Abstract

The Fresnel diffraction of a truncated Gaussian beam is investigated in detail. Our aim is to provide usable analytical expressions of the diffracted field with respect to the variation of propagation and diffraction parameters. Particular attention is paid to the determination of a trade-off between computing accuracy and simplicity. An illustration of the use of the various expressions describing the beam diffraction and propagation is given for the case of mode coupling in a single-mode fiber.

© 2001 Optical Society of America

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References

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  1. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  2. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), pp. 231–235.
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  4. A. Sommerfield, Optics (Academic, Inc., London, 1998), pp. 197, 201, 242.
  5. A. L. Buck, “The radiation pattern of a truncated Gaussian aperture distribution,” Proc. IEEE 55, 448–450 (1967).
    [CrossRef]
  6. J. K. Kauffman, “The calculated radiation patterns of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 473–474 (1965).
    [CrossRef]
  7. A. S. Chai, H. J. Wertz, “The digital computation of the far-field radiation pattern of a truncated Gaussian aperture,” IEEE Trans. Antennas Propag. AP-13, 994–995 (1965).
    [CrossRef]
  8. J. P. Campbell, L. G. DeShazer, “Near fields of truncated Gaussian apertures,” J. Opt. Soc. Am. 59, 1427–1429 (1969).
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  9. Y. Li, “Degeneracy and regeneracy in the axial field of a focused truncated Gaussian beam,” J. Opt. Soc. Am. A 5, 1397–1406 (1988).
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  13. R. G. Schell, G. Tyras, “Irradiance from an aperture with a truncated Gaussian field distribution,” J. Opt. Soc. Am. 61, 31–35 (1971).
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  14. G. Oluremi Olaofe, “Diffraction by Gaussian apertures,” J. Opt. Soc. Am. 60, 1654–1657 (1970).
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  18. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 297.
  19. M. R. Spiegel, Variables Complexes (McGraw-Hill, Paris, 1983), p. 275.
  20. P. Belland, J. P. Crenn, “Changes in the characteristics of a Gaussian beam diffracted by a circular aperture,” Appl. Opt. 21, 522–527 (1982).
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  21. P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. (Bellingham) 39, 387–392 (2000).
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  22. B. Hillerich, “Influence of lens imperfections with LD and LED to single-mode fiber coupling,” J. Lightwave Technol. 7, 77–86 (1989).
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  23. S. Peled, “Near- and far-field characterization of diode laser,” Appl. Opt. 19, 324–328 (1980).
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2000 (1)

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. (Bellingham) 39, 387–392 (2000).
[CrossRef]

1999 (1)

1995 (1)

1993 (1)

1991 (1)

1989 (1)

B. Hillerich, “Influence of lens imperfections with LD and LED to single-mode fiber coupling,” J. Lightwave Technol. 7, 77–86 (1989).
[CrossRef]

1988 (1)

1986 (1)

1982 (1)

1980 (1)

1973 (1)

1971 (1)

1970 (1)

1969 (1)

1967 (1)

A. L. Buck, “The radiation pattern of a truncated Gaussian aperture distribution,” Proc. IEEE 55, 448–450 (1967).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1965 (2)

J. K. Kauffman, “The calculated radiation patterns of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 473–474 (1965).
[CrossRef]

A. S. Chai, H. J. Wertz, “The digital computation of the far-field radiation pattern of a truncated Gaussian aperture,” IEEE Trans. Antennas Propag. AP-13, 994–995 (1965).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 297.

Belland, P.

Brunfeld, A.

Buck, A. L.

A. L. Buck, “The radiation pattern of a truncated Gaussian aperture distribution,” Proc. IEEE 55, 448–450 (1967).
[CrossRef]

Campbell, J. P.

Chai, A. S.

A. S. Chai, H. J. Wertz, “The digital computation of the far-field radiation pattern of a truncated Gaussian aperture,” IEEE Trans. Antennas Propag. AP-13, 994–995 (1965).
[CrossRef]

Chanclou, P.

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. (Bellingham) 39, 387–392 (2000).
[CrossRef]

Crenn, J. P.

DeShazer, L. G.

Ding, D.

Gadonna, M.

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. (Bellingham) 39, 387–392 (2000).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980), pp. 307, 485, 931.

Hillerich, B.

B. Hillerich, “Influence of lens imperfections with LD and LED to single-mode fiber coupling,” J. Lightwave Technol. 7, 77–86 (1989).
[CrossRef]

Kauffman, J. K.

J. K. Kauffman, “The calculated radiation patterns of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 473–474 (1965).
[CrossRef]

Kenney, C. S.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, Y.

Liu, X.

Lostec, J.

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. (Bellingham) 39, 387–392 (2000).
[CrossRef]

Mahajan, V. N.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), pp. 231–235.

Oluremi Olaofe, G.

Overfelt, P. L.

Pavy, D.

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. (Bellingham) 39, 387–392 (2000).
[CrossRef]

Peled, S.

Rose, T. S.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980), pp. 307, 485, 931.

Schell, R. G.

Shamir, J.

