Abstract

The diffraction of Gaussian beams by periodic and aperiodic rulings is considered. The theory of diffraction is based on the Rayleigh–Sommerfeld integral equation with Dirichlet conditions. The transmitted power and the normally diffracted energy are analyzed as a function of the beam radius. Two methods to determine the Gaussian beam radius by means of periodic and aperiodic lamellar gratings are proposed. One is based on the maximum and the minimum transmitted power, and the other one considers the normally diffracted energy. Small and large Gaussian beam radii can be treated with these two methods.

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References

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  1. D. Psaltis, D. G. Stinson, and G. S. Kino, “Optical data storage: three perspectives,” Opt. Photonics News , November 1997, pp. 35-39.
  2. R. M. Herman, J. Pardo, and T. A. Wiggins, “Diffraction and focusing of Gaussian beams,” Appl. Opt. 24, 1346-1354 (1985).
    [CrossRef] [PubMed]
  3. D. Wright, “Beamwidths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, S1129-S1135 (1992).
    [CrossRef]
  4. O. Mata-Mendez, “Diffraction and beam-diameter measurement of Gaussian beams at optical and microwave frequencies,” Opt. Lett. 16, 1629-1631 (1991).
    [CrossRef] [PubMed]
  5. N. Hodgson, T. Haase, R. Kostka, and H. Weber, “Determination of laser beam parameters with the phase space beam analyser,” Opt. Quantum Electron. 24, S927-S949 (1992).
    [CrossRef]
  6. M. A. Karim, A. A. S. Awwal, A. M. Nasiruddin, A. Basit, D. S. Vedak, C. C. Smith, and G. D. Miller, “Gaussian laser-beam-diameter measurement using sinusoidal and triangular rulings,” Opt. Lett. 12, 93-95 (1987).
    [CrossRef] [PubMed]
  7. A. K. Cherri, A. A. S. Awwal, and M. A. Karim, “Generalization of the Ronchi, sinusoidal, and triangular rulings for Gaussian-laser-beam-diameter measurements,” Appl. Opt. 32, 2235-2242 (1993).
    [CrossRef] [PubMed]
  8. A. A. S. Awwal, J. A. Smith, J. Belloto, and G. Bharatram, “Wide range laser beam diameter measurement using a periodic exponential grating,” Opt. Laser Technol. 23, 159-161 (1991).
    [CrossRef]
  9. A. K. Cherri and A. A.S. Awwal, “Periodic exponential ruling for the measurement of Gaussian laser beam diameters,” Opt. Eng. (Bellingham) 32, 1038-1042 (1993).
    [CrossRef]
  10. J. S. Uppal, P. K. Gupta, and R. G. Harrison, “Aperiodic ruling for the measurement of Gaussian laser beam diameters,” Opt. Lett. 14, 683-685 (1989).
    [CrossRef] [PubMed]
  11. O. Mata-Mendez and F. Chavez-Rivas, “Diffraction of Gaussian and Hermite-Gaussian beams by finite gratings,” J. Opt. Soc. Am. A 18, 537-545 (2001).
    [CrossRef]
  12. A. Sommerfeld, “Optics,” in Lectures on Theoretical Physics (Academic, 1964) Vol. IV, Chap. VI, p. 273.
  13. J. Sumaya-Martinez, O. Mata-Mendez, and F. Chavez-Rivas, “Rigorous theory of the diffraction of Gaussian beams by finite gratings: TE polarization,” J. Opt. Soc. Am. A 20, 827-835 (2003).
    [CrossRef]
  14. O. Mata-Mendez, J. Avendaño, and F. Chavez-Rivas, “Rigorous theory of the diffraction of Gaussian beams by finite gratings: TM polarization,” J. Opt. Soc. Am. A 23, 1889-1896 (2006).
    [CrossRef]
  15. O. Mata-Mendez and 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]

2006 (1)

2003 (1)

2001 (1)

1998 (1)

1993 (2)

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

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

1992 (2)

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

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

1991 (2)

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

1989 (1)

1987 (1)

1985 (1)

Avendaño, J.

Awwal, A. A. S.

Awwal, A. A.S.

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

Basit, A.

Belloto, J.

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

Chavez-Rivas, F.

Cherri, A. K.

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

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

Gupta, P. K.

Haase, T.

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

Harrison, R. G.

Herman, R. M.

Hodgson, N.

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

Karim, M. A.

Kino, G. S.

D. Psaltis, D. G. Stinson, and G. S. Kino, “Optical data storage: three perspectives,” Opt. Photonics News , November 1997, pp. 35-39.

