Abstract

We study the diffraction of Hermite–Gaussian beams by N equally spaced slits (finite grating) in a planar screen by means of the Rayleigh–Sommerfeld theory in the scalar diffraction regime. We find in the far field the existence of constant-intensity angles when the incident-beam position on the screen is changed. We have determined the optimal conditions under which this new diffraction property can be achieved. Also, a novel effect called the constant-intensity-angles-collapse effect is presented, in which the constant-intensity angles collapse to the minima of the diffraction pattern when the incident spot size is enlarged. For the grating case, deep dips at the maxima of the diffraction patterns of Hermite–Gaussian beams are predicted. Also, for a grating we have found that large segments of these diffraction patterns are independent of the incident-beam position (called constant-intensity curves). The results of this report may be useful for considering unstable vibrational systems in which constant intensities at some angular direction of the far field are required.

© 1998 Optical Society of America

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References

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  1. O. Mata-Mendez, “Diffraction and beam-diameter measurement of Gaussian beams at optical and microwave frequencies,” Opt. Lett. 16, 1629–1631 (1991).
    [CrossRef] [PubMed]
  2. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–99 (1954).
    [CrossRef]
  3. A. Sasaki, “Fraunhofer diffraction of Gaussian laser beams by a single-slit,” Jpn. J. Appl. Phys. 19, 1195–1196 (1980).
    [CrossRef]
  4. R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. 23, 2227–2231 (1984).
    [CrossRef] [PubMed]
  5. H. K. Pak, S.-H. Park, “Double slit with continuously variable width and center-to-center separation,” Appl. Opt. 32, 3596–3597 (1993).
    [CrossRef] [PubMed]
  6. M. J. McIrvin, “The Fibonacci ruler,” Am. J. Phys. 61, 36–39 (1993).
    [CrossRef]
  7. O. Mata-Mendez, F. Chavez-Rivas, “Diffraction of Hermite–Gaussian beams by a slit,” J. Opt. Soc. Am. A 12, 2440–2445 (1995).
    [CrossRef]
  8. O. Mata-Mendez, M. Cadilhac, R. Petit, “Diffraction of a two-dimensional electromagnetic beam wave by a thick slit pierced in a perfectly conducting screen,” J. Opt. Soc. Am. 73, 328–331 (1983).
    [CrossRef]
  9. Em. E. Kriezis, P. K. Pandelakis, A. G. Papagiannakis, “Diffraction of a Gaussian beam from a periodic planar screen,” J. Opt. Soc. Am. A 11, 630–636 (1994).
    [CrossRef]
  10. G. A. Suedan, E. V. Jull, “Two-dimensional beam diffraction by a half-plane and wide slit,” IEEE Trans. Antennas Propag. AP-35, 1077–1083 (1987).
    [CrossRef]
  11. H. Laabs, B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
    [CrossRef]
  12. M. Padgett, J. Arlt, N. Simpson, L. Allen, “An experiment to observe the intensity and phase structure of Laguerre–Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
    [CrossRef]
  13. T. Kojima, “Diffraction of Hermite–Gaussian beams from a sinusoidal conducting grating,” J. Opt. Soc. Am. A 7, 1740–1744 (1990).
    [CrossRef]
  14. A. Zuñiga-Segundo, O. Mata-Mendez, “Interaction of S-polarized beams with infinitely conducting grooves: enhanced fields and dips in the reflectivity,” Phys. Rev. B 46, 536–539 (1992).
    [CrossRef]
  15. D. Wright, “Beam widths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, S1129–S1135 (1992).
    [CrossRef]
  16. J. T. Foley, E. Wolf, “Note on the far field of a Gaussian beam,” J. Opt. Soc. Am. 69, 761–764 (1979).
    [CrossRef]

1996 (2)

H. Laabs, B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[CrossRef]

M. Padgett, J. Arlt, N. Simpson, L. Allen, “An experiment to observe the intensity and phase structure of Laguerre–Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (2)

1992 (2)

A. Zuñiga-Segundo, O. Mata-Mendez, “Interaction of S-polarized beams with infinitely conducting grooves: enhanced fields and dips in the reflectivity,” Phys. Rev. B 46, 536–539 (1992).
[CrossRef]

