Abstract

An analysis of the effects of relative phase changes on the interference pattern formed by the coherent addition of four plane waves is presented. We focus on the configuration in which four plane waves converge at equal angles along two orthogonal planes, an arrangement that is potentially useful for printing arrays of microstructures in resist. We show that, depending on the set of polarization vectors chosen, the shape of the interference pattern is a strong function of the phase difference between each pair of beams. If all the beams have the same phase constant, an intensity distribution that is perfectly modulated and that exhibits strong contrast is produced. However, if the phase constant of any one of the beams is shifted by π from this condition, a pattern with degraded modulation and significantly weaker contrast is formed. We discuss the implication of these results on lithographic applications of multiple-beam patterns. Further, we show that the sensitivity to phase is a general property of all interference patterns formed by four or more intersecting coherent wave fronts that have collinear electric-field components.

© 1998 Optical Society of America

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References

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  1. C. O. Bozler, C. T. Harris, S. Rabe, D. D. Rathman, M. A. Hollis, H. I. Smith, “Arrays of gated field-emitter cones having 0.32 μm tip-to-tip spacing,” J. Vac. Sci. Technol. B 12, 629–632 (1994).
    [CrossRef]
  2. J. P. Spallas, A. M. Hawryluk, D. R. Kania, “Field emitter array mask patterning using interference lithography,” J. Vac. Sci. Technol. B 13, 1973–1978 (1995).
    [CrossRef]
  3. X. Chen, S. H. Zaidi, S. R. J. Brueck, D. J. Devine, “Interferometric lithography of sub-micrometer sparse hole arrays for field-emission display applications,” J. Vac. Sci. Technol. B 14, 3339–3349 (1996).
    [CrossRef]
  4. A. Fernandez, H. T. Nguyen, J. A. Britten, R. D. Boyd, D. R. Kania, A. M. Hawryluk, M. D. Perry, “The use of interference lithography to pattern arrays of submicron resist structures for field emission flat panel displays,” J. Vac. Sci. Technol. B 15, 729–735 (1997).
    [CrossRef]
  5. S. H. Zaidi, S. R. J. Brueck, “Multiple-exposure interferometric lithography,” J. Vac. Sci. Technol. B 11, 658–666 (1993).
    [CrossRef]
  6. A. Fernandez, P. J. Bedrossian, S. L. Baker, S. P. Vernon, D. R. Kania, “Magnetic force microscopy of single-domain cobalt dots patterned using interference lithography,” IEEE Trans. Magn. 32, 4472–4474 (1996).
    [CrossRef]
  7. A. Fernandez, J. Y. Decker, S. M. Herman, D. W. Phillion, D. W. Sweeney, M. D. Perry, “Methods for fabricating array of holes using interference lithography,” J. Vac. Sci. Technol. B 15, 2439–2443 (1997).
    [CrossRef]
  8. M. L. Schattenburg, R. J. Aucoin, R. C. Fleming, I. Plotnik, J. Porter, H. I. Smith, “Fabrication of high energy x-ray transmission gratings for AXAF,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy V, O. H. Siegmund, J. V. Vallerga, eds., Proc. SPIE2280, 181–190 (1994).
    [CrossRef]

1997 (2)

A. Fernandez, H. T. Nguyen, J. A. Britten, R. D. Boyd, D. R. Kania, A. M. Hawryluk, M. D. Perry, “The use of interference lithography to pattern arrays of submicron resist structures for field emission flat panel displays,” J. Vac. Sci. Technol. B 15, 729–735 (1997).
[CrossRef]

A. Fernandez, J. Y. Decker, S. M. Herman, D. W. Phillion, D. W. Sweeney, M. D. Perry, “Methods for fabricating array of holes using interference lithography,” J. Vac. Sci. Technol. B 15, 2439–2443 (1997).
[CrossRef]

1996 (2)

X. Chen, S. H. Zaidi, S. R. J. Brueck, D. J. Devine, “Interferometric lithography of sub-micrometer sparse hole arrays for field-emission display applications,” J. Vac. Sci. Technol. B 14, 3339–3349 (1996).
[CrossRef]

A. Fernandez, P. J. Bedrossian, S. L. Baker, S. P. Vernon, D. R. Kania, “Magnetic force microscopy of single-domain cobalt dots patterned using interference lithography,” IEEE Trans. Magn. 32, 4472–4474 (1996).
[CrossRef]

1995 (1)

J. P. Spallas, A. M. Hawryluk, D. R. Kania, “Field emitter array mask patterning using interference lithography,” J. Vac. Sci. Technol. B 13, 1973–1978 (1995).
[CrossRef]

1994 (1)

C. O. Bozler, C. T. Harris, S. Rabe, D. D. Rathman, M. A. Hollis, H. I. Smith, “Arrays of gated field-emitter cones having 0.32 μm tip-to-tip spacing,” J. Vac. Sci. Technol. B 12, 629–632 (1994).
[CrossRef]

1993 (1)

S. H. Zaidi, S. R. J. Brueck, “Multiple-exposure interferometric lithography,” J. Vac. Sci. Technol. B 11, 658–666 (1993).
[CrossRef]

Aucoin, R. J.

