Abstract

The analysis of reflections from thin films or dielectric materials can be approached by a matrix method that treats any thin-layer device as a cascade of sequential, zero-thickness reflecting thin-layer surfaces [J. Opt. Soc. Am. A 2, 1363 (1985)] . Our paper presents an alternative method for predicting the reflection/transmission characteristics of such dielectric films in a Fabry–Perot interferometer configuration based on a Gaussian-beam modal analysis within a scattering-matrix framework [in Proceedings of IEE 7th International Conference on Antennas and Propagation (IEE, 1991), Issue 15, p. 201.] We present and validate a scalar Gaussian-beam modal scattering-matrix approach using long-wavelength examples, where diffraction effects are important to model total transmission and reflection characteristics that also include a waveguide modal description of a corrugated horn. For optical beams the same technique is equally applicable, but diffraction is less severe within this framework. This approach is flexible and has many applications within laser optics and in far-infrared or submillimeter-instrumentation optical analysis, where it is possible to incorporate reflections in both waveguide and free space within the description of a whole system. To conclude and verify the accuracy of the technique, experimental measurements taken at 94GHz are compared with theoretical predictions for a dielectric cavity of polyethylene sheets between corrugated source and detector antennas.

© 2008 Optical Society of America

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References

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  1. J. A. Murphy, N. Trappe, and S. Withington, "Gaussian beam mode analysis of partial reflections in simple quasi-optical systems fed by horn antennas," Infrared Phys. Technol. 44, 289-297 (2003).
    [CrossRef]
  2. D. Olver, P. Clarricoats, A. Kishka, and L. Shafai, Microwave Feeds and Horns, Series 39 (IEEE Press, 1994), Chap. 4.
  3. E. Gleeson, J. A. Murphy, and B. Maffei, "Phase centers of far infrared multi-moded horn antennas," Int. J. Infrared Millim. Waves 23, 711-730 (2002).
    [CrossRef]
  4. N. Trappe, A. Murphy, S. Withington, and W. Jellema, "Gaussian beam mode analysis of standing waves between two coupled corrugated horns," IEEE Trans. Antennas Propag. 53, 1755-1761 (2005).
    [CrossRef]
  5. H. Van der Stadt, and J. M. Muller, "Multi-mirror Fabry-Perot Interferometers," J. Opt. Soc. Am. A 2, 1363-1369 (1985).
  6. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chaps. 1 and 7.
  7. D. Martin and J. Bowen, "Long-wave optics," IEEE Trans. Microwave Theory Tech. 41, 1676-1690 (1993).
    [CrossRef]
  8. A. Siegman, Lasers (University Science Books, 1986).
  9. R. Padman and J. A. Murphy, "A scattering matrix formulation for Gaussian beam-mode analysis," in IEE 7th International Conference on Antennas and Propagation (IEE, 1991), pp. 201-204.
  10. N. Trappe, T. Finn, A. Murphy, and W. Jellema, "Gaussian beam mode analysis of standing waves in submillimeter telescopes and receiver systems", Proc. SPIE 5497, 137-145 (2004).
  11. P. Goldsmith, Quasioptical Systems, Vol. x of IEEE Series on Microwave Techniques and Radio Frequencies (IEEE Press, 1998), Chaps. 3, 5, and 9.
    [CrossRef]
  12. J. A. Murphy and S. Withington, "Perturbation analysis of Gaussian beam mode scattering at off-axis ellipsoidal mirrors," Infrared Phys. Technol. 37, 205-219 (1996).
    [CrossRef]
  13. F. Moreno and F. González, "Transmission of a Gaussian beam of low divergence through a high-finesse Fabry-Perot device," J. Opt. Soc. Am. A 9, 2173-2175 (1992).
    [CrossRef]
  14. H. Y. Abu-Safia, "Transmission of a Gaussian beam through a Fabry-Perot interferometer," Appl. Opt. 33, 3805-3811 (1994).
    [CrossRef] [PubMed]

2005 (1)

N. Trappe, A. Murphy, S. Withington, and W. Jellema, "Gaussian beam mode analysis of standing waves between two coupled corrugated horns," IEEE Trans. Antennas Propag. 53, 1755-1761 (2005).
[CrossRef]

2004 (1)

N. Trappe, T. Finn, A. Murphy, and W. Jellema, "Gaussian beam mode analysis of standing waves in submillimeter telescopes and receiver systems", Proc. SPIE 5497, 137-145 (2004).