D. Wright, “Beam widths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, S1129–S1135 (1992).
[CrossRef]

1991 (1)

1990 (1)

1987 (1)

G. A. Suedan, E. V. Jull, “Two-dimensional beam diffraction by a half-plane and wide slit,” IEEE Trans. Antennas Propag. AP-35, 1077–1083 (1987).
[CrossRef]

1984 (1)

1983 (1)

1980 (1)

A. Sasaki, “Fraunhofer diffraction of Gaussian laser beams by a single-slit,” Jpn. J. Appl. Phys. 19, 1195–1196 (1980).
[CrossRef]

1979 (1)

1954 (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–99 (1954).
[CrossRef]

Allen, L.

M. Padgett, J. Arlt, N. Simpson, L. Allen, “An experiment to observe the intensity and phase structure of Laguerre–Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[CrossRef]

Arlt, J.

M. Padgett, J. Arlt, N. Simpson, L. Allen, “An experiment to observe the intensity and phase structure of Laguerre–Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[CrossRef]

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–99 (1954).
[CrossRef]

Cadilhac, M.

Chavez-Rivas, F.

Foley, J. T.

Jull, E. V.

G. A. Suedan, E. V. Jull, “Two-dimensional beam diffraction by a half-plane and wide slit,” IEEE Trans. Antennas Propag. AP-35, 1077–1083 (1987).
[CrossRef]

Kojima, T.

Kriezis, Em. E.

Laabs, H.

H. Laabs, B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[CrossRef]

Mata-Mendez, O.

McCally, R. L.

McIrvin, M. J.

M. J. McIrvin, “The Fibonacci ruler,” Am. J. Phys. 61, 36–39 (1993).
[CrossRef]

Ozygus, B.

H. Laabs, B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[CrossRef]

Padgett, M.

M. Padgett, J. Arlt, N. Simpson, L. Allen, “An experiment to observe the intensity and phase structure of Laguerre–Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[CrossRef]

Pak, H. K.

Pandelakis, P. K.

Papagiannakis, A. G.

Park, S.-H.

Petit, R.

Sasaki, A.

A. Sasaki, “Fraunhofer diffraction of Gaussian laser beams by a single-slit,” Jpn. J. Appl. Phys. 19, 1195–1196 (1980).
[CrossRef]

Simpson, N.

M. Padgett, J. Arlt, N. Simpson, L. Allen, “An experiment to observe the intensity and phase structure of Laguerre–Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[CrossRef]

Suedan, G. A.

G. A. Suedan, E. V. Jull, “Two-dimensional beam diffraction by a half-plane and wide slit,” IEEE Trans. Antennas Propag. AP-35, 1077–1083 (1987).
[CrossRef]

Wolf, E.

Wright, D.

D. Wright, “Beam widths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, S1129–S1135 (1992).
[CrossRef]

Zuñiga-Segundo, A.

A. Zuñiga-Segundo, O. Mata-Mendez, “Interaction of S-polarized beams with infinitely conducting grooves: enhanced fields and dips in the reflectivity,” Phys. Rev. B 46, 536–539 (1992).
[CrossRef]

Am. J. Phys. (2)

M. J. McIrvin, “The Fibonacci ruler,” Am. J. Phys. 61, 36–39 (1993).
[CrossRef]

M. Padgett, J. Arlt, N. Simpson, L. Allen, “An experiment to observe the intensity and phase structure of Laguerre–Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

G. A. Suedan, E. V. Jull, “Two-dimensional beam diffraction by a half-plane and wide slit,” IEEE Trans. Antennas Propag. AP-35, 1077–1083 (1987).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys. (1)

A. Sasaki, “Fraunhofer diffraction of Gaussian laser beams by a single-slit,” Jpn. J. Appl. Phys. 19, 1195–1196 (1980).
[CrossRef]

Opt. Laser Technol. (1)

H. Laabs, B. Ozygus, “Excitation of Hermite Gaussian modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

D. Wright, “Beam widths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, S1129–S1135 (1992).
[CrossRef]

Phys. Rev. B (1)

A. Zuñiga-Segundo, O. Mata-Mendez, “Interaction of S-polarized beams with infinitely conducting grooves: enhanced fields and dips in the reflectivity,” Phys. Rev. B 46, 536–539 (1992).
[CrossRef]

Rep. Prog. Phys. (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–99 (1954).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

System with N equally spaced slits of width l and separation d in an infinitely thin planar screen. The slits are parallel to the Oz axis, θi being the angle of incidence of the beam wave of finite cross section measured from the normal to the screen.