Sommerfield, A.

A. Sommerfield, Optics (Academic, Inc., London, 1998), pp. 197, 201, 242.

Spiegel, M. R.

M. R. Spiegel, Variables Complexes (McGraw-Hill, Paris, 1983), p. 275.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 297.

Thual, M.

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. (Bellingham) 39, 387–392 (2000).
[CrossRef]

Toker, G.

Tyras, G.

Wertz, H. J.

A. S. Chai, H. J. Wertz, “The digital computation of the far-field radiation pattern of a truncated Gaussian aperture,” IEEE Trans. Antennas Propag. AP-13, 994–995 (1965).
[CrossRef]

Williams, C. S.

Yura, H. T.

Appl. Opt. (5)

IEEE Trans. Antennas Propag. (2)

J. K. Kauffman, “The calculated radiation patterns of a truncated Gaussian aperture distribution,” IEEE Trans. Antennas Propag. AP-13, 473–474 (1965).
[CrossRef]

A. S. Chai, H. J. Wertz, “The digital computation of the far-field radiation pattern of a truncated Gaussian aperture,” IEEE Trans. Antennas Propag. AP-13, 994–995 (1965).
[CrossRef]

J. Lightwave Technol. (1)

B. Hillerich, “Influence of lens imperfections with LD and LED to single-mode fiber coupling,” J. Lightwave Technol. 7, 77–86 (1989).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Eng. (Bellingham) (1)

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. (Bellingham) 39, 387–392 (2000).
[CrossRef]

Proc. IEEE (2)

A. L. Buck, “The radiation pattern of a truncated Gaussian aperture distribution,” Proc. IEEE 55, 448–450 (1967).
[CrossRef]

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (5)

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), pp. 231–235.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980), pp. 307, 485, 931.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 297.

M. R. Spiegel, Variables Complexes (McGraw-Hill, Paris, 1983), p. 275.

A. Sommerfield, Optics (Academic, Inc., London, 1998), pp. 197, 201, 242.

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Figures (13)

Fig. 1
Fig. 1

Comparison of expressions (16), (17), and (28) for the calculation of |h|. Curves are plotted for z=20 cm and L=2w0=200 μm.

Fig. 2
Fig. 2

Influence of x and L on |X±|. Curves are plotted for z=20 cm and w0=100 μm.

Fig. 3
Fig. 3

Influence of L/w0 on the sidelobes. Curves are plotted for z=20 cm and w0=100 μm.

Fig. 4
Fig. 4

(a) Local minima of |h| in a complex diagram. The curve winds clockwise as the parameter x increases, but it never reaches zero, except as x. The curve is plotted for z=5 cm,L=2w0=200 μm. (b) Enlargement of (a).

Fig. 5
Fig. 5

Same inflection points as for Fig. 5 are drawn in an amplitude diagram. The curve is plotted for z=5 cm,L=2w0=200 μm.

Fig. 6
Fig. 6

Solid curves argument of h. The 2π phase shift arises from the arctan calculus. The intensity profile (dotted curve, |h|×105) indicates the extrema locations. Curves are plotted for z=5 cm,L=3w0=300 μm.

Fig. 7
Fig. 7

Interval between two successive minima as a function of |x| for different values of L/w0. Curves are plotted for z=5 cm,w0=100 μm.

Fig. 8
Fig. 8

Reconstructed phase deformation of the wave front for a truncated and nontruncated Gaussian beam. Curves are plotted for z=5 cm,L=w0=100 μm.

Fig. 9
Fig. 9

Reconstructed phase difference between a nontruncated and a truncated Gaussian beam with respect to the extrema locations (dotted curve, |h|×102). Curves are plotted for z=5 cm,L=w0=100 μm.

Fig. 10
Fig. 10

Schematic illustration of the propagation into the GRIN-based micro-optic.

Fig. 11
Fig. 11

Principle of the beam profile measurement with use of a far-field technique.

Fig. 12
Fig. 12

Comparison of the intensity profile of the truncated Gaussian beam measured for w0=20.83 μm and L=42.5 μm, by simulation. The curvature correction is obtained for a sphere of radius R=6.5 cm, i.e., with x=R sin θ,z=R cos θ, and θ the angle covered by the fiber.

Fig. 13
Fig. 13

Influence of z on |X±|. Curves are plotted for L=50w0=5 mm.

