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Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 30, Iss. 4 — Apr. 1, 2013
  • pp: 640–644
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Gaussian beam propagation: comparison of the analytical closed-form Fresnel integral solution to the simulations of the Huygens, Fresnel, and Rayleigh–Sommerfeld I approximations

Seyed M. Azmayesh-Fard  »View Author Affiliations


JOSA A, Vol. 30, Issue 4, pp. 640-644 (2013)
http://dx.doi.org/10.1364/JOSAA.30.000640


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Abstract

Simulations of the Huygens, Fresnel, and Rayleigh–Sommerfeld I approximations in the case of free-space propagation of a Gaussian beam are compared with analytical solutions. The most accurate results were obtained by the Rayleigh–Sommerfeld I approximation. The study reveals that the approximations are not uniform throughout the propagation region. While the accuracies of the Huygens and Fresnel methods generally increase as the propagation distance increases, the accuracy of the Rayleigh–Sommerfeld I approximation at first starts to diminish and later recovers as the propagation distance is further increased.

© 2013 Optical Society of America

1. INTRODUCTION

In physical optics, propagation of the field is normally posed as a diffraction problem. In order to solve this problem, diffraction integrals need to be invoked. Because of mathematical difficulties, rigorous solutions to diffraction integrals are rare [1

1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

]. As discussed in this paper, the Gaussian beam is an important special case for which an analytical solution exists. Otherwise, in most cases of practical interest, approximate methods are used [1

1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

]. We use the closed-form analytical solution of the Gaussian beam diffraction in the Fresnel approximation as a reference, in order to determine the level of accuracy obtained from simulations of the Huygens, Fresnel, and Rayleigh–Sommerfeld I approximations.

The propagation of waves can be effectively analyzed using the Huygens and Fresnel integral approaches. Siegman [2

2. A. E. Siegman, Lasers (University Science, 1986).

] takes a brilliant approach and begins with spherical waves as the general solution to the exact wave equation in order to facilitate understanding of the Huygens’ principle and consequently the paraxial Fresnel approximation. We adopt the same strategy and begin our discussion with spherical waves.

2. SPHERICAL WAVES, HUYGENS’ PRINCIPLE, AND THE FRESNEL APPROXIMATION

A source point radiating a uniform diverging spherical wave from position r0 can be expressed as
E˜(r;r0)=exp[jkρ(r,r0)]ρ(r,r0),
(1)
where E˜(r;r0) is the field at point r=(x,y,z) due to a source at point r0=(x0,y0,z0), and where the observation point s, z is separated from the source point s0, z0 by the distance ρ(r,r0) given by
ρ(r,r0)(xx0)2+(yy0)2+(zz0)2.
(2)
According to Huygens’ principle, “Every point on a propagating wavefront serves as the source of spherical secondary wavelets, such that the wavefront at some later time is the envelope of these wavelets” [3

3. C. Huygens, Treatise on Light (Dover, 1962).

]. In other words, each of the Huygens’ wavelets is a spherical wave of the form given by Eq. (1), and a sum of all these wavelets leads to Huygens’ integral equation of the form [2

2. A. E. Siegman, Lasers (University Science, 1986).

]
E˜(s,z)=jλS0E˜0(s0,z0)exp[jkρ(r,r0)]ρ(r,r0)cosθ(r,r0)ds0,
(3)
where E˜0(s0,z0) is the incident field distribution, ds0 is an incremental element of the surface S0 at the point s0, z0, and cosθ(r,r0) is an “obliquity factor” defined by the angle θ(r,r0) between the line ρ(r,r0) and the normal to the surface element ds0 (Fig. 1).

Fig. 1. Geometry for evaluation of the Huygens’ integral [after 2].