M. L. Schattenburg, R. J. Aucoin, R. C. Fleming, I. Plotnik, J. Porter, H. I. Smith, “Fabrication of high energy x-ray transmission gratings for AXAF,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy V, O. H. Siegmund, J. V. Vallerga, eds., Proc. SPIE2280, 181–190 (1994).
[CrossRef]

Baker, S. L.

A. Fernandez, P. J. Bedrossian, S. L. Baker, S. P. Vernon, D. R. Kania, “Magnetic force microscopy of single-domain cobalt dots patterned using interference lithography,” IEEE Trans. Magn. 32, 4472–4474 (1996).
[CrossRef]

Bedrossian, P. J.

A. Fernandez, P. J. Bedrossian, S. L. Baker, S. P. Vernon, D. R. Kania, “Magnetic force microscopy of single-domain cobalt dots patterned using interference lithography,” IEEE Trans. Magn. 32, 4472–4474 (1996).
[CrossRef]

Boyd, R. D.

A. Fernandez, H. T. Nguyen, J. A. Britten, R. D. Boyd, D. R. Kania, A. M. Hawryluk, M. D. Perry, “The use of interference lithography to pattern arrays of submicron resist structures for field emission flat panel displays,” J. Vac. Sci. Technol. B 15, 729–735 (1997).
[CrossRef]

Bozler, C. O.

C. O. Bozler, C. T. Harris, S. Rabe, D. D. Rathman, M. A. Hollis, H. I. Smith, “Arrays of gated field-emitter cones having 0.32 μm tip-to-tip spacing,” J. Vac. Sci. Technol. B 12, 629–632 (1994).
[CrossRef]

Britten, J. A.

A. Fernandez, H. T. Nguyen, J. A. Britten, R. D. Boyd, D. R. Kania, A. M. Hawryluk, M. D. Perry, “The use of interference lithography to pattern arrays of submicron resist structures for field emission flat panel displays,” J. Vac. Sci. Technol. B 15, 729–735 (1997).
[CrossRef]

Brueck, S. R. J.

X. Chen, S. H. Zaidi, S. R. J. Brueck, D. J. Devine, “Interferometric lithography of sub-micrometer sparse hole arrays for field-emission display applications,” J. Vac. Sci. Technol. B 14, 3339–3349 (1996).
[CrossRef]

S. H. Zaidi, S. R. J. Brueck, “Multiple-exposure interferometric lithography,” J. Vac. Sci. Technol. B 11, 658–666 (1993).
[CrossRef]

Chen, X.

X. Chen, S. H. Zaidi, S. R. J. Brueck, D. J. Devine, “Interferometric lithography of sub-micrometer sparse hole arrays for field-emission display applications,” J. Vac. Sci. Technol. B 14, 3339–3349 (1996).
[CrossRef]

Decker, J. Y.

A. Fernandez, J. Y. Decker, S. M. Herman, D. W. Phillion, D. W. Sweeney, M. D. Perry, “Methods for fabricating array of holes using interference lithography,” J. Vac. Sci. Technol. B 15, 2439–2443 (1997).
[CrossRef]

Devine, D. J.

X. Chen, S. H. Zaidi, S. R. J. Brueck, D. J. Devine, “Interferometric lithography of sub-micrometer sparse hole arrays for field-emission display applications,” J. Vac. Sci. Technol. B 14, 3339–3349 (1996).
[CrossRef]

Fernandez, A.

A. Fernandez, H. T. Nguyen, J. A. Britten, R. D. Boyd, D. R. Kania, A. M. Hawryluk, M. D. Perry, “The use of interference lithography to pattern arrays of submicron resist structures for field emission flat panel displays,” J. Vac. Sci. Technol. B 15, 729–735 (1997).
[CrossRef]

A. Fernandez, J. Y. Decker, S. M. Herman, D. W. Phillion, D. W. Sweeney, M. D. Perry, “Methods for fabricating array of holes using interference lithography,” J. Vac. Sci. Technol. B 15, 2439–2443 (1997).
[CrossRef]

A. Fernandez, P. J. Bedrossian, S. L. Baker, S. P. Vernon, D. R. Kania, “Magnetic force microscopy of single-domain cobalt dots patterned using interference lithography,” IEEE Trans. Magn. 32, 4472–4474 (1996).
[CrossRef]

Fleming, R. C.