2003 (1)

J. A. Murphy, N. Trappe, and S. Withington, "Gaussian beam mode analysis of partial reflections in simple quasi-optical systems fed by horn antennas," Infrared Phys. Technol. 44, 289-297 (2003).
[CrossRef]

2002 (1)

E. Gleeson, J. A. Murphy, and B. Maffei, "Phase centers of far infrared multi-moded horn antennas," Int. J. Infrared Millim. Waves 23, 711-730 (2002).
[CrossRef]

1999 (1)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chaps. 1 and 7.

1998 (1)

P. Goldsmith, Quasioptical Systems, Vol. x of IEEE Series on Microwave Techniques and Radio Frequencies (IEEE Press, 1998), Chaps. 3, 5, and 9.
[CrossRef]

1996 (1)

J. A. Murphy and S. Withington, "Perturbation analysis of Gaussian beam mode scattering at off-axis ellipsoidal mirrors," Infrared Phys. Technol. 37, 205-219 (1996).
[CrossRef]

1994 (2)

D. Olver, P. Clarricoats, A. Kishka, and L. Shafai, Microwave Feeds and Horns, Series 39 (IEEE Press, 1994), Chap. 4.

H. Y. Abu-Safia, "Transmission of a Gaussian beam through a Fabry-Perot interferometer," Appl. Opt. 33, 3805-3811 (1994).
[CrossRef] [PubMed]

1993 (1)

D. Martin and J. Bowen, "Long-wave optics," IEEE Trans. Microwave Theory Tech. 41, 1676-1690 (1993).
[CrossRef]

1992 (1)

1991 (1)

R. Padman and J. A. Murphy, "A scattering matrix formulation for Gaussian beam-mode analysis," in IEE 7th International Conference on Antennas and Propagation (IEE, 1991), pp. 201-204.

1986 (1)

A. Siegman, Lasers (University Science Books, 1986).

1985 (1)

Abu-Safia, H. Y.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chaps. 1 and 7.

Bowen, J.

D. Martin and J. Bowen, "Long-wave optics," IEEE Trans. Microwave Theory Tech. 41, 1676-1690 (1993).
[CrossRef]

Clarricoats, P.

D. Olver, P. Clarricoats, A. Kishka, and L. Shafai, Microwave Feeds and Horns, Series 39 (IEEE Press, 1994), Chap. 4.

Finn, T.

N. Trappe, T. Finn, A. Murphy, and W. Jellema, "Gaussian beam mode analysis of standing waves in submillimeter telescopes and receiver systems", Proc. SPIE 5497, 137-145 (2004).

Gleeson, E.

E. Gleeson, J. A. Murphy, and B. Maffei, "Phase centers of far infrared multi-moded horn antennas," Int. J. Infrared Millim. Waves 23, 711-730 (2002).
[CrossRef]

Goldsmith, P.

P. Goldsmith, Quasioptical Systems, Vol. x of IEEE Series on Microwave Techniques and Radio Frequencies (IEEE Press, 1998), Chaps. 3, 5, and 9.
[CrossRef]

González, F.

Jellema, W.

N. Trappe, A. Murphy, S. Withington, and W. Jellema, "Gaussian beam mode analysis of standing waves between two coupled corrugated horns," IEEE Trans. Antennas Propag. 53, 1755-1761 (2005).
[CrossRef]

N. Trappe, T. Finn, A. Murphy, and W. Jellema, "Gaussian beam mode analysis of standing waves in submillimeter telescopes and receiver systems", Proc. SPIE 5497, 137-145 (2004).

Kishka, A.

D. Olver, P. Clarricoats, A. Kishka, and L. Shafai, Microwave Feeds and Horns, Series 39 (IEEE Press, 1994), Chap. 4.

Maffei, B.

E. Gleeson, J. A. Murphy, and B. Maffei, "Phase centers of far infrared multi-moded horn antennas," Int. J. Infrared Millim. Waves 23, 711-730 (2002).
[CrossRef]

Martin, D.