Fig. 2
Fig. 2

One slit. Diffraction patterns normalized to the incident energy of a normally incident Gaussian beam. With λ/l=0.01 and L/l=3/2. The beam positions are b/l=0.5, 0.25, 0.125, and -0.03 (the slit is always covered by the incident beam).

Fig. 3
Fig. 3

One slit. Diffraction patterns normalized to the incident energy of normally incident Hermite–Gaussian beams of order (a) n=1 and (b) n=2. Same parameters as in Fig. 2.

Fig. 4
Fig. 4

Two slits. Diffraction patterns normalized to the incident energy of a normally incident Gaussian beam. With λ/l=0.01, d/l=1, and L/l=5/2. The beam positions are b/l=0.75, 1.0, 1.5, and 1.85 (the two slits are always covered by the incident beam).

Fig. 5
Fig. 5

Two slits. Diffraction patterns normalized to the incident energy of a normally incident Hermite–Gaussian beam of order n=1. Same parameters as in Fig. 4.

Fig. 6
Fig. 6

Two slits. Behavior of the (a), (b), and (c) CIA’s of Fig. 4 as a function of the wavelength, from λ/l=0.001 to λ/l=0.2 (the scalar–vectorial transition point). Top panel, angular position; bottom panel, corresponding normalized intensity. Same parameters as in Fig. 4.

Fig. 7
Fig. 7

Two slits. Behavior of the (a), (b), and (c) CIA’s of Fig. 4 as a function of the normalized Gaussian beam diameter (L/l). Top panel, angular position; bottom panel, corresponding normalized intensity. Same parameters as in Fig. 4.

Fig. 8
Fig. 8

Two slits. Behavior of the (a), (b), and (c) CIA’s of Fig. 4 as a function of the normalized slit separation (d/l). Top panel, angular position; bottom panel, normalized intensity. Same parameters as in Fig. 4.

Fig. 9
Fig. 9

Grating. Diffraction patterns generated by normally incident Gaussian beams on a grating, with λ/l=0.1 and d/l=1. The beam widths are (a) L/l=4/2, (b) L/l=9/2, and (c) L/l=20/2.

Fig. 10
Fig. 10

Grating. Same as Fig. 9 but for normally incident Hermite–Gaussian beams of order n=1.

Fig. 11
Fig. 11

Intensity of the first CIA’s of Fig. 2 as a function of the beam position calculated by RST and two rigorous theories for TE and TM polarizations.

Equations (10)

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Ei(x, y=0)=Hn2L(x-b)exp-2(x-b)2L2,
E(x0, y0)=f(θ)exp(ikr0)/r0,
f(θ)=k exp(-iπ/4)cos θEˆi(k sin θ, 0).
Eˆi(α, 0)12π-Ei(x, 0)exp(-iαx)dx=12πj=1Najaj+lEi(x, 0)exp(-iαx)dx,
Ei(x, y)=12π-kkA(α)exp[i(αx-βy)]dα,
Eˆi(k sin θ, 0)=l2π-kkA(α)exp[i(α-k sin θ)l/2]×exp[i(N-1)×(α-k sin θ)(d+l)/2]×sin[(α-k sin θ)l/2](α-k sin θ)l/2×sin[N(α-k sin θ)(d+l)/2]sin[(α-k sin θ)(d+l)/2]dα.
I(θ)=k2 cos2 θI0|Eˆi(k sin θ, 0)|2,
I0=-kkβ|A(α)|2dα.
A(α)=L2(i)nHn-L2q1(θi)q2(θi)×exp(-iαb)exp[-q1(θi)2L2/8],
|θ|=|θmin|±λ221L-12L2,

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