M. L. Schattenburg, R. J. Aucoin, R. C. Fleming, I. Plotnik, J. Porter, H. I. Smith, “Fabrication of high energy x-ray transmission gratings for AXAF,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy V, O. H. Siegmund, J. V. Vallerga, eds., Proc. SPIE2280, 181–190 (1994).
[CrossRef]

Harris, C. T.

C. O. Bozler, C. T. Harris, S. Rabe, D. D. Rathman, M. A. Hollis, H. I. Smith, “Arrays of gated field-emitter cones having 0.32 μm tip-to-tip spacing,” J. Vac. Sci. Technol. B 12, 629–632 (1994).
[CrossRef]

Hawryluk, A. M.

A. Fernandez, H. T. Nguyen, J. A. Britten, R. D. Boyd, D. R. Kania, A. M. Hawryluk, M. D. Perry, “The use of interference lithography to pattern arrays of submicron resist structures for field emission flat panel displays,” J. Vac. Sci. Technol. B 15, 729–735 (1997).
[CrossRef]

J. P. Spallas, A. M. Hawryluk, D. R. Kania, “Field emitter array mask patterning using interference lithography,” J. Vac. Sci. Technol. B 13, 1973–1978 (1995).
[CrossRef]

Herman, S. M.

A. Fernandez, J. Y. Decker, S. M. Herman, D. W. Phillion, D. W. Sweeney, M. D. Perry, “Methods for fabricating array of holes using interference lithography,” J. Vac. Sci. Technol. B 15, 2439–2443 (1997).
[CrossRef]

Hollis, M. A.

C. O. Bozler, C. T. Harris, S. Rabe, D. D. Rathman, M. A. Hollis, H. I. Smith, “Arrays of gated field-emitter cones having 0.32 μm tip-to-tip spacing,” J. Vac. Sci. Technol. B 12, 629–632 (1994).
[CrossRef]

Kania, D. R.

A. Fernandez, H. T. Nguyen, J. A. Britten, R. D. Boyd, D. R. Kania, A. M. Hawryluk, M. D. Perry, “The use of interference lithography to pattern arrays of submicron resist structures for field emission flat panel displays,” J. Vac. Sci. Technol. B 15, 729–735 (1997).
[CrossRef]

A. Fernandez, P. J. Bedrossian, S. L. Baker, S. P. Vernon, D. R. Kania, “Magnetic force microscopy of single-domain cobalt dots patterned using interference lithography,” IEEE Trans. Magn. 32, 4472–4474 (1996).
[CrossRef]

J. P. Spallas, A. M. Hawryluk, D. R. Kania, “Field emitter array mask patterning using interference lithography,” J. Vac. Sci. Technol. B 13, 1973–1978 (1995).
[CrossRef]

Nguyen, H. T.

A. Fernandez, H. T. Nguyen, J. A. Britten, R. D. Boyd, D. R. Kania, A. M. Hawryluk, M. D. Perry, “The use of interference lithography to pattern arrays of submicron resist structures for field emission flat panel displays,” J. Vac. Sci. Technol. B 15, 729–735 (1997).
[CrossRef]

Perry, M. D.

A. Fernandez, J. Y. Decker, S. M. Herman, D. W. Phillion, D. W. Sweeney, M. D. Perry, “Methods for fabricating array of holes using interference lithography,” J. Vac. Sci. Technol. B 15, 2439–2443 (1997).
[CrossRef]

A. Fernandez, H. T. Nguyen, J. A. Britten, R. D. Boyd, D. R. Kania, A. M. Hawryluk, M. D. Perry, “The use of interference lithography to pattern arrays of submicron resist structures for field emission flat panel displays,” J. Vac. Sci. Technol. B 15, 729–735 (1997).
[CrossRef]

Phillion, D. W.

A. Fernandez, J. Y. Decker, S. M. Herman, D. W. Phillion, D. W. Sweeney, M. D. Perry, “Methods for fabricating array of holes using interference lithography,” J. Vac. Sci. Technol. B 15, 2439–2443 (1997).
[CrossRef]

Plotnik, I.

M. L. Schattenburg, R. J. Aucoin, R. C. Fleming, I. Plotnik, J. Porter, H. I. Smith, “Fabrication of high energy x-ray transmission gratings for AXAF,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy V, O. H. Siegmund, J. V. Vallerga, eds., Proc. SPIE2280, 181–190 (1994).
[CrossRef]

Porter, J.