Tables (3)

Tables Icon

Table 1 Truncated Gaussian Beam Expression As a Function of L/w0

Tables Icon

Table 2 Variation of N0 with Respect to z for L/w0=1,=10-2, and =10-3

Tables Icon

Table 3 Variation of N0 with Respect to L/w0, for z=15 cm,=0.01

Equations (62)

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g(x, y)=exp-x2+y2w02
f(x, y, z)=exp[-iK(x2+y2)]
2h+k2h=0,
h(x, y, z)=w0w(z)exp-ikz-iπr2λR(z)-r2w2(z)+iϕ
=11-i/(Kw02)×exp-ikz-iπr2λR(z)-r2w2(z),
ϕ=arctan1Kw02=1iln1+i/(Kw02)1-i/(Kw02)1/2,
r2=x2+y2;R(z)=z[1+(Kw02)2],
w2(z)=w021+1Kw022.
U(P)=iλexp(-ikr)rcos(n, r)dS,
r=[(x-x)2+(y-y)2+z2]1/2z1+(x-x)22z2+(y-y)2z2,
z3 π4λ [(x-x)4+(y-y)4]max
h(x, z)=α(g(x)*f(x))x=α-exp-x2w02×exp[-iK(x-x)2]dx,
α=exp(-ikz)-iλz.
β=w0241-iKw021+K2w04,γ=-2iKx,
h(x, z)=α exp(-iKx2)-exp-x24β-γxdx.
0exp-t24β-γtdt=πβexp(βγ2)
×[1-erf(γβ)]
ifRe(β)>0,
-exp-t24β-γtdt=2πβexp(βγ2).
h(x, z)=2απβexp(-iKx2+βγ2)
=λz-iπ(1-iKw02)1/2w(z)×exp-ikz-iπx2λR(z)-x2w2(z)
=exp(-ikz)[1-i/(Kw02)]1/2×exp-iπx2λR(z)-x2w2(z),
h(x, y, z)=exp(-ikz)1-i/(Kw02)exp-iπr2λR(z)-r2w2(z).
h(ν)=exp(-ikz)-iλz- f(x) exp-2iπxxλzdx=πw0exp(-ikz)-iλzexp(-π2w02ν2).
h(x, z)={[g(x)2L(x)]*f(x)}x
Lexp-t24β-γtdt=πβexp(βγ2)×1-erfγβ+L2β 
ifRe(β)>0andL>0.
h(x, z)=αerf-γβ+L2β+erfγβ+L2β×πβexp(-iKx2+βγ2).
DFT-1[DFT(g)×Frenel'skernelsampledintheFourierspace],
erf(X)=1-exp(-X)πk=0n-1(-1)k×Γ(k+12)Xk+1/2+exp(-X)π Rn,
|Rn|<Γ(n+1/2)|X|n+1/2cos ϕ/2.
erf(X)=2πn=0(-1)nX2n+1n!(2n+1),
erf(X+)+erf(X-)
X±=L2β±γβ=Lw0+iKw0(Lx)×(1-iKw02)1/2(1+K2w02)1/2=|X±|exp(iϕ±),
(1-iKw02)1/2=(1+K2w04)1/4exp-i2arctan Kw02.
|X±|=(1+K2w04)-1/4Lw02+K2w02(Lx)21/2.
|X±|=Lw02+πw0λz (Lx)21/2.
arg(X±)=ϕ(X±)=arctan(L/w0) sin(θ)+Kw0(Lx)cos(θ)(L/w0) cos(θ)-Kw0(Lx)sin(θ)
 arg(X±)=ϕ(X±)=-π
+arctan(L/w0) sin(θ)+Kw0(Lx)cos(θ)(L/w0) cos(θ)-Kw0(Lx)sin(θ)
ϕ(X±)=arctanKw02LxL
ϕ(X±)=-π+arctanKw02LxL
h(x, z)=NTG1-1π-xw [1-iKw02]1/2sin2πxLλz+Lw0 (1+iKw02)1/2cos2πxLλzx2/w2+L2/w02-i(πw02/λz)(x2/w2-L2/w02)×expx2w2-L2w02-i πw02λzx2w2+L2w02.
h(x, z)=NTG1-N exp(A-B)A+B,
Nx, z, Lw0=1π-xw (1-iKw02)1/2sin2πxLλz+Lw0 (1+iKw02)1/2cos2πxLλz,
A(x, z)=x2w2 (1-iKw02),
Bz, Lw0=L2w02 (1+iKw02),
h(ν)=αw0πexp(-π2w02ν2)1-exp(-L2/w02+π2w02ν2)π×(L/w0) cos(2πνL)-πνw0sin(2πνL)L2/w02+π2w02ν2.
h(ν)=exp(-ikz)-iλz-f(x)2L(x)exp(-2iπxν)dx
=αw0π2exp(-π2w02ν2)erfLw0+iπνw0+erfLw0-iπνw0.
h(x, z)=12[erf(Ki(L+x))+erf(Ki(L-x))].
hN0(x, z)=2NGTπn=0N0(-1)nn!(2n+1)
×q=0E[(2n+1)/2]C2n+12qL2β2n+1-2q(γβ)2q,
Cnp=n!(n-p)!p!.
h(x, z)=abexp(-iKt2)dt=F(b)-F(a),
 arg(X±)=ϕ(X±)=-π
+arctan(L/w0) sin(θ)+Kw0(Lx)cos(θ)(L/w0)cos(θ)-Kw0(Lx)sin(θ),
Lw0cos(θ)-Kw0(Lx)sin(θ)<0,
LKw02<(Lx)tan θ.
2LK2w04<(-L±x).
z2<π2w042Lλ2 (|x|-L).
|h(x)|-|hN0(x)||h(x)|<forxΔx,

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