Opt. 18, 2187–2191 (1979).
    [CrossRef]
  4. E. M. Vaskas, “Note on the Wasserman–Wolf method for designing aspherical surfaces,” J. Opt. Soc. Am. 47, 669–670 (1957).
    [CrossRef]
  5. C. Zhao and J. H. Burge, “Application of the pupil astigmatism criteria in optical design,” Appl. Opt. 41, 7288–7293 (2002).
    [CrossRef]
  6. H. Howden and J. A. Clarke, “Refracting replica aspheric optics,” Opt. Eng. 15, 197–201 (1976).
  7. J. Haisma, E. Hugues, and C. Babolat, “Realization of a bi-aspherical objective lens for the Philips video long play system,” Opt. Lett. 4, 70–72 (1979).
    [CrossRef]
  8. G. Boothroyd, Fundamentals of Metal Machining and Machine Tools (McGraw-Hill, 1975), Chap. 7.
  9. A. Q. Biddut, M. Rahman, K. S. Neo, K. M. Rezaur, M. Sawa, and Y. Maeda, “Performance of single crystal diamond tools with different rake angles during micro-grooving on electroless nickel plated die materials,” Int. J. Adv. Manuf. Technol. 33, 891–899 (2007).
    [CrossRef]
  10. R. P. Paul, Robot Manipulators: Mathematics, Programming, and Control (MIT, 1982).

2007 (1)

A. Q. Biddut, M. Rahman, K. S. Neo, K. M. Rezaur, M. Sawa, and Y. Maeda, “Performance of single crystal diamond tools with different rake angles during micro-grooving on electroless nickel plated die materials,” Int. J. Adv. Manuf. Technol. 33, 891–899 (2007).
[CrossRef]

2006 (1)

T. Takaoka, N. Kawano, Y. Awatsuji, and T. Kubota, “Design of a reflective aspherical surface of a compact beam-shaping device,” Opt. Rev. 13, 77–86 (2006).
[CrossRef]

2002 (1)

1979 (2)

1976 (1)

H. Howden and J. A. Clarke, “Refracting replica aspheric optics,” Opt. Eng. 15, 197–201 (1976).

1965 (1)

1957 (1)

Awatsuji, Y.

T. Takaoka, N. Kawano, Y. Awatsuji, and T. Kubota, “Design of a reflective aspherical surface of a compact beam-shaping device,” Opt. Rev. 13, 77–86 (2006).
[CrossRef]

Babolat, C.

Biddut, A. Q.

A. Q. Biddut, M. Rahman, K. S. Neo, K. M. Rezaur, M. Sawa, and Y. Maeda, “Performance of single crystal diamond tools with different rake angles during micro-grooving on electroless nickel plated die materials,” Int. J. Adv. Manuf. Technol. 33, 891–899 (2007).
[CrossRef]

Boothroyd, G.

G. Boothroyd, Fundamentals of Metal Machining and Machine Tools (McGraw-Hill, 1975), Chap. 7.

Braat, J. J. M.

Burge, J. H.

Clarke, J. A.

H. Howden and J. A. Clarke, “Refracting replica aspheric optics,” Opt. Eng. 15, 197–201 (1976).

Frieden, B. R.

Greve, P. F.

Haisma, J.

Howden, H.

H. Howden and J. A. Clarke, “Refracting replica aspheric optics,” Opt. Eng. 15, 197–201 (1976).

Hugues, E.

Kawano, N.

T. Takaoka, N. Kawano, Y. Awatsuji, and T. Kubota, “Design of a reflective aspherical surface of a compact beam-shaping device,” Opt. Rev. 13, 77–86 (2006).
[CrossRef]

Kubota, T.

T. Takaoka, N. Kawano, Y. Awatsuji, and T. Kubota, “Design of a reflective aspherical surface of a compact beam-shaping device,” Opt. Rev. 13, 77–86 (2006).
[CrossRef]

Maeda, Y.

A. Q. Biddut, M. Rahman, K. S. Neo, K. M. Rezaur, M. Sawa, and Y. Maeda, “Performance of single crystal diamond tools with different rake angles during micro-grooving on electroless nickel plated die materials,” Int. J. Adv. Manuf. Technol. 33, 891–899 (2007).
[CrossRef]

Neo, K. S.

A. Q. Biddut, M. Rahman, K. S. Neo, K. M. Rezaur, M. Sawa, and Y. Maeda, “Performance of single crystal diamond tools with different rake angles during micro-grooving on electroless nickel plated die materials,” Int. J. Adv. Manuf. Technol. 33, 891–899 (2007).
[CrossRef]

Paul, R. P.

R. P. Paul, Robot Manipulators: Mathematics, Programming, and Control (MIT, 1982).

Rahman, M.

A. Q. Biddut, M. Rahman, K. S. Neo, K. M. Rezaur, M. Sawa, and Y. Maeda, “Performance of single crystal diamond tools with different rake angles during micro-grooving on electroless nickel plated die materials,” Int. J. Adv. Manuf. Technol. 33, 891–899 (2007).
[CrossRef]

Rezaur, K. M.

A. Q. Biddut, M. Rahman, K. S. Neo, K. M. Rezaur, M. Sawa, and Y. Maeda, “Performance of single crystal diamond tools with different rake angles during micro-grooving on electroless nickel plated die materials,” Int. J. Adv. Manuf. Technol. 33, 891–899 (2007).
[CrossRef]

Sawa, M.

A. Q. Biddut, M. Rahman, K. S. Neo, K. M. Rezaur, M. Sawa, and Y. Maeda, “Performance of single crystal diamond tools with different rake angles during micro-grooving on electroless nickel plated die materials,” Int. J. Adv. Manuf. Technol. 33, 891–899 (2007).
[CrossRef]

Takaoka, T.