M. L. Schattenburg, R. J. Aucoin, R. C. Fleming, I. Plotnik, J. Porter, H. I. Smith, “Fabrication of high energy x-ray transmission gratings for AXAF,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy V, O. H. Siegmund, J. V. Vallerga, eds., Proc. SPIE2280, 181–190 (1994).
[CrossRef]

Rabe, S.

C. O. Bozler, C. T. Harris, S. Rabe, D. D. Rathman, M. A. Hollis, H. I. Smith, “Arrays of gated field-emitter cones having 0.32 μm tip-to-tip spacing,” J. Vac. Sci. Technol. B 12, 629–632 (1994).
[CrossRef]

Rathman, D. D.

C. O. Bozler, C. T. Harris, S. Rabe, D. D. Rathman, M. A. Hollis, H. I. Smith, “Arrays of gated field-emitter cones having 0.32 μm tip-to-tip spacing,” J. Vac. Sci. Technol. B 12, 629–632 (1994).
[CrossRef]

Schattenburg, M. L.

M. L. Schattenburg, R. J. Aucoin, R. C. Fleming, I. Plotnik, J. Porter, H. I. Smith, “Fabrication of high energy x-ray transmission gratings for AXAF,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy V, O. H. Siegmund, J. V. Vallerga, eds., Proc. SPIE2280, 181–190 (1994).
[CrossRef]

Smith, H. I.

C. O. Bozler, C. T. Harris, S. Rabe, D. D. Rathman, M. A. Hollis, H. I. Smith, “Arrays of gated field-emitter cones having 0.32 μm tip-to-tip spacing,” J. Vac. Sci. Technol. B 12, 629–632 (1994).
[CrossRef]

M. L. Schattenburg, R. J. Aucoin, R. C. Fleming, I. Plotnik, J. Porter, H. I. Smith, “Fabrication of high energy x-ray transmission gratings for AXAF,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy V, O. H. Siegmund, J. V. Vallerga, eds., Proc. SPIE2280, 181–190 (1994).
[CrossRef]

Spallas, J. P.

J. P. Spallas, A. M. Hawryluk, D. R. Kania, “Field emitter array mask patterning using interference lithography,” J. Vac. Sci. Technol. B 13, 1973–1978 (1995).
[CrossRef]

Sweeney, D. W.

A. Fernandez, J. Y. Decker, S. M. Herman, D. W. Phillion, D. W. Sweeney, M. D. Perry, “Methods for fabricating array of holes using interference lithography,” J. Vac. Sci. Technol. B 15, 2439–2443 (1997).
[CrossRef]

Vernon, S. P.

A. Fernandez, P. J. Bedrossian, S. L. Baker, S. P. Vernon, D. R. Kania, “Magnetic force microscopy of single-domain cobalt dots patterned using interference lithography,” IEEE Trans. Magn. 32, 4472–4474 (1996).
[CrossRef]

Zaidi, S. H.

X. Chen, S. H. Zaidi, S. R. J. Brueck, D. J. Devine, “Interferometric lithography of sub-micrometer sparse hole arrays for field-emission display applications,” J. Vac. Sci. Technol. B 14, 3339–3349 (1996).
[CrossRef]

S. H. Zaidi, S. R. J. Brueck, “Multiple-exposure interferometric lithography,” J. Vac. Sci. Technol. B 11, 658–666 (1993).
[CrossRef]

IEEE Trans. Magn. (1)

A. Fernandez, P. J. Bedrossian, S. L. Baker, S. P. Vernon, D. R. Kania, “Magnetic force microscopy of single-domain cobalt dots patterned using interference lithography,” IEEE Trans. Magn. 32, 4472–4474 (1996).
[CrossRef]

J. Vac. Sci. Technol. B (6)

A. Fernandez, J. Y. Decker, S. M. Herman, D. W. Phillion, D. W. Sweeney, M. D. Perry, “Methods for fabricating array of holes using interference lithography,” J. Vac. Sci. Technol. B 15, 2439–2443 (1997).
[CrossRef]

C. O. Bozler, C. T. Harris, S. Rabe, D. D. Rathman, M. A. Hollis, H. I. Smith, “Arrays of gated field-emitter cones having 0.32 μm tip-to-tip spacing,” J. Vac. Sci. Technol. B 12, 629–632 (1994).
[CrossRef]

J. P. Spallas, A. M. Hawryluk, D. R. Kania, “Field emitter array mask patterning using interference lithography,” J. Vac. Sci. Technol. B 13, 1973–1978 (1995).
[CrossRef]