We are interested in Huygens’ integral in one dimension since all our simulations are one-dimensional. Considering only one transverse dimension, r=(x,z); r0=(x0,z0); and z0=0, and replacing ρ(r,r0) by L, the Eq. (2) becomes
L(xx0)2+z2.
(4)
For free-space propagation and in the paraxial approximation where the point source x0 is not too far off the z axis, L can be approximated as independent of x0 and the one-dimensional Huygens’ integral [2

2. A. E. Siegman, Lasers (University Science, 1986).

] can be given by
E˜(x,z)=jLλE˜0(x0,z0)exp(jkL)cosθdx0.
(5)
To obtain the Fresnel integral in the paraxial approximation, we expand Eq. (2) as a power series
ρ(r,r0)=zz0+(xx0)2+(yy0)22(zz0)+.
(6)
In rewriting the spherical wave of Eq. (1), we disregard all terms higher than quadratic terms in the power series expansion for the phase shift factor exp[jkρ(r,r0)], and in the denominator we simply replace ρ(r,r0) by zz0; then we have what we may consider a “paraxial–spherical wave” in the Fresnel approximation [2

2. A. E. Siegman, Lasers (University Science, 1986).

]:
E˜(x,y,z)1zz0exp[jk(zz0)jk(xx0)2+(yy0)22(zz0)].
(7)
The two-dimensional diffraction integral in the Fresnel approximation can be written as [2

2. A. E. Siegman, Lasers (University Science, 1986).

]
E˜(x,y,z)j(zz0)λE˜0(x0,y0,z0)exp[jk(zz0)jk(xx0)2+(yy0)22(zz0)]cosθdx0dy0.
(8)
With z0=0, the one-dimensional Fresnel integral has the form [2

2. A. E. Siegman, Lasers (University Science, 1986).

]
E˜(x,z)jzλE˜0(x0,z0)exp[jkzjk(xx0)22z]cosθdx0.
(9)

3. RAYLEIGH–SOMMERFELD I DIFFRACTION FORMULA

For the case of the Rayleigh–Sommerfeld I formulation of diffraction [4

4. A. Sommerfeld, Optics, Vol. 3 of Lectures on Theoretical Physics (Academic, 1964).

], the two-dimensional diffraction integral can be written in terms of the Hankel function as outlined by Totzeck in [5

5. M. Totzeck, “Validity of the scalar Kirchhoff and Rayleigh-Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A 8, 27–32 (1991). [CrossRef]

]. The example case chosen for derivation of the Rayleigh–Sommerfeld I formula is the classical problem of two-dimensional diffraction of a monochromatic, cylindrical scalar wave by a slit, although we are going to use the obtained integral formula for propagation (diffraction) of a Gaussian beam. Figure 2 illustrates the geometry under consideration.

Fig. 2. Diffraction of an incident cylindrical wave by a slit aperture [after 5].

The slit width is 2b and the aperture screen is opaque, completely absorbing, and perfectly conducting. According to the Rayleigh–Sommerfeld theory the field at any point beyond the aperture screen can be calculated from the field or its normal derivative on the planar screen. In the first Rayleigh–Sommerfeld integral the boundary condition is on the field, while in the second Rayleigh–Sommerfeld formula the boundary condition is on the normal derivative of the field. For the Rayleigh–Sommerfeld I formulation the diffraction integral can be written as [4

4. A. Sommerfeld, Optics, Vol. 3 of Lectures on Theoretical Physics (Academic, 1964).

,6

6. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954). [CrossRef]

]
UI(r)=12πbbU0(r)n[exp(ik|rr|)|rr|]dxdy
(10)
with U0 as the field at the aperture.

The following equation can be used to convert the three-dimensional diffraction integral into a two-dimensional diffraction integral [7

7. L. Eyges, The Classical Electromagnetic Field (Addison-Wesley, 1972), p. 263.

]:
exp(ik|rr|)|rr|dy=iπH0(k|ρρ|).
(11)
Here H0 is the zero-order Hankel function of the first kind, ρ=(x,0,z), and ρ=(x,0,z).

By substituting Eq. (11) in the integral of Eq. (10), we obtain
UI(ρ)=i2bbU0(ρ)nH0(k|ρρ|)dx.
(12)
Taking the normal derivative of the Hankel function H0, we have
nH0(k|ρρ|)=kH0(k|ρρ|)k|ρρ|(ρρ)|ρρ|·n=kH1(k|ρρ|)cos(ϑ),
(13)
where H1 is the first-order Hankel function of the first kind and ϑ is the angle between n and (ρρ) (see Fig. 2).