D. Martin and J. Bowen, "Long-wave optics," IEEE Trans. Microwave Theory Tech. 41, 1676-1690 (1993).
[CrossRef]

Moreno, F.

Muller, J. M.

Murphy, A.

N. Trappe, A. Murphy, S. Withington, and W. Jellema, "Gaussian beam mode analysis of standing waves between two coupled corrugated horns," IEEE Trans. Antennas Propag. 53, 1755-1761 (2005).
[CrossRef]

N. Trappe, T. Finn, A. Murphy, and W. Jellema, "Gaussian beam mode analysis of standing waves in submillimeter telescopes and receiver systems", Proc. SPIE 5497, 137-145 (2004).

Murphy, J. A.

J. A. Murphy, N. Trappe, and S. Withington, "Gaussian beam mode analysis of partial reflections in simple quasi-optical systems fed by horn antennas," Infrared Phys. Technol. 44, 289-297 (2003).
[CrossRef]

E. Gleeson, J. A. Murphy, and B. Maffei, "Phase centers of far infrared multi-moded horn antennas," Int. J. Infrared Millim. Waves 23, 711-730 (2002).
[CrossRef]

J. A. Murphy and S. Withington, "Perturbation analysis of Gaussian beam mode scattering at off-axis ellipsoidal mirrors," Infrared Phys. Technol. 37, 205-219 (1996).
[CrossRef]

R. Padman and J. A. Murphy, "A scattering matrix formulation for Gaussian beam-mode analysis," in IEE 7th International Conference on Antennas and Propagation (IEE, 1991), pp. 201-204.

Olver, D.

D. Olver, P. Clarricoats, A. Kishka, and L. Shafai, Microwave Feeds and Horns, Series 39 (IEEE Press, 1994), Chap. 4.

Padman, R.

R. Padman and J. A. Murphy, "A scattering matrix formulation for Gaussian beam-mode analysis," in IEE 7th International Conference on Antennas and Propagation (IEE, 1991), pp. 201-204.

Shafai, L.

D. Olver, P. Clarricoats, A. Kishka, and L. Shafai, Microwave Feeds and Horns, Series 39 (IEEE Press, 1994), Chap. 4.

Siegman, A.

A. Siegman, Lasers (University Science Books, 1986).

Trappe, N.

N. Trappe, A. Murphy, S. Withington, and W. Jellema, "Gaussian beam mode analysis of standing waves between two coupled corrugated horns," IEEE Trans. Antennas Propag. 53, 1755-1761 (2005).
[CrossRef]

N. Trappe, T. Finn, A. Murphy, and W. Jellema, "Gaussian beam mode analysis of standing waves in submillimeter telescopes and receiver systems", Proc. SPIE 5497, 137-145 (2004).

J. A. Murphy, N. Trappe, and S. Withington, "Gaussian beam mode analysis of partial reflections in simple quasi-optical systems fed by horn antennas," Infrared Phys. Technol. 44, 289-297 (2003).
[CrossRef]

Van der Stadt, H.

Withington, S.

N. Trappe, A. Murphy, S. Withington, and W. Jellema, "Gaussian beam mode analysis of standing waves between two coupled corrugated horns," IEEE Trans. Antennas Propag. 53, 1755-1761 (2005).
[CrossRef]

J. A. Murphy, N. Trappe, and S. Withington, "Gaussian beam mode analysis of partial reflections in simple quasi-optical systems fed by horn antennas," Infrared Phys. Technol. 44, 289-297 (2003).
[CrossRef]

J. A. Murphy and S. Withington, "Perturbation analysis of Gaussian beam mode scattering at off-axis ellipsoidal mirrors," Infrared Phys. Technol. 37, 205-219 (1996).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chaps. 1 and 7.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

N. Trappe, A. Murphy, S. Withington, and W. Jellema, "Gaussian beam mode analysis of standing waves between two coupled corrugated horns," IEEE Trans. Antennas Propag. 53, 1755-1761 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

D. Martin and J. Bowen, "Long-wave optics," IEEE Trans. Microwave Theory Tech. 41, 1676-1690 (1993).
[CrossRef]

Infrared Phys. Technol. (2)

