Abstract

The diffraction of Hermite–Gaussian beams by N equally spaced slits in a planar screen (a lamellar finite grating) at the scalar diffraction regime is analyzed. We start from the Rayleigh–Sommerfeld theory for two-dimensional problems with Dirichlet conditions and Kirchhoff’s approximation. The theory is presented for beams at oblique incidence; however, in the numerical simulation mainly normal incidence is considered. For Gaussian beams the ratio of the intensity diffracted at normal direction E at minimum and maximum transmitted power is studied in this paper as a function of the beam width L. Also, for Gaussian beams, the angular positions θmin of the first minimum of the diffraction patterns as a function of the L-spot diameter is analyzed. Two methods to determine L for Gaussian beams are proposed. For Hermite–Gaussian beams an interesting diffraction property previously presented for one slit [J. Opt. Soc. Am. A 12, 2440 (1995)] is generalized to finite gratings, namely, τ=λ(E/N), where τ is the transmission coefficient and λ is the wavelength of the incident radiation. We study this property in detail and analyze its validity conditions in terms of the optogeometrical parameters of the system. This property might be useful in determining the total energy transmitted by a lamellar grating from knowledge of the intensity diffracted in the normal direction.

© 2001 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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. 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]
  3. H. G. Kraus, “Huygens–Fresnel–Kirchhoff wave-front diffraction formulation: paraxial and exact Gaussian laser beams,” J. Opt. Soc. Am. A 7, 47–65 (1990).
    [CrossRef]
  4. E. 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]
  5. 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]
  6. B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am. A 8, 705–717 (1991).
    [CrossRef]
  7. P. L. Overfelt, C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel–Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8, 732–745 (1991).
    [CrossRef]
  8. O. Mata-Mendez, F. Chavez-Rivas, “Diffraction of Hermite–Gaussian beams by a slit,” J. Opt. Soc. Am. A 12, 2440–2445 (1995).
    [CrossRef]
  9. T. Kojima, “Diffraction of Hermite–Gaussian beams from a sinusoidal conducting grating,” J. Opt. Soc. Am. A 7, 1740–1744 (1990).
    [CrossRef]
  10. O. Mata-Mendez, F. Chavez-Rivas, “New property in the diffraction of Hermite–Gaussian beams by a finite grating in the scalar diffraction regime: constant-intensity angles in the far field when the beam center is displaced through the grating,” J. Opt. Soc. Am. A 15, 2698–2704 (1998).
    [CrossRef]
  11. D. S. Marx, D. Psaltis, “Optical diffraction of focused spots and subwavelength structures,” J. Opt. Soc. Am. A 14, 1268–1278 (1997).
    [CrossRef]
  12. R. Csomor, “Techniques for measuring 1-μm diam Gaussian beams: comment,” Appl. Opt. 24, 2295–2298 (1985).
    [CrossRef] [PubMed]
  13. M. A. Karim, A. A. Awwal, A. M. Nasiruddin, A. Basit, D. S. Vedak, C. C. Smith, G. D. Miller, “Gaussian laser-beam-diameter measurement using sinusoidal and triangular rulings,” Opt. Lett. 12, 93–95 (1987).
    [CrossRef] [PubMed]
  14. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–99 (1954).
    [CrossRef]
  15. J. T. Foley, E. Wolf, “Note on the far field of a Gaussian beam,” J. Opt. Soc. Am. 69, 761–764 (1979).
    [CrossRef]
  16. 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]
  17. 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]
  18. O. Mata-Mendez, “Scattering of electromagnetic beams from rough surfaces,” Phys. Rev. B 37, 8182–8189 (1988).
    [CrossRef]
  19. A. A. S. Awwal, J. A. Smith, J. Belloto, G. Bharatram, “Wide-range laser-beam-diameter measurement using a periodic exponential grating,” Opt. Laser Technol. 23, 159–161 (1991).
    [CrossRef]
  20. A. K. Cherri, A. S. Awwal, “Periodic exponential ruling for the measurement of Gaussian laser beam diameters,” Opt. Eng. 32, 1038–1042 (1993).
    [CrossRef]
  21. J. S. Uppal, P. K. Gupta, R. G. Harrison, “Aperiodic ruling for the measurement of Gaussian laser beam diameters,” Opt. Lett. 14, 683–685 (1989).
    [CrossRef] [PubMed]
  22. A. K. Cherri, A. A. Awwal, A. A. Karim, “Generalization of the Ronchi, sinusoidal, and triangular ruling for Gaussian-laser-beam-diameter measurements,” Appl. Opt. 32, 2235–2242 (1993).
    [CrossRef] [PubMed]

1998 (1)

1997 (1)

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 (3)

1991 (4)

1990 (2)

1989 (1)

1988 (1)

O. Mata-Mendez, “Scattering of electromagnetic beams from rough surfaces,” Phys. Rev. B 37, 8182–8189 (1988).
[CrossRef]

1987 (1)

1985 (1)

1983 (1)

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]

Awwal, A. A.