By substituting Eq. (13) into Eq. (12), we obtain the first Rayleigh–Sommerfeld diffraction formula in terms of the Hankel function [5

5. M. Totzeck, “Validity of the scalar Kirchhoff and Rayleigh-Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A 8, 27–32 (1991). [CrossRef]

]:
UI(ρ)=ik2bbH1(k|ρρ|)cos(ϑ)U0(ρ)dx.
(14)

4. PROPAGATION OF A GAUSSIAN BEAM

A Gaussian beam is an optical beam such that the amplitude of the wave function u(x,y,z) associated with it has a Gaussian distribution at each cross section. Laser beams emerging from cylindrically symmetric cavities have this character. For convenience we relegate the dependence on z to a subscript and introduce
uz(x,y)u(x,y,z).
(15)
To study the propagation of a Gaussian beam we may consider a “signal” in an input plane (z=0) with a Gaussian distribution
u0(x,y)=cexp((x2+y2)ω02),
(16)
where c and ω0 are constants. Its square amplitude, known as irradiance or intensity, is also Gaussian:
I(x,y)=|u0(x,y)|2=c2exp(2(x2+y2)ω02).
(17)
Using the Fresnel diffraction formula we will obtain an expression for uz(x,y), the complex optical signal at a plane z units distant from the input plane. It will be shown that the amplitude distribution remains Gaussian for all z and the radius of the beam increases with z.

In the Fresnel approximation we have
uz(x,)=jλexp(jkz)zcexp((x2+y2)ω02)exp(jk2z[(xx)2+(yy)2])dxdy.
(18)
Separating the integral with respect to x,
exp(jk2zx2)exp((1ω02+jk2z)x2)exp(jkzxx)dx.
(19)
We note that the integral is of the form of a Fourier transform of a Gaussian function
exp(αξ2)exp(±jβξ)dξ=παexp(β24α).
(20)
Here
α=(1ω02+jk2z)=2z+jkω022ω02z;β=kxz.
(21)
After some algebra we find for the x part
2πω02z2z+jkω02exp(j2kz4z2+k2ω04x2)exp(k2ω024z2+k2ω04x2).
(22)
A similar result is found for the integral with respect to y, except that x is replaced by y.

Thus we obtain
uz(x,y)=cjλexp(jkz)z2πω02z2z+jkω02exp(j2kz4z2+k2ω04(x2+y2))exp(k2ω024z2+k2ω04(x2+y2)),uz(x,y)=cjkω022z+jkω02exp(jkz)exp(j2kz4z2+k2ω04(x2+y2))exp(k2ω024z2+k2ω04(x2+y2)).
(23)
Note that at z=0, this reduces to cexp(((x2+y2)/ω02)), as it should.

If the real exponential is written
exp(k2ω024z2+k2ω04(x2+y2))exp((x2+y2)ω2(z)),
(24)
then for (x2+y2)=ω2(z) the amplitude of uz(x,y) is down to e1 (and its square is down to e2=13.5%) of its value at x=0, y=0. Thus ω(z) is a measure of the beam radius (half width) at any plane z (Fig. 3).

Fig. 3. Gaussian beam with radius of the beam (ω0) at e2 or 13.5% of its maximum intensity.

From the above definition we have
ω(z)4z2+k2ω04k2ω02=ω01+(2zkω02)2.
(25)
Another parameter frequently used is the Rayleigh range, z0,
z0kω022=πω02λ.
(26)
In terms of z0 we may write the beam radius and radius of curvature of the wavefront as
ω(z)=ω01+(zz0)2,
(27)
R(z)=z[1+(z0z)2].
(28)
Finally, the closed-form solution of the one-dimensional Fresnel diffraction integral that can be used for the propagation of a Gaussian beam is given by
uz(x)=cjzλ2πω02z2z+jkω02exp(jkz)exp(j2kz4z2+k2ω04x2)exp(k2ω024z2+k2ω04x2).
(29)

