Abstract

We demonstrate that published vectorial laws of reflection and refraction of light based solely on the cross product do not, in general, uniquely determine the direction of the reflected and refracted waves without additional information. This is because the cross product does not have a unique inverse operation, which is explained in this Letter in linear algebra terms. However, a vector is in fact uniquely determined if both the cross product (vector product) and dot product (scalar product) with a known vector are specified, which can be written as a single equation with a left-invertible matrix. It is thus possible to amend the vectorial laws of reflection and refraction to incorporate both the cross and dot products for a complete specification with unique solution. This enables highly efficient, unambiguous computation of reflected and refracted wave vectors from the incident wave and surface normal.

© 2012 Optical Society of America

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References

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  1. E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987).
  2. A. Glassner, An Introduction to Ray Tracing (Morgan Kaufmann, 1989).
  3. P. Bhattacharjee, Eur. J. Phys. 26, 901 (2005).
    [CrossRef]
  4. P. Bhattacharjee, Optik 120, 642 (2009).
    [CrossRef]
  5. P. Bhattacharjee, Optik 121, 2128 (2010).
    [CrossRef]
  6. P. Bhattacharjee, Optik123, 381 (2012).
  7. V. Shalaev, Nature Photonics 1, 41 (2007).
    [CrossRef]

2010

P. Bhattacharjee, Optik 121, 2128 (2010).
[CrossRef]

2009

P. Bhattacharjee, Optik 120, 642 (2009).
[CrossRef]

2007

V. Shalaev, Nature Photonics 1, 41 (2007).
[CrossRef]

2005

P. Bhattacharjee, Eur. J. Phys. 26, 901 (2005).
[CrossRef]

Bhattacharjee, P.

P. Bhattacharjee, Optik 121, 2128 (2010).
[CrossRef]

P. Bhattacharjee, Optik 120, 642 (2009).
[CrossRef]

P. Bhattacharjee, Eur. J. Phys. 26, 901 (2005).
[CrossRef]

P. Bhattacharjee, Optik123, 381 (2012).

Glassner, A.

A. Glassner, An Introduction to Ray Tracing (Morgan Kaufmann, 1989).

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987).

Shalaev, V.

V. Shalaev, Nature Photonics 1, 41 (2007).
[CrossRef]

Eur. J. Phys.

P. Bhattacharjee, Eur. J. Phys. 26, 901 (2005).
[CrossRef]

Nature Photonics

V. Shalaev, Nature Photonics 1, 41 (2007).
[CrossRef]

Optik

P. Bhattacharjee, Optik 120, 642 (2009).
[CrossRef]

P. Bhattacharjee, Optik 121, 2128 (2010).
[CrossRef]

Other

P. Bhattacharjee, Optik123, 381 (2012).

E. Hecht, Optics, 2nd ed. (Addison-Wesley, 1987).

A. Glassner, An Introduction to Ray Tracing (Morgan Kaufmann, 1989).

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Figures (2)

Fig. 1.
Fig. 1.

(a) Typical case of reflection and refraction for n2>n1. (b) Reflected vector q=k2knn=(q·n)n+(n×q)×n.

Fig. 2.
Fig. 2.

Ambiguity if a cross product alone is used in determination of refracted vector. (a) All wavevectors displayed in bold have the same cross product with the outward unit normal n. (b) For all vectors R displayed, the equation μ(n×R)=n×i is true. Note that two of the solutions are radii of the unit circle.

Equations (22)

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n=[nxnynz],nT=[nxnynz],Ncross=[0nznynz0nxnynx0].
B=(BAA+C×A)/|A|2.
q=(qy)y+(y×q)×y.
q=k+cn,
q=k+cn.
c=kn±sqrt(kn2).
c=kn±sqrt(kn2+(μ21)|k|2).
n×q=n×k,
nq=nk.
Ncrossq=NcrossknTq=nTk.
Mq=b,whereM=[0nznynz0nxnynx0nxnynz]andb=[nzky+nykznzkxnxkznykx+nxky(nxkx+nyky+nzkz)].
q=MTb.
q=k2knn.
n×q=n×k
nq=sqrt(|k|2(μ21)+(nk)2).
Mq=b,withb=[nzky+nykznzkxnxkznykx+nxky(nxkx+nyky+nzkz)2+(μ21)(kx2+ky2+kz2)].
q=MTb
q=(qn)nn×(n×q)=k(sqrt(kn2+(μ21)|k|2)+kn)n.
μ(n×R)=n×i,
μ(nR)=sqrt(μ21+in2).
R=μ1i+(μ1cosθicosθR)n,withcosθi=nk/|k|,andcosθR=sqrt(1μ2(1cos2θi)).
RR=1

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