J. A. Murphy and S. Withington, "Perturbation analysis of Gaussian beam mode scattering at off-axis ellipsoidal mirrors," Infrared Phys. Technol. 37, 205-219 (1996).
[CrossRef]

J. A. Murphy, N. Trappe, and S. Withington, "Gaussian beam mode analysis of partial reflections in simple quasi-optical systems fed by horn antennas," Infrared Phys. Technol. 44, 289-297 (2003).
[CrossRef]

Int. J. Infrared Millim. Waves (1)

E. Gleeson, J. A. Murphy, and B. Maffei, "Phase centers of far infrared multi-moded horn antennas," Int. J. Infrared Millim. Waves 23, 711-730 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

Proc. SPIE (1)

N. Trappe, T. Finn, A. Murphy, and W. Jellema, "Gaussian beam mode analysis of standing waves in submillimeter telescopes and receiver systems", Proc. SPIE 5497, 137-145 (2004).

Other (5)

P. Goldsmith, Quasioptical Systems, Vol. x of IEEE Series on Microwave Techniques and Radio Frequencies (IEEE Press, 1998), Chaps. 3, 5, and 9.
[CrossRef]

A. Siegman, Lasers (University Science Books, 1986).

R. Padman and J. A. Murphy, "A scattering matrix formulation for Gaussian beam-mode analysis," in IEE 7th International Conference on Antennas and Propagation (IEE, 1991), pp. 201-204.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Chaps. 1 and 7.

D. Olver, P. Clarricoats, A. Kishka, and L. Shafai, Microwave Feeds and Horns, Series 39 (IEEE Press, 1994), Chap. 4.

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Figures (16)

Fig. 1
Fig. 1

Optical system represented by the mode-matching concept.

Fig. 2
Fig. 2

Cascade of N submatrices yields a full 4 × 4 matrix that represents the optical system.

Fig. 3
Fig. 3

Reflection of a quasi-optical beam from a dielectric sheet.

Fig. 4
Fig. 4

Effect on the Gaussian-beam parameters of propagation through a dielectric interface.

Fig. 5
Fig. 5

Propagation of an EM wave through a homogenous film. The arrows represent light rays propagating through the respective media.

Fig. 6
Fig. 6

Reflectivity of a dielectric film of refractive index n as a function of optical thickness for the first configuration (see text).

Fig. 7
Fig. 7

Reflectivity of a dielectric film of refractive index n as a function of optical thickness for the second configuration (see text).

Fig. 8
Fig. 8

Reflectivity response of a quarter-wavelength-thick dielectric as a function of the refractive index of the second medium.

Fig. 9
Fig. 9

Fabry–Perot interferometer with an incident Gaussian beam. The effect of the multiple reflections is to confine a longer free-space propagation to a smaller distance.

Fig. 10
Fig. 10

Amplitudes at partially transmitting sheets i and i + 1 for a succession of N sheets.

Fig. 11
Fig. 11

Transmission response of the FPI for a variety of dielectric substances and wavelengths.

Fig. 12
Fig. 12

Transmission profile for a multimirror FPI where all mirrors have the same reflectivity ( R = 0.5 ) .

Fig. 13
Fig. 13

(a) Transmission profile for a multimirror FPI where the middle mirror has a reflectivity of R = 0.1 . (b) Transmission profile for a multimirror FPI where the middle mirror has a reflectivity of R = 0.88 .

Fig. 14
Fig. 14

Response of the multimirror FPI as a bandpass filter with the appropriate choice of reflectivity values.

Fig. 15
Fig. 15

Experimental setup of two corrugated horns coupled via a FPI.

Fig. 16
Fig. 16

Power coupling of two corrugated horns coupled via a FPI.