Kostka, R.

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

Mata-Mendez, O.

Miller, G. D.

Nasiruddin, A. M.

Pardo, J.

Psaltis, D.

D. Psaltis, D. G. Stinson, and G. S. Kino, “Optical data storage: three perspectives,” Opt. Photonics News , November 1997, pp. 35-39.

Smith, C. C.

Smith, J. A.

A. A. S. Awwal, J. A. Smith, J. Belloto, and 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 (Academic, 1964) Vol. IV, Chap. VI, p. 273.

Stinson, D. G.

D. Psaltis, D. G. Stinson, and G. S. Kino, “Optical data storage: three perspectives,” Opt. Photonics News , November 1997, pp. 35-39.

Sumaya-Martinez, J.

Uppal, J. S.

Vedak, D. S.

Weber, H.

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

Wiggins, T. A.

Wright, D.

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

Appl. Opt. (2)

J. Opt. Soc. Am. A (4)

Opt. Eng. (Bellingham) (1)

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

Opt. Laser Technol. (1)

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

Opt. Lett. (3)

Opt. Photonics News (1)

D. Psaltis, D. G. Stinson, and G. S. Kino, “Optical data storage: three perspectives,” Opt. Photonics News , November 1997, pp. 35-39.

Opt. Quantum Electron. (2)

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

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

Other (1)

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

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Figures (11)

Fig. 1
Fig. 1

Our system. An aperiodic ruling made of alternate opaque and transparent zones of widths d and l, respectively, with an opaque zone of width d . The ruling is parallel to the O z axis. The observation point is given by P ( x 0 , y 0 ) .

Fig. 2
Fig. 2

Transmitted power and the normally diffracted energy are plotted as a function of the beam position b D . We have a normally incident Gaussian beam with λ l = 0.1 , d l = 0.8 , and r 0 l = 1.0 .

Fig. 3
Fig. 3

Minimum ( E min ) and maximum ( E max ) values of the normally diffracted energy are plotted as functions of the inverse of the wavelength.

Fig. 4
Fig. 4

Diffraction patterns at minimum ( P min ) and maximum ( P max ) transmitted power, where λ D = 0.03 , l D = 0.5 , d D = 0.5 , and r 0 D = 0.3535 .

Fig. 5
Fig. 5

Ratio P is plotted as a function of the field amplitude radius normalized to the grating period ( r 0 D ) for a normally incident Gaussian beam, where q = d l = 0.1 1.9 , 0.5 1.5 , 1 1 , 1.5 0.5 , and 1.9 0.1 .

Fig. 6
Fig. 6

Same as Fig. 5 but for the ratio K.

Fig. 7
Fig. 7

Transmitted power and the normally diffracted energy are plotted as a function of the spot position ( b D ) , with λ = 0.1 , l = 1.5 , d = 0.5 , and d = 3.0 . Several values of the field amplitude radius are considered: r 0 D = 0.5 and 2.0.

Fig. 8
Fig. 8

Ratio P is plotted as a function of the field amplitude radius ( r 0 D ) , for several values of the opaque discontinuity ( d D = 2.5 , 5.0 , 7.5 ) , when l = 0.5 and d = 0.5 .

Fig. 9
Fig. 9

Same as Fig. 8 but for the ratio K.

Fig. 10
Fig. 10

P and K are plotted as a function of r 0 D when d D = 7.5 .

Fig. 11
Fig. 11

Diffraction patterns at minimum and maximum transmitted power when λ = 0.1 , d D = 7.5 , l = 0.5 , d = 0.5 , and r 0 D = 10 .

Equations (14)

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E ( x ) = t ( x ) E i ( x ) .
E ( x 0 , y 0 ) = i 2 E ( x ) y 0 H 0 1 ( k r ) d x = i 2 t ( x ) E i ( x ) 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 E ̂ ( k sin θ ) ,
E ̂ ( α ) = 1 2 π + E ( x ) exp ( i α x ) d x = 1 2 π + t ( x ) E i ( x ) exp ( i α x ) d x .
I ( θ ) = 1 2 π k cos 2 θ + t ( x ) E i ( x ) exp ( i k sin θ x ) d x 2 .
E = k 2 π + t ( x ) E i ( x ) d x 2 ,
P T = π 2 π 2 I ( θ ) d θ .
E ( x , y = 0 ) = exp [ ( x b ) 2 r 0 2 ] ,
K = E min E max ,
P = P min P max ,
E min 1 λ , E max 1 λ .

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