Abstract

A simple model for the path of a light beam through an optical coating is the path of a ray predicted by Snell's law. By determining the exit point of a beam for various types of coating, one finds that the simple model is a good approximation in the case of antireflection coatings, but not for coatings of other designs. An approximate method for determining the correct path of a beam through the coating is derived and the path is illustrated using a Gaussian incident beam and tracing the position of the peak field of the beam as it traverses the coating.

© 2006 Optical Society of America

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References

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  1. P. W. Baumeister, Optical Coating Technology (SPIE Press, 2004), p. 10-47.
  2. H. A. Mcleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
    [CrossRef]
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  4. This formula applies to a Gaussian beam with a minimum half 1/e2 width w0 and an angular half spectral extent of Delta kx and is approximately true for beams of other profiles.
  5. B. R. Horowitz and T. Tamir, "Lateral displacement of a light beam at a dielectric surface," J. Opt. Soc. Am. 61, 586-594 (1971).
    [CrossRef]
  6. M. McGuirk and C. K. Carniglia, "An angular spectrum representation approach to the Goos-Haenchen shift," J. Opt. Soc. Am. 67, 103-107 (1977).
    [CrossRef]
  7. C. W. Hsue and T. Tamir, "Lateral displacement and distortion of beams incident upon a transmitting-layer configuration," J. Opt. Soc. Am. A 2, 978-988 (1985).
    [CrossRef]
  8. C. K. Carniglia, "Diffraction effects in interference filters," in OSA Annual Meeting, Vol. 11 of OSA Technical Digest Series, paper WS2 (1988).
  9. M. Gerken and D. A. B. Miller, "Multilayer thin-film structures with high spatial dispersion," Appl. Opt. 42, 1330-1345 (2003).
    [CrossRef] [PubMed]
  10. C. K. Carniglia and K. R. Brownstein, "Focal shift and ray model for total internal reflection," J. Opt. Soc. Am. 67, 121-122 (1977).
    [CrossRef]
  11. C. K. Carniglia, D. G. Jensen, and A. J. Fielding, "Lateral shift and internal electric fields in multicavity narrow-bandpass filters," in Optical Interference Coatings, OSA Technical Digest Series (2004).
  12. C. K. Carniglia, R. E. Klinger, C. A. Hulse, and R. B. Sargent, "Beam displacement and distortion effects in narrowband optical thin-film filters," Appl. Opt. (to be published).
    [PubMed]
  13. C. Andrew Hulse, R. E. Klinger, and R. B. Sargent, "Optical coupler device for dense wavelength division multiplexing," U.S. patent 6,215,924 B1 (10 April 2001).

2003 (1)

1985 (1)

1977 (2)

1971 (1)

Baumeister, P. W.

P. W. Baumeister, Optical Coating Technology (SPIE Press, 2004), p. 10-47.

Brownstein, K. R.

Carniglia, C. K.

C. K. Carniglia and K. R. Brownstein, "Focal shift and ray model for total internal reflection," J. Opt. Soc. Am. 67, 121-122 (1977).
[CrossRef]

M. McGuirk and C. K. Carniglia, "An angular spectrum representation approach to the Goos-Haenchen shift," J. Opt. Soc. Am. 67, 103-107 (1977).
[CrossRef]

C. K. Carniglia, "Diffraction effects in interference filters," in OSA Annual Meeting, Vol. 11 of OSA Technical Digest Series, paper WS2 (1988).

C. K. Carniglia, R. E. Klinger, C. A. Hulse, and R. B. Sargent, "Beam displacement and distortion effects in narrowband optical thin-film filters," Appl. Opt. (to be published).
[PubMed]

C. K. Carniglia, D. G. Jensen, and A. J. Fielding, "Lateral shift and internal electric fields in multicavity narrow-bandpass filters," in Optical Interference Coatings, OSA Technical Digest Series (2004).

Fielding, A. J.

C. K. Carniglia, D. G. Jensen, and A. J. Fielding, "Lateral shift and internal electric fields in multicavity narrow-bandpass filters," in Optical Interference Coatings, OSA Technical Digest Series (2004).