Equations (30)

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Ψ m n ( x , y ) = h m ( 2 x W ) h n ( 2 y W ) exp [ j k ( z + r 2 2 R ) ] exp ( j ϕ m n ) ,
h m ( 2 x W ) = 1 2 m 1 2 m ! π W 2 H m ( 2 x W ) exp ( x 2 W 2 ) ,
h n ( 2 y W ) = 1 2 n 1 2 n ! π W 2 H n ( 2 y W ) exp ( y 2 W 2 ) ,
[ [ B ] [ D ] ] = [ [ S 11 ] [ S 12 ] [ S 21 ] [ S 22 ] ] [ [ A ] [ C ] ] ,
[ S a ] = [ [ S 11 a ] [ S 12 a ] [ S 21 a ] [ S 22 a ] ] , [ S b ] = [ [ S 11 b ] [ S 12 b ] [ S 21 b ] [ S 22 b ] ] ,
[ S c ] = [ S 11 c S 12 c S 21 c S 22 c ] .
[ S 11 c ] = [ S 12 a ] [ [ I ] [ S 11 b ] [ S 22 a ] ] 1 [ S 11 b ] [ S 21 a ] + [ S 11 a ] ,
[ S 12 c ] = [ S 12 a ] [ [ I ] [ S 11 b ] [ S 22 a ] ] 1 [ S 12 b ] ,
[ S 21 c ] = [ S 21 b ] [ [ I ] [ S 22 a ] [ S 11 b ] ] 1 [ S 21 a ] ,
[ S 22 c ] = [ S 21 b ] [ [ I ] [ S 22 a ] [ S 11 b ] ] 1 [ S 22 a ] [ S 12 b ] + [ S 22 b ] ,
V m n = exp [ i k z + i ( m + n + 1 ) Δ ϕ m n ] ,
S 11 S 12 S 21 S 22 prop = 0 V V 0 .
S m n = A ( ψ m exp [ j k r 2 2 ( 1 R i n c 2 R c u r v ) ] ) * ψ n exp [ j k r 2 2 ( 1 R i n c ) ] d x d y ,
R ̂ = R n ,
R ̂ 2 = A R ̂ 1 + B C R ̂ 1 + D ,
1 q ̂ n q n R ( z ) j n λ π W ( z ) 2 = 1 R ̂ j λ 0 π W ( z ) 2 ,
q ̂ 2 = A q ̂ 1 + B C q ̂ 1 + D ,
[ r i j S 11 S 12 t i j t i j S 21 S 22 r i j ] ,
r i j = n i n j n i + n j , t i j = 2 n i n i + n j ,
R = r 2 = r i j 2 + r j k 2 + 2 r i j r j k cos ( 2 β ) 1 + r i j 2 r j k 2 + 2 r i j r j k cos ( 2 β ) T = t 2 = n 3 cos ( θ 3 ) n 3 cos ( θ 1 ) t i j 2 t j k 2 1 + r i j 2 r j k 2 + 2 r i j r j k cos ( 2 β ) ,
Δ h = λ 0 2 n 2 cos θ 2 ,
If ( 1 ) m ( n 1 n 2 ) ( n 2 n 3 ) > 0 , we find a maximum.
If ( 1 ) m ( n 1 n 2 ) ( n 2 n 3 ) < 0 , we find a minimum.
F = π r 2 1 r 2 .
t = E N + 1 + E 1 + = t N E N + = t 1 t 2 t N E N + A E N + + B E N = t 1 t 2 t N A + r N B ,
T = t 2 = t 1 2 t 2 2 1 + r 1 2 r 2 2 + 2 r 1 r 2 cos ( 2 β 1 ) ,
Sheet Interface Medium Sheet Interface Air Sheet Interface Medium Sheet Interface.
T = t 1 2 t 2 2 t 3 2 D 3 ,
D 3 = 1 + ( r 1 r 2 ) 2 + ( r 2 r 3 ) 2 + ( r 1 r 3 ) 2 + 2 r 1 r 2 ( 1 + r 3 2 ) cos ( 2 ϕ 1 ) + 2 r 2 r 3 ( 1 + r 1 2 ) cos ( 2 ϕ 2 ) + 2 r 1 r 3 cos ( 2 ϕ 1 + 2 ϕ 2 ) + 2 r 1 r 2 2 r 3 cos ( 2 ϕ 1 2 ϕ 2 ) ,
T = ( 1 r 1 2 ) 2 ( 1 r 2 2 ) ( 1 r 1 2 ) 2 ( 1 r 2 2 ) + [ r 2 ( 1 + r 1 2 ) + 2 r 1 x ] 2 ,

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