Abstract

Diffraction of Hermite–Gaussian beams of arbitrary order by a slit is treated. We start from the Rayleigh integral equation for two-dimensional diffraction problems with Dirichlet conditions and the Kirchhoff approximation. We analyze the transmission coefficient τ, the intensity diffracted at normal direction ℰ, and the ratio of the minimum transmitted power to the maximum transmitted power, κ. New analytical expressions for τ and κ are given in simple form as a function of the position of the incident beam wave (with respect to the slit) and the spot size. Also, an interesting diffraction property of ℰ and τ given by ℰ = τ/λ is presented, where λ is the wavelength of the incident beam wave. We have found that the diffraction patterns at minimum transmitted power have an unusual shape: at vertical incidence the diffracted energy at normal direction is always null.

© 1995 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. 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]
  3. B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am A 8, 705–717 (1991).
    [CrossRef]
  4. 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]
  5. T. Kojima, “Diffraction of Hermite–Gaussian beams from a sinusoidal conducting grating,” J. Opt. Soc. Am. A 7, 1740–1744 (1990).
    [CrossRef]
  6. T. Kudou, M. Yokota, O. Fukumitsu, “Reflection and transmission of a Hermite–Gaussian beam incident upon a curved dielectric layer,” J. Opt. Soc. Am. A 8, 718–723 (1991).
    [CrossRef]
  7. 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]
  8. N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyzer,” Opt. Quantum Electron. 24, S927–S949 (1992).
    [CrossRef]
  9. G. A. Suedan, E. V. Jull, “Two-dimensional beam diffraction by a half-plane and wide slit,” IEEE Trans. Antennas Propag., AP-35, 1007–1083 (1987).
  10. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1975), Chap. 6.
  11. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–99 (1954).
    [CrossRef]
  12. A. Sommerfeld, “Optics,” in Lectures on Theoretical Physics, A. Sommerfeld, ed. (Academic, New York, 1964), Vol. IV, Chap. VI, p. 273.
  13. D. Wright, “Beam widths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, S1129–S1135 (1992).
    [CrossRef]
  14. J. T. Foley, E. Wolf, “Note on the far field of a Gaussian beam,” J. Opt. Soc. Am. 69, 761–764 (1979).
    [CrossRef]
  15. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965) p.838, formula (7.376.1).
  16. We call attention to the following typographical error in Eq. (4) of Ref. 6 and Eq. (13) of Ref. 8. The argument of the Hermite polynomial must be that given in Eq. (12) of this paper.

1992 (3)

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]

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyzer,” Opt. Quantum Electron. 24, S927–S949 (1992).
[CrossRef]

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

1991 (5)

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, 1007–1083 (1987).

1979 (1)

1954 (1)

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

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]

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]

Foley, J. T.

Fukumitsu, O.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965) p.838, formula (7.376.1).

Haase, T.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyzer,” Opt. Quantum Electron. 24, S927–S949 (1992).
[CrossRef]

Hafizi, B.

B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am A 8, 705–717 (1991).
[CrossRef]

Hodgson, N.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyzer,” Opt. Quantum Electron. 24, S927–S949 (1992).
[CrossRef]

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, 1007–1083 (1987).

Kenney, C. S.

Kojima, T.

Kostka, R.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyzer,” Opt. Quantum Electron. 24, S927–S949 (1992).
[CrossRef]

Kudou, T.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1975), Chap. 6.

Mata-Mendez, O.

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]

O. Mata-Mendez, “Diffraction and beam-diameter measurement of Gaussian beams at optical and microwave frequencies,” Opt. Lett. 16, 1629–1631 (1991).
[CrossRef] [PubMed]

Overfelt, P. L.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965) p.838, formula (7.376.1).

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]

Sommerfeld, A.

A. Sommerfeld, “Optics,” in Lectures on Theoretical Physics, A. Sommerfeld, ed. (Academic, New York, 1964), Vol. IV, Chap. VI, p. 273.

Sprangle, P.

B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am A 8, 705–717 (1991).
[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, 1007–1083 (1987).

Weber, H.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyzer,” Opt. Quantum Electron. 24, S927–S949 (1992).
[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]

Yokota, M.

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]

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, 1007–1083 (1987).

J. Opt. Soc. Am A (1)

B. Hafizi, P. Sprangle, “Diffraction effects in directed radiation beams,” J. Opt. Soc. Am A 8, 705–717 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Laser Technol. (1)

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]

Opt. Lett. (1)

Opt. Quantum Electron. (2)

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyzer,” Opt. Quantum Electron. 24, S927–S949 (1992).
[CrossRef]

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]

Other (4)

A. Sommerfeld, “Optics,” in Lectures on Theoretical Physics, A. Sommerfeld, ed. (Academic, New York, 1964), Vol. IV, Chap. VI, p. 273.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965) p.838, formula (7.376.1).

