Abstract

We propose a technique for ray tracing, based in the propagation of a Gaussian shape invariant under the Fresnel diffraction integral. The technique uses two driving independent terms to direct the ray and is based on the fact that at any arbitrary distance, the center of the propagated Gaussian beam corresponds to the geometrical projection of the center of the incident beam. We present computer simulations as examples of the use of the technique consisting in the calculation of rays through lenses and optical media where the index of refraction varies as a function of position.

© 2009 Optical Society of America

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References

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  1. V. N. Mahajan, Optical Imaging and Aberrations Part I Ray Geometrical Optics, (SPIE Press USA, 1998).
  2. M. Herzberger, Modern Geometrical Optics, (Interscience Publishers, Inc. N.Y., 1958).
  3. O. N. Stavroudis, "Simpler derivation of the formulas for generalized ray tracing," J. Opt. Soc. Am. 66, 1330-1333 (1976).
    [CrossRef]
  4. G. H. Spencer and M. V. R. K. Murty, "General ray-tracing procedure," J. Opt. Soc. Am. 52, 672-678 (1962).
    [CrossRef]
  5. O. N. Stavroudis, The Optics of Rays,Wavefronts, and Caustics, (Academic Press, Inc. N.Y., 1972).
  6. A. Sharma, D. V. Kumar and A. K. Ghatak, "Tracing rays through graded-index media: a new method," App. Opt.,  21, 984-987 (1982).
    [CrossRef]
  7. S. Nolte, M. Will, J. Burghoff, and A. Tünerman, "Ultrafast laser processing: new options for three-dimensional photonic structures," J. Mod. Opt. 51, 2533-2542 (2004).
    [CrossRef]
  8. S. Szatmári and G. Kühnle, "Pulse front and pulse duration distortion in refractive optics, and its compensation," Opt. Commun. 69, 60-65 (1988).
    [CrossRef]
  9. U. Fuchs, U. D. Zeitner, and A. Tűnnermann, "Ultra-short pulse propagation in complex optical systems," Opt. Express 13, 3852-3861 (2005).
    [CrossRef] [PubMed]
  10. A. Rohani, A. A. Shishegar and S. Safavi-Naeini, "A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces," Opt. Commun. 232, 1-10 (2004).
    [CrossRef]
  11. J. J. Stamnes, Waves in Focal Regions, (IOP Publishing Limited, 1986).
  12. L. Novotny and B. Hecht, Principles of Nano-Optics, (Cambridge University Press, 2006).
  13. K. Iizuka, Engineering Optics, (Springer Verlag, 1986).

2005

2004

S. Nolte, M. Will, J. Burghoff, and A. Tünerman, "Ultrafast laser processing: new options for three-dimensional photonic structures," J. Mod. Opt. 51, 2533-2542 (2004).
[CrossRef]

A. Rohani, A. A. Shishegar and S. Safavi-Naeini, "A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces," Opt. Commun. 232, 1-10 (2004).
[CrossRef]

1988

S. Szatmári and G. Kühnle, "Pulse front and pulse duration distortion in refractive optics, and its compensation," Opt. Commun. 69, 60-65 (1988).
[CrossRef]

1982

A. Sharma, D. V. Kumar and A. K. Ghatak, "Tracing rays through graded-index media: a new method," App. Opt.,  21, 984-987 (1982).
[CrossRef]

1976

1962

Burghoff, J.

S. Nolte, M. Will, J. Burghoff, and A. Tünerman, "Ultrafast laser processing: new options for three-dimensional photonic structures," J. Mod. Opt. 51, 2533-2542 (2004).
[CrossRef]

Fuchs, U.

Ghatak, A. K.

A. Sharma, D. V. Kumar and A. K. Ghatak, "Tracing rays through graded-index media: a new method," App. Opt.,  21, 984-987 (1982).
[CrossRef]

Kühnle, G.

S. Szatmári and G. Kühnle, "Pulse front and pulse duration distortion in refractive optics, and its compensation," Opt. Commun. 69, 60-65 (1988).
[CrossRef]

Kumar, D. V.

A. Sharma, D. V. Kumar and A. K. Ghatak, "Tracing rays through graded-index media: a new method," App. Opt.,  21, 984-987 (1982).
[CrossRef]

Murty, M. V. R. K.

Nolte, S.

S. Nolte, M. Will, J. Burghoff, and A. Tünerman, "Ultrafast laser processing: new options for three-dimensional photonic structures," J. Mod. Opt. 51, 2533-2542 (2004).
[CrossRef]

Rohani, A.

A. Rohani, A. A. Shishegar and S. Safavi-Naeini, "A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces," Opt. Commun. 232, 1-10 (2004).
[CrossRef]

Safavi-Naeini, S.

A. Rohani, A. A. Shishegar and S. Safavi-Naeini, "A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces," Opt. Commun. 232, 1-10 (2004).
[CrossRef]

Sharma, A.

