Change of the paraxiality of a Gaussian beam diffracted by a circular aperture

Guoquan Zhou

doi:10.1364/OE.17.008417

1. Introduction

The propagation of laser beams, especially Gaussian beams, through apertured optical systems has received considerable interest [1-4]. By using the theory of boundary diffraction wave, the diffraction of a Gaussian beam that is normally incident on a circular aperture has been investigated [5]. The behavior of the diffraction field of a Gaussian beam through an aperture depends on the incident condition and the radius of aperture [6]. When a Gaussian beam is weakly diffracted by a circular aperture, its far-field expression is still approximately Gaussian [7,8]. When a Gaussian beam is limited by an aperture, there exists a relative phase shift [9]. Based on the vectorial Rayleigh diffraction integral, the propagation of a Gaussian beam diffracted by a circular aperture has been investigated within the non-paraxial framework [10,11]. By means of the angular spectrum representation and the method of stationary phase, the far-field expression of a non-paraxial Gaussian beam diffracted by a circular aperture has been derived [12]. The diffraction integral of a truncated Gaussian beam can be described by the Lommel functions, which offers an efficient way to calculate the optical field near the focus [13]. By expanding an aperture function into three different models, approximate analytical propagation equation for a Gaussian beam through an apertured ABCD optical system has been presented [14]. The Fresnel diffraction of a Gaussian beam truncated by a circular aperture can be analytically described by Bessel functions [15]. By using the superposition of Gaussian beams instead of the aperture function, the approximate formula for the far-field diffraction of a Gaussian beam through a circular aperture has been derived, and the explicit expression of the beam divergence has also been presented [16]. The vectorial structure of a Gaussian beam diffracted by a circular aperture has been examined in the far field [17]. An analytical propagation expression for a decentered elliptical Gaussian beam through an axially nonsymmetrical optical system with an elliptical aperture has been derived by using vector integration [18].

On the other hand, the paraxiality of a laser beam has also been of concern since the advent of laser. The paraxial optics result has been verified to be just the zeorth-order solution of Maxwell’s equations, and the equations which yield higher-order corrections have been presented [19]. Starting with the Hermholtz equation, the electric field associated with the Gaussian beam has been shown to consist of the paraxial result and higher-order non-Gaussian correction terms [20]. The propagation of Laguerre-Gaussian and Hermite-Gaussian beams beyond the paraxial approximation can be handled by superposing the corrected beams [21]. Non-paraxial Gaussian beam that is in Gaussian form but violates the paraxial condition is introduced, and in this case the paraxial approximation completely fails and the first-order correction never works [22]. How can one judge a Gaussian beam to be paraxial or non-paraxial? The degree of paraxiality is proposed to evaluate how paraxial a laser beam is, and the paraxiality of any beams can be quantitatively calculated [23,24]. According to the degree of paraxiality, one can judge whether the propagation of the beam should be treated within the framework of paraxial approximation or beyond the paraxial approximation. In this paper, the change of the paraxiality of a Gaussian beam diffracted by a circular aperture is investigated.

2. Change of the paraxiality of a Gaussian beam diffracted by a circular aperture

In Cartesian coordinate system, the z-axis is taken to be the propagation direction. The degree of paraxiality is defined as [23,24]

P = \frac{\iint_{p^{2} + q^{2} < 1} ∣ {A_{E}^{(0)} (p, q) ∣}^{2} {(1 - p^{2} - q^{2})}^{1 / 2} dpdq}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ∣ {A_{E}^{(0)} (p, q) ∣}^{2} dpdq},

where A_E ⁽⁰⁾ (p,q) is the vector angular spectrum of the input optical field in the source plane. The values of p ² +q ² <1 correspond to the homogeneous waves propagating at angles sin^-1(p ² + q ²)^1/2 with respect to the z-axis.

A Gaussian beam in the source plane z=0 takes the form as

{\begin{array}{c} E_{x} (ρ_{0}, 0) = \exp (- \frac{ρ_{0}^{2}}{w_{0}^{2}}) \exp (- iωt) \\ E_{y} (ρ_{0}, 0) = 0 \end{array},

where ρ ₀ = (x ₀ ² + y ₀ ²)^1/2 · w ₀ is the waist size of a Gaussian beam, and ω is the circular frequency. The degree of paraxiality for this propagating Gaussian beam yields

P_{0} = 1 - i \sqrt{\frac{π}{2}} f \exp (- \frac{1}{2 f^{2}}) \erf (\frac{1}{i \sqrt{2} f}),

where erf(.) is the error function. f=1/kw ₀, and k=2π/λ with λ the incident wavelength.