We call attention to the following typographical error in Eq. (4) of Ref. 6 and Eq. (13) of Ref. 8. The argument of the Hermite polynomial must be that given in Eq. (12) of this paper.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1975), Chap. 6.

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Figures (7)

Fig. 1
Fig. 1

The model. The slit (of width l) is parallel to the z axis, i.e., perpendicular to the plane of the figure. The position of the incident Hermite–Gaussian beam is fixed by the parameters b and h.

Fig. 2
Fig. 2

Diffraction pattern dependence on the m mode (m = 0, 1, 2, 3) of the Hermite–Gaussian beam for θ0 = 30° with λ/l = 0.1, L/l = 4, and b/l = 0.5 (at the edge of the slit).

Fig. 3
Fig. 3

Global maximum transmitted power normalized to the total incident energy (Pmax/I0) as a function of the basic beam diameter normalized to the width slit (L/l) for several m modes (m = 0–4) and θ0 = 0°.

Fig. 4
Fig. 4

Transmission coefficient τ as a function of the normalized position (b/l) of the incident Hermite–Gaussian beam for several m modes (m = 0–3) with θ0 = 0°, L/l = 4, and λ/l = 0.1. Continuous curves are from Eq. (8) and dotted curves from Eq. (16).

Fig. 5
Fig. 5

Intensity diffracted at normal direction normalized to the total incident energy (ℰ/I0) as a function of the position of the incident Hermite–Gaussian beam normalized to the slit width (b/l) for several m modes (m = 0–3), with θ0 = 0°, L/l = 4, and λ/l = 0.1. Continuous curves are from Eq. (8) and dotted curves from Eqs. (16) and (19).

Fig. 6
Fig. 6

Intensity ratio κ (ratio between the global minimum and the global maximum transmitted power) as a function of the basic beam diameter normalized to the width slit (L/l) for several m modes and θ0 = 0°. Continuous curves are from Eq. (8) and dotted curves from Eq. (20).

Fig. 7
Fig. 7

Diffraction patterns of Hermite–Gaussian beams at normal incidence, with λ/l = 0.01, L / l = 3.3 / 2, and θ0 = 0°. (a) Global maximum transmitted power, where the positions b/l are determined from Eq. (17); (b) global minimum transmitted power, with b/l = 0.0, 0.82, 1.40, 1.91 for m = 1, 2, 3, 4, respectively.

Tables (1)

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Table 1 Variation of Parameters

Equations (21)

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E ( x 0 , y 0 ) = i 2 E ( x , 0 ) y 0 H 0 1 ( k r ) d x ,
E ( x 0 , y 0 ) = i 2 l / 2 l / 2 E i ( x , 0 ) y 0 H 0 1 ( k r ) d x ,
H 0 1 ( k r ) 2 / π k r exp ( i π / 4 ) exp ( i k r )
y 0 H 0 1 ( k r ) i 2 k / π exp ( i π / 4 ) × exp [ i k ( r 0 x sin θ ) ] r 0 1 / 2 cos θ ,
E ( x 0 , y 0 ) = f ( θ ) exp ( i k r 0 ) / r 0 .
f ( θ ) = k exp ( i π / 4 ) cos θ Ȇ i ( k sin θ , 0 ) ,
Ȇ i ( α , 0 ) = 1 2 π l / 2 l / 2 E i ( x , 0 ) exp ( i α x ) d x .
I ( θ ) = k 2 cos 2 θ | Ȇ i ( k sin θ , 0 ) | 2 ,
E m i ( x , y = 0 ) = H m [ 2 L ( x b ) ] exp [ 2 ( x b ) 2 L 2 ] ,
E i ( x , y ) = 1 2 π k k A ( α ) exp [ i ( α x β y ) ] d α ,
exp ( i x y ) exp ( x 2 / 2 ) H m ( x ) d x = ( 2 π ) 1 / 2 ( i ) m exp ( y 2 / 2 ) H m ( y ) ,
A ( α ) = L 2 ( i ) m H m [ L 2 ( α cos θ 0 β sin θ 0 ) ] × ( cos θ 0 + α β sin θ 0 ) [ exp i ( α b + β h ) ] × exp [ ( α cos θ 0 β sin θ 0 ) 2 L 2 / 8 ] ,
I 0 = k k β | A ( α ) | 2 d α ,
k sin [ θ 0 arcsin ( 8.4852 L k ) ] α k sin [ θ 0 + arcsin ( 8.4852 L k ) ] .
Δ θ 20 ( λ / l ) .
τ ( b ) = P max [ H m ( 2 b / L ) exp ( 2 b 2 / L 2 ) H m ( 2 b max / L ) exp ( 2 b max 2 / L 2 ) ] 2 ,
| b max | = a L + c ,
P max = 1 q 1 L + q 2 .
( b / l ) = τ ( b / l ) λ / l ,
κ = e [ L / l ] f ,
P min = κ P max = e ( L / l ) f q 1 L / l + q 2 .

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