5. RESULTS AND DISCUSSION

We have implemented the analytical closed-form solution of the Fresnel diffraction integral for an input Gaussian field [Eq. (29)], together with Huygens, Fresnel, and Rayleigh–Sommerfeld I approximations [Eqs. (5), (9), and (14), respectively], in a MATLAB program for comparison. The parameters of the simulation are the beam radius ω0 of the initial Gaussian field, the optical wavelength λ, and the propagation distance z. In all simulations the parameters ω0 and λ were kept constant. We have increased z from 1 to 1,000,000 μm in increments that translate to half a unit in the logarithmic scale (i.e., 1,3,10,32). For each approximate method we recorded the values for the maximum intensity of the output Gaussian beam as the propagation distance z was incrementally increased. We then compared these values with the maximum intensities obtained from exact integration of the Fresnel diffraction integral. The results are compiled in the logarithmic graphs of Fig. 4. According to our simulations, in the Fresnel approximation a Gaussian beam that has a radius of ω0=20μm requires a minimum propagation distance of at least 10λ (Fresnel near-field criteria). We have implemented two versions of the Fresnel integral. In the first version (Fresnel 1) the obliquity factor is (1+cosθ)/2, while in the second version (Fresnel 2) the obliquity factor is (cosθ). Based on our simulations, (1+cosθ)/2 provides more accurate results [see Fig. 4(a)], as we expect the Rayleigh–Sommerfeld formula provides the most accurate approximation. For example, at a propagation distance of 1 μm the relative error is less than 0.00000001%. Contrary to the other approximations used in this study, in the Rayleigh–Sommerfeld approximation the level of accuracy falls off rapidly at the beginning as the propagation distance increases, but even in the worst case the relative error is still around 0.001% [see Fig. 4(b)]. To evaluate the Hankel function, in the Rayleigh–Sommerfeld I integral we have used the built-in MATLAB function “besselh.” In the program we have also included an integral simulation in which an asymptotic expansion for large arguments (32) is used in lieu of the Hankel function (see Abramowitz and Stegun [8

8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 7th ed. (National Bureau of Standards, 1968).

], p. 364, Section 9.2.3):
H11(x)2/(πx)exp(j(x0.75π)).
(32)
Looking at Fig. 4(c), we notice that using the above expansion instead of the Hankel function, we obtain results that are identical to the ones obtained by Huygens’ approximation. As indicated by the logarithmic graphs of Fig. 4, the level of accuracy for each method fluctuates throughout the propagation region. While accuracies of the Fresnel method are generally increased as the propagation distance increases, the accuracy of the Rayleigh–Sommerfeld I approximation for propagation distances in the range of 1–3162 μm diminishes at first and later recovers as the propagation distance is further increased. The relative percent error for the ending point of all five simulations (at z=1m) is in the 0.000003% range, which is an excellent approximation.

Fig. 4. Logarithmic graphs of percent relative error versus propagation distance for maximum Gaussian beam intensity obtained from various approximations as compared to the values obtained from the exact integration. (a) Fresnel 1 and 2 approximations. (b) Rayleigh–Sommerfeld I approximation. (c) Huygens and asymptotic approximations.

ACKNOWLEDGMENTS

The author is grateful for the contributions and helpful advice of the late Professor James Neil McMullin in preparation of this manuscript. The author acknowledges the financial support of the Natural Sciences and Engineering Research Council of Canada.

REFERENCES

1.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

2.

A. E. Siegman, Lasers (University Science, 1986).

3.

C. Huygens, Treatise on Light (Dover, 1962).

4.

A. Sommerfeld, Optics, Vol. 3 of Lectures on Theoretical Physics (Academic, 1964).

5.

M. Totzeck, “Validity of the scalar Kirchhoff and Rayleigh-Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A 8, 27–32 (1991). [CrossRef]

6.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954). [CrossRef]

7.

L. Eyges, The Classical Electromagnetic Field (Addison-Wesley, 1972), p. 263.

8.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 7th ed. (National Bureau of Standards, 1968).

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(050.1960) Diffraction and gratings : Diffraction theory

ToC Category:
Diffraction and Gratings

History
Original Manuscript: January 17, 2013
Manuscript Accepted: February 2, 2013
Published: March 18, 2013

Citation
Seyed M. Azmayesh-Fard, "Gaussian beam propagation: comparison of the analytical closed-form Fresnel integral solution to the simulations of the Huygens, Fresnel, and Rayleigh–Sommerfeld I approximations," J. Opt. Soc. Am. A 30, 640-644 (2013)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-4-640


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References

  1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  2. A. E. Siegman, Lasers (University Science, 1986).
  3. C. Huygens, Treatise on Light (Dover, 1962).
  4. A. Sommerfeld, Optics, Vol. 3 of Lectures on Theoretical Physics (Academic, 1964).
  5. M. Totzeck, “Validity of the scalar Kirchhoff and Rayleigh-Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A 8, 27–32 (1991). [CrossRef]
  6. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954). [CrossRef]
  7. L. Eyges, The Classical Electromagnetic Field (Addison-Wesley, 1972), p. 263.
  8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 7th ed. (National Bureau of Standards, 1968).

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