If a co-axial circular aperture with radius R is positioned in the source plane. The aperture plane is just the x-y plane, and the center of the circular aperture coincides with the original point. In this case, the boundary condition is described by

{\begin{array}{c} E_{x} (ρ_{0}, 0) = \exp (- \frac{ρ_{0}^{2}}{w_{0}^{2}}) circ (ζ) \exp (- iωt) \\ E_{y} (ρ_{0}, 0) = 0 \end{array},

where ζ=ρ ₀/R. circ(ζ) is the aperture function and given by

circ (ζ) = {\begin{array}{c} 1 & 0 \leq ζ < 1 \\ 0 & ζ \geq 1 \end{array} .

The diffracted Gaussian beam propagating toward the half free space z≥0 reads as

E (ρ, z) = \exp (- iωt) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} A_{E}^{(0)} (p, q) \exp [ik (px + qy + γz)] dpdq,

with A_E ⁽⁰⁾ (p,q) given by

A_{E}^{(0)} (p . q) = \frac{i}{λ^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} E_{x} (ρ_{0}, 0) \exp [- ik (p x_{0} + q y_{0})] d x_{0} d y_{0} = \frac{π w_{0}^{2}}{λ^{2}} \exp (- β) \sum_{n = 1}^{\infty} \frac{{(2 β)}^{n} J_{n} (αb)}{{(αb)}^{n}} i,

where ρ = (x ² + y ²)^1/2 , b = (p ² +q ²)^1/2, γ = (1-b ²)^1/2, α = kR , and β = R ²/w ₀ ² · J_n(·) is the n-th order Bessel function of the first kind, and n is an integer. i is the unit vector in the x-direction. In the derivation of the above equation, the following mathematic formula is used

x^{n + 1} J_{n} (x) = \frac{d}{dx} [x^{n + 1} J_{n + 1} (x)] .

Substituting Eq. (7) into Eq. (1), the degree of paraxiality for the diffracted Gaussian beam is found to be

P_{1} = \frac{\int_{0}^{1} (\sum_{n = 1}^{\infty} \frac{{(2 β)}^{2 n} J_{n}^{2} (αb)}{{(αb)}^{2 n}} + 2 \sum_{n = 1}^{\infty} \sum_{l = 1}^{n - 1} \frac{{(2 β)}^{n + l} J_{n} (αb) J_{l} (αb)}{{(αb)}^{n + l}}) {(1 - b^{2})}^{1 / 2} bdb}{\int_{0}^{\infty} (\sum_{n = 1}^{\infty} \frac{{(2 β)}^{2 n} J_{n}^{2} (αb)}{{(αb)}^{2 n}} + 2 \sum_{n = 1}^{\infty} \sum_{l = 1}^{n - 1} \frac{{(2 β)}^{n + l} J_{n} (αb) J_{l} (αb)}{{(αb)}^{n + l}}) bdb} .

As we note that the following integral is satisfied [25 ]

\int_{0}^{\infty} J_{n} (αb) J_{l} (αb) b^{- (n + l - 1)} db = \frac{α^{n + l - 2}}{2^{n + l - 2} (l - 1)! (n - 1)! (n + l - 2)},

the degree of paraxiality for the diffracted Gaussian beam can be expressed as

P_{1} = \frac{α^{2}}{2} \frac{\int_{0}^{1} (\sum_{n = 1}^{\infty} \frac{{(2 β)}^{2 n} J_{n}^{2} (αb)}{{(αb)}^{2 n}} + 2 \sum_{n = 1}^{\infty} \sum_{l = 1}^{n - 1} \frac{{(2 β)}^{n + l} J_{n} (αb) J_{l} (αb)}{{(αb)}^{n + l}}) {(1 - b^{2})}^{1 / 2} bdb}{\sum_{n = 1}^{\infty} \frac{β^{2 n}}{{[(n - 1)!]}^{2} (2 n - 1)} + \sum_{n = 1}^{\infty} \sum_{l = 1}^{n - 1} \frac{{2 β}^{n + l}}{(n - 1)! (l - 1)! (n + l - 1)}} .

To obtain the analytical expression of the degree of paraxiality, J_n(αb)J_l(αb) and J_n ²(αb) are separately expanded in the series form as

J_{n} (αb) J_{l} (αb) = \frac{1}{l!} \sum_{m = 0}^{\infty} \frac{{(- 1)}^{m} F (- m, - n - m; l + 1; 1)}{m! (n + m)!} {(\frac{αb}{2})}^{2 m + n + l},

J_{n}^{2} (αb) = \sum_{m = 0}^{\infty} \frac{{(- 1)}^{m}}{m! (2 n + m)! (n + m)!} {(\frac{αb}{\sqrt{2}})}^{2 n + 2 m},

where F(-m,-n-m;l+1;1) is a Gaussian hypergeometric function. Then, the replacement of variable is conducted as follow:

b = \sin θ, 0 \leq θ \leq π / 2,

γ = \sqrt{1 - b^{2}} = \sqrt{1 - \sin^{2} θ} = \cos θ .