Gerken, M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Horowitz, B. R.

Hsue, C. W.

Hulse, C. A.

C. K. Carniglia, R. E. Klinger, C. A. Hulse, and R. B. Sargent, "Beam displacement and distortion effects in narrowband optical thin-film filters," Appl. Opt. (to be published).
[PubMed]

Hulse, C. Andrew

C. Andrew Hulse, R. E. Klinger, and R. B. Sargent, "Optical coupler device for dense wavelength division multiplexing," U.S. patent 6,215,924 B1 (10 April 2001).

Jensen, D. G.

C. K. Carniglia, D. G. Jensen, and A. J. Fielding, "Lateral shift and internal electric fields in multicavity narrow-bandpass filters," in Optical Interference Coatings, OSA Technical Digest Series (2004).

Klinger, R. E.

C. Andrew Hulse, R. E. Klinger, and R. B. Sargent, "Optical coupler device for dense wavelength division multiplexing," U.S. patent 6,215,924 B1 (10 April 2001).

C. K. Carniglia, R. E. Klinger, C. A. Hulse, and R. B. Sargent, "Beam displacement and distortion effects in narrowband optical thin-film filters," Appl. Opt. (to be published).
[PubMed]

McGuirk, M.

Mcleod, H. A.

H. A. Mcleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
[CrossRef]

Miller, D. A. B.

Sargent, R. B.

C. Andrew Hulse, R. E. Klinger, and R. B. Sargent, "Optical coupler device for dense wavelength division multiplexing," U.S. patent 6,215,924 B1 (10 April 2001).

C. K. Carniglia, R. E. Klinger, C. A. Hulse, and R. B. Sargent, "Beam displacement and distortion effects in narrowband optical thin-film filters," Appl. Opt. (to be published).
[PubMed]

Tamir, T.

Appl. Opt. (2)

C. K. Carniglia, R. E. Klinger, C. A. Hulse, and R. B. Sargent, "Beam displacement and distortion effects in narrowband optical thin-film filters," Appl. Opt. (to be published).
[PubMed]

M. Gerken and D. A. B. Miller, "Multilayer thin-film structures with high spatial dispersion," Appl. Opt. 42, 1330-1345 (2003).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Other (7)

C. Andrew Hulse, R. E. Klinger, and R. B. Sargent, "Optical coupler device for dense wavelength division multiplexing," U.S. patent 6,215,924 B1 (10 April 2001).

P. W. Baumeister, Optical Coating Technology (SPIE Press, 2004), p. 10-47.

H. A. Mcleod, Thin-Film Optical Filters, 3rd ed. (Institute of Physics, 2001).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

This formula applies to a Gaussian beam with a minimum half 1/e2 width w0 and an angular half spectral extent of Delta kx and is approximately true for beams of other profiles.

C. K. Carniglia, "Diffraction effects in interference filters," in OSA Annual Meeting, Vol. 11 of OSA Technical Digest Series, paper WS2 (1988).

C. K. Carniglia, D. G. Jensen, and A. J. Fielding, "Lateral shift and internal electric fields in multicavity narrow-bandpass filters," in Optical Interference Coatings, OSA Technical Digest Series (2004).

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Figures (9)

Fig. 1
Fig. 1

Schematic view of a multilayer coating, showing the entrance medium on the left with refractive index na and the substrate with index ns on the right. A typical layer is labeled with its refractive index ni and thickness di. The front surface of the coating corresponds with the x axis; the z axis runs through the coating. A typical incident wave has a wave vector that makes an angle θ a with the z axis. The path of the light shown here is the path predicted by Snell's law and the angle of the beam θ i in the ith layer is given by Eq. (1) in the text. The displacement of the beam at the exit surface is labeled X 0 in this case. The displacement of the beam perpendicular to its direction of propagation is denoted D as shown.

