Abstract

In this work, Fermat’s principle is applied to derive a simple exact formula for refraction (reflection) in terms of the lengths of the incident and refracted rays. This formula is a nontrigonometric alternative to Snell’s law and is general for all optical surfaces. It is used to derive the paraxial optics equations in a more simple and direct way than that often used in the literature. It’s also applied to derive a new single, exact ray tracing formula for the nonparaxial refraction (reflection) at a single optical surface. The obtained formulas are used to develop a simple ray tracing procedure for meridional refraction through systems of spherical surfaces without the need to use any form of Snell’s law. Numerical examples are provided and discussed.

© 2012 Optical Society of America

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References

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  1. R. P. Feynman, R. B. Leighton, and M. L. Sands, “Optics: the principle of least time,” in The Feynman Lectures on Physics(Addison Wesley Longman, 1970), Vol. 1, Chap. 26.
  2. F. L. Pedrotti and L. S. Pedrotti, “Geometrical optics,” in Introduction to Optics (Prentice-Hall, 1993), pp. 34–36.
  3. P. Mouroulis and J. Macdonald, “Rays and foundations of geometrical optics,” in Geometrical Optics and Optical Design (Oxford University, 1997), pp. 11–13.
  4. R. Kingslake and R. B. Johnson, “Meridional ray tracing,” in Lens Design Fundamentals (Academic, 2010), pp. 25–45.
  5. D. Malacara and Z. Malacara, Handbook of Optical Design(Marcel Dekker, 2004).
  6. W. J. Smith, “Optical computation,” in Modern Optical Engineering (McGraw-Hill, 1990), pp. 308–404.
  7. J. E. Harvey, “Exact ray trace procedure,” http://ebookbrowse.com/3-5-exact-ray-trace-procedure-pdf-d354859582 .

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. L. Sands, “Optics: the principle of least time,” in The Feynman Lectures on Physics(Addison Wesley Longman, 1970), Vol. 1, Chap. 26.

Johnson, R. B.

R. Kingslake and R. B. Johnson, “Meridional ray tracing,” in Lens Design Fundamentals (Academic, 2010), pp. 25–45.

Kingslake, R.

R. Kingslake and R. B. Johnson, “Meridional ray tracing,” in Lens Design Fundamentals (Academic, 2010), pp. 25–45.

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. L. Sands, “Optics: the principle of least time,” in The Feynman Lectures on Physics(Addison Wesley Longman, 1970), Vol. 1, Chap. 26.

Macdonald, J.

P. Mouroulis and J. Macdonald, “Rays and foundations of geometrical optics,” in Geometrical Optics and Optical Design (Oxford University, 1997), pp. 11–13.

Malacara, D.

D. Malacara and Z. Malacara, Handbook of Optical Design(Marcel Dekker, 2004).

Malacara, Z.

D. Malacara and Z. Malacara, Handbook of Optical Design(Marcel Dekker, 2004).

Mouroulis, P.

P. Mouroulis and J. Macdonald, “Rays and foundations of geometrical optics,” in Geometrical Optics and Optical Design (Oxford University, 1997), pp. 11–13.

Pedrotti, F. L.

F. L. Pedrotti and L. S. Pedrotti, “Geometrical optics,” in Introduction to Optics (Prentice-Hall, 1993), pp. 34–36.

Pedrotti, L. S.

F. L. Pedrotti and L. S. Pedrotti, “Geometrical optics,” in Introduction to Optics (Prentice-Hall, 1993), pp. 34–36.

Sands, M. L.

R. P. Feynman, R. B. Leighton, and M. L. Sands, “Optics: the principle of least time,” in The Feynman Lectures on Physics(Addison Wesley Longman, 1970), Vol. 1, Chap. 26.

Smith, W. J.

W. J. Smith, “Optical computation,” in Modern Optical Engineering (McGraw-Hill, 1990), pp. 308–404.

Other

R. P. Feynman, R. B. Leighton, and M. L. Sands, “Optics: the principle of least time,” in The Feynman Lectures on Physics(Addison Wesley Longman, 1970), Vol. 1, Chap. 26.

F. L. Pedrotti and L. S. Pedrotti, “Geometrical optics,” in Introduction to Optics (Prentice-Hall, 1993), pp. 34–36.

P. Mouroulis and J. Macdonald, “Rays and foundations of geometrical optics,” in Geometrical Optics and Optical Design (Oxford University, 1997), pp. 11–13.

R. Kingslake and R. B. Johnson, “Meridional ray tracing,” in Lens Design Fundamentals (Academic, 2010), pp. 25–45.

D. Malacara and Z. Malacara, Handbook of Optical Design(Marcel Dekker, 2004).

W. J. Smith, “Optical computation,” in Modern Optical Engineering (McGraw-Hill, 1990), pp. 308–404.

J. E. Harvey, “Exact ray trace procedure,” http://ebookbrowse.com/3-5-exact-ray-trace-procedure-pdf-d354859582 .

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Figures (10)

Fig. 1.
Fig. 1.