Awwal, A. A. S.

A. A. S. Awwal, J. A. Smith, J. Belloto, G. Bharatram, “Wide-range laser-beam-diameter measurement using a periodic exponential grating,” Opt. Laser Technol. 23, 159–161 (1991).
[CrossRef]

Awwal, A. S.

A. K. Cherri, A. S. Awwal, “Periodic exponential ruling for the measurement of Gaussian laser beam diameters,” Opt. Eng. 32, 1038–1042 (1993).
[CrossRef]

Basit, A.

Belloto, J.

A. A. S. Awwal, J. A. Smith, J. Belloto, G. Bharatram, “Wide-range laser-beam-diameter measurement using a periodic exponential grating,” Opt. Laser Technol. 23, 159–161 (1991).
[CrossRef]

Bharatram, G.

A. A. S. Awwal, J. A. Smith, J. Belloto, G. Bharatram, “Wide-range laser-beam-diameter measurement using a periodic exponential grating,” Opt. Laser Technol. 23, 159–161 (1991).
[CrossRef]

Bouwkamp, C. J.

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

Cadilhac, M.

Chavez-Rivas, F.

Cherri, A. K.

A. K. Cherri, A. A. Awwal, A. A. Karim, “Generalization of the Ronchi, sinusoidal, and triangular ruling for Gaussian-laser-beam-diameter measurements,” Appl. Opt. 32, 2235–2242 (1993).
[CrossRef] [PubMed]

A. K. Cherri, A. S. Awwal, “Periodic exponential ruling for the measurement of Gaussian laser beam diameters,” Opt. Eng. 32, 1038–1042 (1993).
[CrossRef]

Csomor, R.

Foley, J. T.

Gupta, P. K.

Hafizi, B.

Harrison, R. G.

Karim, A. A.

Karim, M. A.

Kenney, C. S.

Kojima, T.

Kraus, H. G.

Kriezis, E. 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]

Marx, D. S.

Mata-Mendez, O.

Miller, G. D.

Nasiruddin, A. M.

Overfelt, P. L.

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.

Psaltis, D.

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]

Smith, C. C.

Smith, J. A.

A. A. S. Awwal, J. A. Smith, J. Belloto, G. Bharatram, “Wide-range laser-beam-diameter measurement using a periodic exponential grating,” Opt. Laser Technol. 23, 159–161 (1991).
[CrossRef]

Sprangle, P.

Uppal, J. S.

Vedak, D. S.

Wolf, E.

Am. J. Phys. (1)

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. (3)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (8)

Opt. Eng. (1)

A. K. Cherri, A. S. Awwal, “Periodic exponential ruling for the measurement of Gaussian laser beam diameters,” Opt. Eng. 32, 1038–1042 (1993).
[CrossRef]

Opt. Laser Technol. (2)

A. A. S. Awwal, J. A. Smith, J. Belloto, G. Bharatram, “Wide-range laser-beam-diameter measurement using a periodic exponential grating,” Opt. Laser Technol. 23, 159–161 (1991).
[CrossRef]

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. (3)

Phys. Rev. B (1)

O. Mata-Mendez, “Scattering of electromagnetic beams from rough surfaces,” Phys. Rev. B 37, 8182–8189 (1988).
[CrossRef]

Rep. Prog. Phys. (1)

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1

Our configuration composed of N slits of width l, separation d, and period D=l+d in an infinitely thin planar screen (finite lamellar grating). The slits are parallel to the Oz axis.

Fig. 2
Fig. 2

Energy diffracted normally to the grating E as a function of the normalized beam position b/l when l=1, D/l=2, L/l=0.8, 1.2, 1.5, and λ/l=0.03.

Fig. 3
Fig. 3

Intensity ratio K=Emin/Emax as a function of the normalized beam width L/l when l=1 and D=2.

Fig. 4
Fig. 4

Diffraction patterns at minimum Pmin and maximum Pmax transmitted powers calculated by means of the Rayleigh–Sommerfeld theory when l=1, D=2, and λ=0.03: (a) L/l=1.5, (b) L/l=2.5.

Fig. 5
Fig. 5

First minimum θmin (degree) of the diffraction patterns at minimum transmitted power versus the normalized beam width (L/l) for D/λ=350, D/λ=66.666, and D/λ=15. The solid curves are the result of Eq. (16), and the points were calculated by means of the Rayleigh–Sommerfeld theory.