Fig. 2
Fig. 2

Lateral shift in micrometers of the transmitted beam versus angle in degrees for a single-layer film with an index of 2.35 and a thickness of 450 nm for a wavelength of 500 nm. Curve X 0, the Snell's law prediction X 01; curve X S, X for s-polarized incident light; curve X P, X for p-polarized incident light.

Fig. 3
Fig. 3

Lateral shift versus wavelength for the edge filter given by expression (33) for an incidence angle of 30°. The curve labeled X is the calculated beam shift for s-polarized light according to Eq. (21) in the text, plotted to the left-hand scale. The flat curve labeled X 0, plotted to the left-hand scale, is the shift predicted by Snell's law. For reference the curve labeled T and plotted to the right-hand scale gives the transmittance for s polarization of the coating as a function of wavelength.

Fig. 4
Fig. 4

Lateral shift X versus angle for the narrow-bandpass filter plotted to the left-hand scale. The curve labeled T is the transmittance versus angle plotted to the right-hand scale. Usually the transmittance is plotted as a function of wavelength.

Fig. 5
Fig. 5

Contour map of the internal electric field for a 60 μm wide Gaussian beam incident from the left on the edge filter discussed in the text at an angle of 30° with a wavelength of 534 nm. The incident beam is illustrated between layers −10 and 0. The coating extends from layer 0 to layer 55 and the beam in the substrate is from layer 55 to 60. The plot is actually of the log of the time-averaged square of the electric field. The reddest regions have the highest field and the violet region represents anything below about 1∕850 of the maximum field. The white curve represents the path of the peak electric field through the coating. The black line represents the z axis. In this case, the beam path is only barely distinguishable from the straight-through line.

Fig. 6
Fig. 6

Similar to Fig. 5 for a beam with a wavelength of 466 nm. This wavelength is in the edge region of the filter. In this case, the white curve, representing the path of the light differs significantly from the black curve representing the z axis. The reflected beam is illustrated in the region from −20 to −10 layers and is shifted in the positive x direction from the incident beam.

Fig. 7
Fig. 7

Similar to Fig. 6 for a beam with a wavelength of 400 nm. This wavelength is in the HR region of the filter. The filter extends to layer 55, but the field damps out quickly and is only visible over the first ten layers or so. The white curve tracing the path of the light through the coating can still be drawn, even though the field is not visible in this representation. At this scale the black curve representing the z axis is coincident with the beam path.

Fig. 8
Fig. 8

Similar to Fig. 5 for a 400 μm wide Gaussian beam incident on the NBP DWDM filter at an angle of 3° and with a wavelength of 1550 nm. The regions of large constructive interference within the five cavities are apparent. The white curve representing the path of the actual beam differs significantly from the black curve representing the z axis. Surprisingly, the actual path appears to go straight through the spacer layers of the cavities and shift in the reflectors between the cavities.

Fig. 9
Fig. 9

Path of the beam of light as a function of depth z through the NBP filter calculated according to approximation (44). The rectangles illustrate the location of the low-index spacer layers relative to the path.

Equations (47)