X. Chen, S. H. Zaidi, S. R. J. Brueck, D. J. Devine, “Interferometric lithography of sub-micrometer sparse hole arrays for field-emission display applications,” J. Vac. Sci. Technol. B 14, 3339–3349 (1996).
[CrossRef]

A. Fernandez, H. T. Nguyen, J. A. Britten, R. D. Boyd, D. R. Kania, A. M. Hawryluk, M. D. Perry, “The use of interference lithography to pattern arrays of submicron resist structures for field emission flat panel displays,” J. Vac. Sci. Technol. B 15, 729–735 (1997).
[CrossRef]

S. H. Zaidi, S. R. J. Brueck, “Multiple-exposure interferometric lithography,” J. Vac. Sci. Technol. B 11, 658–666 (1993).
[CrossRef]

Other (1)

M. L. Schattenburg, R. J. Aucoin, R. C. Fleming, I. Plotnik, J. Porter, H. I. Smith, “Fabrication of high energy x-ray transmission gratings for AXAF,” in EUV, X-Ray, and Gamma-Ray Instrumentation for Astronomy V, O. H. Siegmund, J. V. Vallerga, eds., Proc. SPIE2280, 181–190 (1994).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

2D color map of the intensity distribution for the TE–TM four-beam interference pattern when the net relative phase ϕ0 is zero and the incident angle is 30°. The distribution is normalized to E 0 2 = 1, and the spatial scales are normalized to k = 1. This same distribution occurs whenever the net relative phase is an integral multiple of 2π.

Fig. 2
Fig. 2

2D color map of the intensity distribution for the TE–TE four-beam interference pattern, plotted with the same color scale and normalization as that used in Fig. 1.

Fig. 3
Fig. 3

Plot of intensity profiles from the TE–TM (solid curve) and TE–TE (dashed curve) distributions, taken through the maxima along a line that runs parallel to the y axis for the TE–TM pattern and that runs diagonal to the xy axes for the TE–TE pattern. The intensity values are normalized to the maximum intensity for the corresponding distribution, and the horizontal scale is normalized to the periodicity of that pattern.

Fig. 4
Fig. 4

2D color maps of the intensity distribution for the TE–TM four-beam interference pattern when the net relative phase ϕ0 is (a) π/2 and (b) π, and the incident angle is 30°. The intensity and the spatial scales are normalized in a fashion similar to that of Fig. 1. These plots illustrate the strong sensitivity of the TE–TM pattern to phase.

Fig. 5
Fig. 5

Plot of the TE–TM intensity distribution taken along the y axis when a tilt error of π/100 is added to the beam incident in the -y direction.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

E 1 = E 0 p ˆ 1 cos k y   sin   θ - z   cos   θ - ω t + ϕ 1 , E 2 = E 0 p ˆ 2 cos k - y   sin   θ - z   cos   θ - ω t + ϕ 2 , E 3 = E 0 p ˆ 3 cos k x   sin   θ - z   cos   θ - ω t + ϕ 3 , E 4 = E 0 p ˆ 4 cos k - x   sin   θ - z   cos   θ - ω t + ϕ 4 ,
I x ,   y = E 0 2 2 + cos 2 ky   sin   θ + ϕ 1 - ϕ 2 + cos 2 kx   sin   θ + ϕ 3 - ϕ 4 .
I x ,   y = E 0 2 2 + cos   2 θ   cos 2 kx   sin   θ + cos 2 ky   sin   θ + ϕ 0 + 2   cos   θ   cos kx   sin   θ cos ky   sin   θ + cos ky   sin   θ + ϕ 0 ,
I x ,   y ,   z = E 0 2 2 + cos   2 θ   cos 2 kx   sin   θ + 2   cos   θ   cos kx   sin   θ cos ky   sin   θ + cos kz δ   sin   θ + ky δ   cos   θ + ky   sin   θ + ϕ 0 + 2   cos kz δ   sin   θ + ky δ   cos   θ + 2 ky   sin   θ + ϕ 0 .
Δ x = ϕ 4 - ϕ 3 / 2 k   sin   θ , Δ y = ϕ 2 - ϕ 3 + ϕ 4 / 2 / k   sin   θ ,
E 1 = E 0 p ˆ 1 cos k y   sin   θ - z   cos   θ - ω t + ϕ 0 + ψ 0 , E 2 = E 0 p ˆ 2 cos k - y   sin   θ - z   cos   θ - ω t + ψ 0 , E 3 = E 0 p ˆ 3 cos k x   sin   θ - z   cos   θ - ω t + ψ 0 , E 4 = E 0 p ˆ 4 cos k - x   sin   θ - z   cos   θ - ω t + ψ 0 ,

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