Refraction by a single spherical surface.

Fig. 2.
Fig. 2.

Line OI can meet the radius at any point C.

Fig. 3.
Fig. 3.

Refraction at a planar surface. The line OI can meet the normal N at any point C.

Fig. 4.
Fig. 4.

Refraction of a meridional ray through the last two surfaces of a centered system of m spherical surfaces.

Fig. 5.
Fig. 5.

Meridional reflection at a convex spherical mirror.

Fig. 6.
Fig. 6.

Refraction of a paraxial ray through a thick lens.

Fig. 7.
Fig. 7.

Refraction of a paraxial ray through a thick lens.

Fig. 8.
Fig. 8.

Reflection by a single spherical surface.

Fig. 9.
Fig. 9.

Refraction through the last two surfaces of a centered system of spherical surfaces.

Fig. 10.
Fig. 10.

Centered system of five spherical surfaces.

Equations (91)

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OPL=noOP+n1PI.
A=OP=R2+K22RKcosψ,
A=PI=R2+K2+2RKcosψ.
ψOPL=noRKsinψAn1RKsinψA=0.
AA=noKn1K.
AA=noKn1K.
AA=KK,
AA=KK.
AA=±noKn1K.
AmAm=nm1KmnmKm.
noSo+n1S1=n1noR.
1So+1S1=2R.
nmSm+nm1tm1Sm1=nmnm1Rm.
Am=Km+Rm,
Am=Km+Rm.
nmKm+nm1Km=nmnm1Rm.
n1K1+n0K1=n1n0R1.
n2S2+n1t1S1=n2n1R2,
1fv+nt1S1=1nR2,
1fvnS1[1t1/S1]=1nR2,
S1=nR1n1,
E=[1t1/S1]=[1(n1)t1nR1].
EfvnS1=1nR2E.
1fp=(n1)[1R11R2]+(n1)2[t1nR1R2],
n2K2+n1K2=n2n1R2.
n2K1e+n1K2=n2n1R2,
e=t[R1R2].
1/K1=(n11)/R1.
1K1[1e/K1]+nK2=(1nR2).
C=[1e/K1].
1K1+nCK2=(1nR2)C.
C=[1(n1R1)e].
CK2=(n1n)[1R11R2]+(n1)2[enR1R2],
1fc=(n1n)[1R11R2]+(n1)2[enR1R2].
CK2=Efv=1fp.
A=βK,
β=nAK,
A2=R2+K2+2RKcosψ,
cosψ=±R2h2R.
K=±β2R2h2+R2h2β21.
K=±R2β2R2h2R2h2,
S1=R±R2β2R2h2R2h2.
β=nKh2+[K±R2h2]2.
S1=R±R2nR2K2[K2+R2±2KR2h2](hn)2R2h2.
1K=βR1R,
β=nnRK.
nK+1K=n1R.
S1=R±R2n2R2h2R2h2.
L=Rn1R2n2R2h2R2h2.
K=So+R.
r1=Ksinθ,
A=Kcosθ±R2r2,
h=Asinθ.
β=ncosθ±n(R/K)2sin2θ.
h=Koβnsinθ.
β=ncosθ±n(R/K)2sin2θ,
h=Kβnsinθ,
S1=R±R2β2R2h2R2h2,
A=βK,
β=AK=1Kh2+[K±R2h2]2.
K=±R2β2R2h2+R2h2.
K=±R2R2K2[K2+R2±2KR2h2]h2+R2h2.
K=R2nR2K2[K2+R2±2KR2h2](hn)2R2h2.
1K+1K=2R1(hR)2,
1RS0+1RS1=2R1(hR)2.
1K1+1K2=2R,
1RSo+1RS1=2R.
K=±R22R2h2.
S1=RR22R2h2.
L=R2±R22R2h2.
h=(cosθ±(R/K)2sin2θ)Ksinθ,
1K+1K=2R1(hR)2.
r1=Kmsinθm1,
Am=Kmcosθm1±Rm2r12.
Am=Kmcosθm1±Rm2Km2sin2θm1.
Am=(Kmcosθm1±Rm2Km2sin2θm1),
hm=Amsinθm1.
βm=nmAmnm1Km.
βm=nmnm1(cosθm1±(Rm/Km)2sin2θm1),
hm=(nm1nm)Kmβmsinθm1,
sinθm1=(nm2Km1nm1Km1)sinθm2.
r2=Km1sinθm1,
r1=Km1sinθm2,
r2r1=sinβ2sinβ1=nm2nm1.
Km=Km1tm1+Rm1Rm.
sinθm1=(nm2Km1nm1Km1)sinθm2,
Km=Km1tm1+Rm1Rm,
βm=nmnm1(cosθm1±(Rm/Km)2sin2θm1),
hm=(nm1nm)Kmβmsinθm1,
Km=±Rm2βm2Rm2hm2Rm2hm2.
Sm=Km+Rm.

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