A. Sharma, D. V. Kumar and A. K. Ghatak, "Tracing rays through graded-index media: a new method," App. Opt.,  21, 984-987 (1982).
[CrossRef]

Shishegar, A. A.

A. Rohani, A. A. Shishegar and S. Safavi-Naeini, "A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces," Opt. Commun. 232, 1-10 (2004).
[CrossRef]

Spencer, G. H.

Stavroudis, O. N.

Szatmári, S.

S. Szatmári and G. Kühnle, "Pulse front and pulse duration distortion in refractive optics, and its compensation," Opt. Commun. 69, 60-65 (1988).
[CrossRef]

Tünerman, A.

S. Nolte, M. Will, J. Burghoff, and A. Tünerman, "Ultrafast laser processing: new options for three-dimensional photonic structures," J. Mod. Opt. 51, 2533-2542 (2004).
[CrossRef]

Tunnermann, A.

Will, M.

S. Nolte, M. Will, J. Burghoff, and A. Tünerman, "Ultrafast laser processing: new options for three-dimensional photonic structures," J. Mod. Opt. 51, 2533-2542 (2004).
[CrossRef]

Zeitner, U. D.

App. Opt.

A. Sharma, D. V. Kumar and A. K. Ghatak, "Tracing rays through graded-index media: a new method," App. Opt.,  21, 984-987 (1982).
[CrossRef]

J. Mod. Opt.

S. Nolte, M. Will, J. Burghoff, and A. Tünerman, "Ultrafast laser processing: new options for three-dimensional photonic structures," J. Mod. Opt. 51, 2533-2542 (2004).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

S. Szatmári and G. Kühnle, "Pulse front and pulse duration distortion in refractive optics, and its compensation," Opt. Commun. 69, 60-65 (1988).
[CrossRef]

A. Rohani, A. A. Shishegar and S. Safavi-Naeini, "A fast Gaussian beam tracing method for reflection and refraction of general vectorial astigmatic Gaussian beams from general curved surfaces," Opt. Commun. 232, 1-10 (2004).
[CrossRef]

Opt. Express

Other

V. N. Mahajan, Optical Imaging and Aberrations Part I Ray Geometrical Optics, (SPIE Press USA, 1998).

M. Herzberger, Modern Geometrical Optics, (Interscience Publishers, Inc. N.Y., 1958).

O. N. Stavroudis, The Optics of Rays,Wavefronts, and Caustics, (Academic Press, Inc. N.Y., 1972).

J. J. Stamnes, Waves in Focal Regions, (IOP Publishing Limited, 1986).

L. Novotny and B. Hecht, Principles of Nano-Optics, (Cambridge University Press, 2006).

K. Iizuka, Engineering Optics, (Springer Verlag, 1986).

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Figures (5)

Fig. 1.
Fig. 1.

Ray tracing in a gap between two reflective planes (units in m).

Fig. 2.
Fig. 2.

Ray tracing in a selfoc media as described in the text.

Fig. 3.
Fig. 3.

Ray tracing through a convex lens as described in the text.

Fig. 4.
Fig. 4.

Ray tracing through an aspheric lens as described in the text.

Fig. 5.
Fig. 5.

Ray tracing through an aspheric lens and an inhomogeneous media placed near the back focal plane of the lens as described in the text.

Equations (12)

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Ψ(x,y=0)=A exp (i2πtan(θ)λx) exp (x2r02) ,
ΨF(ξ,z)=exp(i2πzλ)iλzΨ(x,0)exp(ìπλz(xξ)2)dx.
ΨF(ξ,z)=Aiλzπr02λzλziπr02exp(iπro4α24λz)exp((ξξA)2R02)exp(iπλRc(ξξB)2),
where α=2πtan(θ)λ,ξA=aλz2π,R0=λ2z2+π2r04πr0,ξB=πr04α2λzandRc=λ2z2+π2r04πλ2z.
Ψn(x)=Anexp(iαnx)exp(iβnx2)exp((xxAn)2rn2)exp(iγn[xxBn]2),
Ψn+1(ξ,z)=An+1exp(iαn+1ξ)exp(iβn+1ξ2)exp((ξξAn+1)2rn+12)exp(iγn+1[ξξBn+1]2),
An+1=Anexp(i2πzλ)iλz πrn2λzλzirn2(βnλz+γnλz+π) ×
exp (iγn(xBn)2) exp (iλz(xAn)2rn4(βnλz+γnλz+π)) ,
αn+1=0,βn+1=πλz,Dn=λ2z2+rn4(βnλz+γnλz+π)2,
γn+1=π2rn4Dnλz(βnλz+γnλz+π).
rn+1=Dnπrn,ξAn+1=xAn+αnλz2πγnλz(xBn)π+(βn+γn)λz(xAn)π
ξBn+1=αnλz2πγnλz(xBn)πλ2z2βnλz+γnλz+πxAnπrn4.

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