T. Takaoka, N. Kawano, Y. Awatsuji, and T. Kubota, “Design of a reflective aspherical surface of a compact beam-shaping device,” Opt. Rev. 13, 77–86 (2006).
[CrossRef]

Vaskas, E. M.

Zhao, C.

Appl. Opt. (3)

Int. J. Adv. Manuf. Technol. (1)

A. Q. Biddut, M. Rahman, K. S. Neo, K. M. Rezaur, M. Sawa, and Y. Maeda, “Performance of single crystal diamond tools with different rake angles during micro-grooving on electroless nickel plated die materials,” Int. J. Adv. Manuf. Technol. 33, 891–899 (2007).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

H. Howden and J. A. Clarke, “Refracting replica aspheric optics,” Opt. Eng. 15, 197–201 (1976).

Opt. Lett. (1)

Opt. Rev. (1)

T. Takaoka, N. Kawano, Y. Awatsuji, and T. Kubota, “Design of a reflective aspherical surface of a compact beam-shaping device,” Opt. Rev. 13, 77–86 (2006).
[CrossRef]

Other (2)

G. Boothroyd, Fundamentals of Metal Machining and Machine Tools (McGraw-Hill, 1975), Chap. 7.

R. P. Paul, Robot Manipulators: Mathematics, Programming, and Control (MIT, 1982).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

Schematic illustration showing that unit normal vector n¯i of refractive surface at incidence point P¯i can be determined by rotating unit directional vector ¯i1 about m¯i by an angle θi.

Fig. 2.
Fig. 2.

Schematic illustration showing that unit normal vector n¯i of a reflective surface at incidence point P¯i can be determined by rotating unit directional vector ¯i1 about m¯i by an angle θi.

Fig. 3.
Fig. 3.

Use of aspherical mirror to correct all spherical aberration of any order by means of Fermat’s principle.

Fig. 4.
Fig. 4.

Use of plano-aspherical lens to correct all spherical aberration of any order by means of Fermat’s principle.

Fig. 5.
Fig. 5.

Generation curve of single lens having one flat refractive surface and one aspherical refractive surface (λ1/axis=200mm, λ2/axis=10mm, λ3/axis=50mm, ξ0=ξ2=1, and ξ1=1.65).

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

¯i1·¯i=C(θiθ̲i)=CθiCθ̲i+SθiSθ̲i,
Sθ̲i=(ξi1/ξi)Sθi=NiSθi,
Sθi=1(¯i1·¯i)2Ni2+12Ni(¯i1·¯i),
Cθi=|Ni(¯i1·¯i)|Ni2+12Ni(¯i1·¯i),
Sθ̲i=Ni1(¯i1·¯i)2Ni2+12Ni(¯i1·¯i),
Cθ̲i=|1Ni(¯i1·¯i)|Ni2+12Ni(¯i1·¯i).
m¯i=[mixmiymiz0]T=¯i1ׯiS(θiθ̲i).
S(θiθ̲i)(m¯iׯi1)=(¯i1ׯi)ׯi1=¯i(¯i1·¯i1)¯i1(¯i·¯i1)=¯i¯i1C(θiθ̲i).
n¯i=[nixniyniz0]=[mix2(1Cθi)+Cθimiymix(1Cθi)mizSθimizmix(1Cθi)+miySθi0mixmiy(1Cθi)+mizSθimiy2(1Cθi)+Cθimizmiy(1Cθi)mixSθi0mixmiz(1Cθi)miySθimiymiz(1Cθi)+mixSθimiz2(1Cθi)+Cθi00001][i1xi1yi1z0].
n¯i=[nixniyniz0]=[C(θiθ̲i)SθiS(θiθ̲i)Cθi][i1xi1yi1z0][SθiS(θiθ̲i)][ixiyiz0]=[(CθiCθ̲i+SθiSθ̲i)SθiSθiCθ̲iCθiSθ̲iCθi][i1xi1yi1z0][SθiSθiCθ̲iCθiSθ̲i][ixiyiz0],
m¯i=[mixmiymiz0]T=¯iׯi1S(2θi),
C(2θi)=¯i1·¯i.
S(2θi)(m¯iׯi1)=(¯iׯi1)ׯi1=¯i1(¯i·¯i1)¯i(¯i1·¯i1)=C(2θi)¯i1¯i.
n¯i=[nixniyniz0]=12Cθi([ixiyiz0][i1xi1yi1z0])=12(1¯i1·¯i)([ixiyiz0][i1xi1yi1z0]).
ε¯i=[0εiyεiz0]T=±[0nizniy0]T.
¯0=[00y0z0]=[0Cβ0Sβ00],
¯1=[01y1z0]=1(P2yP1y)2+(P2zP1z)2[0P2yP1yP2zP1z0]=1(λ2/axis+λ1/axisCβ0λ1)2+(Sβ0λ1)2[0λ2/axis+λ1/axisCβ0λ1Sβ0λ10],
OPLsystem=ξ0λ1+ξ0λ2=ξ0λ1+ξ0(λ2/axis+λ1/axisCβ0λ1)2+(Sβ0λ1)2=ξ0(λ1/axis+λ2/axis+2λ3/axis)=constant,
(y0+150)2(300)2+z02(5011)2=1.
¯2=[02y2z0]=1(P3yP2y)2+(P3zP2z)2[0P3yP2yP3zP2z0]=1[λ2/axis+λ3/axis(P1y+1yλ2)]2+(P1z+1zλ2)2[0λ2/axis+λ3/axis(P1y+1yλ2)(P1z+1zλ2)0],
OPLsystem=ξ0λ1+ξ1λ2+ξ2λ3=ξ0λ1/axis+ξ1λ2/axis+ξ2λ3/axis.

Metrics