Fig. 6
Fig. 6

Same as Fig. 5 but for the normalized wavelength (λ/l), with D/L=1.1312, D/L=0.75, and D/L=1.66.

Fig. 7
Fig. 7

First minimum θmin (degree) of the diffraction patterns at minimum transmitted power versus the normalized grating period (D/l), for m=0 and λ/l=0.03. The solid curves are the result of Eq. (16), and the points were calculated by means of the Rayleigh–Sommerfeld theory.

Fig. 8
Fig. 8

Comparison of the two methods proposed in Subsections 3.A (L1/l) and 3.B (L2/l). L1/l and L2/l are plotted as a function of L/l for D/l=2, d/l=1, and λ/l=0.03. The straight line has an inclination of 45°.

Fig. 9
Fig. 9

One slit. Ratio τ/λ (solid curves) and E (dotted curves) as a function of the normalized beam width (L/l) with λ/l=0.01, θi=0°, and b/l=0.5: (a) Hermite–Gaussian beams, (b) distorted beams.

Fig. 10
Fig. 10

One slit. Ratio τ/λ (solid curves) and E (dotted curves) as a function of the normalized beam position (b/l) with λ/l=0.01, L/l=4, θi=0°, and m=1: (a) Hermite–Gaussian beam, (b) distorted beam.

Fig. 11
Fig. 11

One slit. Ratio τ/λ (solid curves) and E (dotted curves) as a function of the normalized wavelength (λ/l) with θi=0° and b/l=0.5: (a) Hermite–Gaussian beams with L/l=4, (b) distorted beams with L/l=7.

Fig. 12
Fig. 12

One slit. Ratio τ/λ (solid curves) and E (dotted curves) as a function (a) of the normalized wavelength (λ/l) and (b) for the normalized beam width (L/l) with L/l=4 and λ/l=0.13, respectively, for θi=45°, b/l=0.5, and m=0.

Fig. 13
Fig. 13

Finite grating. Nτ/λ (solid curves) and E (dotted curves) as a function of the normalized beam width (L/l) with N=8, θi=0°, d/l=π, b/l=14.995, and λ/l=0.01.

Fig. 14
Fig. 14

Finite grating. Nτ/λ (solid curves) and E (dotted curves) as a function of the normalized beam position (b/l) with N=8, θi=0°, d/l=π, λ/l=0.01, and L/l=100.

Fig. 15
Fig. 15

Finite grating. Same as Fig. 14 but for L/l=500 and m=3.

Fig. 16
Fig. 16

Grating. Nτ/λ (solid curve) and E (dotted curve) as a function of the normalized beam position (b/l) with N=1000, L/l=50,000, m=1, λ/l=0.1, d/l=1, and θi=0°.

Equations (25)

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

E(x0, y0)=i2-E(x, 0)y0H01(kr)dx,
E(x0, y0)=i2j=1Najaj+lEi(x, 0)y0H01(kr)dx,
H01(kr)2/πkrexp(iπ/4)exp(ikr).
y0H01(kr)-i2k/πexp(-iπ/4)×exp[ik(r0-x sin θ)]r01/2cos θ,
E(x0, y0)=f(θ)exp(ikr0)/r0,
f(θ)=kexp(-iπ/4)cos θE^i(k sin θ, 0),
E^i(α, 0)=12πj=1Najaj+lEi(x, 0)exp(-iαx)dx.
I(θ)=k2cos2 θ|E^i(k sin θ, 0)|2,
Ei(x, y)=12π-kkA(α)exp[i(αx-βy)]dα,
E^i(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.
I0=-Im-Ei*Eiydx=-kkβ|A(α)|2dα,
Ei(x, y=0)=Hm2L(x-b)exp-2(x-b)2L2,
A(α)=L2(i)mHm-L2q1(θi)q2(θi)exp[-iαb]×exp[-q1(θi)2L2/8],
q1(θi)=α cos θi-β sin θi
q2(θi)=cos θi+(α/β)sin θi.
Ad(α)=A(α)exp(α).
K=Emin/Emax
θmin=λ2D1+0.3709(L/l)2.
E=Nτλ,
E^i(α, 0)=12πj=1N-kkA(α)dα×ajaj+lexp[i(α-α)x]dx,
ajaj+lexp[i(α-α)x]dx
=l exp[(α-α)(aj+l/2)]×sin[(α-α)l/2](α-α)l/2,
E^i(α, 0)=l2π-kkA(α)sin[(α-α)l/2](α-α)l/2×exp[i(α-α)l/2]dα×j=1Nexp[i(α-α)aj].
j=1Nexp[i(α-α)aj]
=j=1Nexp[i(α-α)(l+d)(j-1)]=1-exp[iN(α-α)(l+d)]1-exp[i(α-α)(l+d)].

Metrics