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n i sin θ i = n a sin θ a .
X 0 = i = 1 m d i tan θ i ,
k a = | k a | = k a x         2 + k a y         2 + k a z         2 = n a ω c = 2 π n a λ ,
E i ( x , y , z , t ) = A ( k a x , k a y ) exp [ i ( ω t k a x x k a y y k a z z ) ] d k a x d k a y .
k ax         2 + k ay         2  <  k a ,
E i ( x , z ) = A ( k x ) exp [ i ( k x x + k a z z ) ] d k x , for   z 0.
k x k j x = k j sin θ j = 2 π n j λ sin θ j ,
for   j = a , s ,   and   i = 1   to   m .
k j z = k j     2 k x     2 , for   j = a , s ,   and   i = 1   to   m .
A ( k x ) = 1 2 π E ( x , 0 ) exp ( i k x x ) d x .
k x 0 k a sin θ a 0 .
| k x k x 0 | < Δ k x .
w 0 > 2 Δ k x .
d = i = 1 m d i .
E t ( x , z ) = A ( k x ) t ( k x ) exp { i [ k x x + k s z ( z d ) ] } d k x , for   z d .
t ( k x ) | t ( k x 0 ) | exp { i [ φ ( k x 0 ) + k x φ ( k x 0 ) ] } = t ( k x 0 ) exp [ i k x φ ( k x 0 ) ] ,
φ ( k x 0 ) d φ d k x | k x = k x 0 .
E t ( x , z ) t ( k x 0 ) A ( k x ) exp ( i { k x [ x φ ( k x 0 ) ] + k s z ( z d ) } ) d k x , for   z d .
E t ( x , d ) t ( k x 0 ) A ( k x ) exp { i k x [ x φ ( k x 0 ) ] } d k x ,
E t ( x , d ) t ( k x 0 ) E i [ x φ ( k x 0 ) , 0 ] .
X = d φ d k x | k x = k x 0
X = λ 2 π n a cos θ a d φ d θ a | θ a = θ a 0 .
D = λ cos θ s 2 π n a cos θ a d φ d θ a | θ a = θ a 0 .
t 1 = t a 1 t 1 s exp ( i δ 1 ) 1 r a 1 r 1 s exp ( i 2 δ 1 ) ,
δ 1 = 2 π n 1 d 1 cos θ 1 / λ
t i j = 2 η i η i + η j , for   i , j = a , 1   and   1 , s ,
r i j = η i η j η i + η j , for   i , j = a , 1   and   1 , s .
η i = n i cos θ i , for   s   polarization ,
η i = n i / cos θ i , for   p   polarization .
X 01 = d 1 tan θ 1 = d 1 tan θ s .
φ ( θ a ) = δ 1 = 2 π n s d 1 cos θ s / λ = 2 π d 1 λ n s     2 n a     2 sin 2 θ a .
d φ d θ a = 2 π d 1 n a cos θ a tan θ s λ .
X = d 1 tan θ s ,
substrate / 5.7895 L   1 .1642H   0 .3098L   1.2878 H   1 .3138L   0 .1767H   ( 0 .5H   L   0 .5 H ) 20 0.2637 H   1 .2316L   1 .0042H   0 .7158L   0 .9299H   2 .0023L / air ,
( LH ) 8 10 L   ( HL ) 16  H   10L   ( HL ) 17  H   10L ( HL ) 17 H   10L   ( HL ) 16  H   10L   ( HL ) 8 ,
L ( HL ) 7 H   10L   H ( LH ) 7  L ( HL ) 7 H   10L   H ( LH ) 7 L ( HL ) 8 H   10L   H ( LH ) 8  L ( HL ) 7 H   10L   H ( LH ) 7 L ( HL ) 7 H   10L   H ( LH ) 7  L .
( t E ( k x , z ) t H ( k x , z ) ) = [ M ( k x , z d ) ] ( 1 η s ) t ( k x ) ,
A ( k x , z ) = A ( k x ) t E ( k x , z ) .
E ( x , z ) = A ( k x ) t E ( k x , z ) exp ( i k x x ) d k x , for 0 < z < d .
t E ( k x , z ) t E ( k x 0 , z ) exp [ i k x φ E ( k x 0 , z ) ] ,
φ E ( k x 0 , z ) d φ E ( k x , z ) d k x | k x = k x 0 .
E ( x , z ) t E ( k x 0 , z ) A ( k x ) exp { i k x [ x φ E ( k x 0 , z ) ] } d k x , for 0 < z < d ,
E ( x , z )  ≅  t E ( k x 0 , z ) E i [ x φ E ( k x 0 , z ) , 0 ] , for   0 < z < d .
X ( z ) d φ E ( k x , z ) d k x | k x = k x 0 ,
X ( z ) λ 2 π n a cos θ a d φ E ( θ a , z ) d θ a | θ a = θ a 0 .
E ( x ) = E 0 exp ( x 2 / w 0     2 ) .
A ( k x ) = π w 0 E 0 exp [ ( k x w 0 / 2 ) 2 ] .

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