Finally, we use the integral formula [25]

\int_{0}^{π / 2} \sin^{2 m + 1} θdθ = \frac{2^{2 m} {(m!)}^{2}}{(2 m + 1)!}

where m is an arbitrary integer. As a result, the analytical expression of the degree of paraxiality for the diffracted Gaussian beam turns out to be

P_{1} = \frac{\sum_{n = 1}^{\infty} \sum_{m = 0}^{\infty} \frac{{(- 1)}^{m} 2^{m + n} (m + 1)! α^{2 m} β^{2 n}}{(2 n + m)! (n + m)! (2 m + 3)!} + \sum_{n = 1}^{\infty} \sum_{l = 1}^{n - 1} \sum_{m = 0}^{\infty} \frac{2 {(- 1)}^{m} (m + 1)! F (- m, - n - m; l + 1; 1) α^{2 m} β^{n + l}}{l! (n + m)! (2 m + 3)!}}{\frac{1}{α^{2}} (\sum_{n = 1}^{\infty} \frac{β^{2 n}}{{[(n - 1)!]}^{2} (2 n - 1)} + \sum_{n = 1}^{\infty} \sum_{l = 1}^{n - 1} \frac{{2 β}^{n + l}}{(n - 1)! (l - 1)! (n + l - 1)})}

Though Eq. (17) is complicated, it is derived without any approximation. The degree of paraxiality for the diffracted Gaussian beam is determined by two ratios R/λ and R/w ₀. When a Gaussian beam is diffracted by a circular aperture, its paraxiality decreases. The absolutely decreasing quantity of the degree of paraxiality is evaluated by

Δ P = P_{0} - P_{1} .

We can also calculate the relative change for the degree of paraxiality by

\frac{Δ P}{P_{0}} = \frac{P_{0} - P_{1}}{P_{0}} \times 100 % .

3. Numerical calculations and discussions

For the sake of intuition, the degree of paraxiality, the absolute change for the degree of paraxiality, and the relative change for the degree of paraxiality of different Gaussian beams diffracted by a circular aperture as a function of the ratio R/λ are plotted in Fig. 1. The solid, the long-dash, the short-dash and the dotted curves correspond to w ₀=0.5λ, 2λ, 5λ, and 10λ, respectively. The insets show the detail variation within the range of 0≤R/λ=1. When the ratio R/λ is very small, the degree of the paraxiality of the diffracted Gaussian beams is independent of w ₀ and determined by the ratio R/λ. In fact, Eq. (17) reduces to P ₁=k ² R ²/6 when R/λ tends to be zero. As the ratio R/λ increases, the degree of paraxiality for a diffracted Gaussian beam first quickly increases and then tends to be a saturated value. The saturated value is just the degree of paraxiality of a unapertured Gaussian beam. The curves of w ₀=5λ and 10λ nearly overlap. When w ₀=5λ and 10λ, the degree of paraxiality of unapertured Gaussian beams is approximately equal to 1. When Gaussian beams with adequately high paraxiality are diffracted by a circular aperture, the influence of the circular aperture on the degree of paraxiality of the diffracted Gaussian beams is equivalent. The smaller w ₀ is, the smaller the radius under which the degree of paraxiality reaches a saturated value is. As the ratio R/λ increases, the absolute and the relative changes for the degree of paraxiality decrease steeply. As w0 decreases, the descending speed of the absolute and the relative changes for the degree of paraxiality versus the ratio R/λ augments. Figure 2 represents the degree of paraxiality, the absolute change for the degree of paraxiality, and the relative change for the degree of paraxiality of different diffracted Gaussian beams as a function of the ratio R/w ₀. The four curves are apparently detached from each other. When Gaussian beams with low paraxiality are diffracted by a circular aperture, the variation of the absolute and the relative changes for the degree of paraxiality versus the ratio R/w ₀ is laggardly. Otherwise, the variation of the absolute and the relative changes for the degree of paraxiality versus the ratio R/w ₀ is hasty. When the ratio R/w ₀ is large enough, the change for the degree of paraxiality of a diffracted Gaussian beam is small enough to be neglected. Accordingly, the change for the degree of paraxiality of a diffracted Gaussian beam is determined by two ratios R/λ and R/w ₀. In other words, the change for the degree of paraxiality is determined by the Gaussian-beam’s self and the circular aperture.

4. Conclusions

The change of the paraxiality of a Gaussian beam diffracted by a circular aperture has been investigated. An analytical expression of the degree of paraxiality for a diffracted Gaussian beam has been derived without any approximation. The change of the paraxiality of a diffracted Gaussian beam is graphically illustrated with numerical examples, and the influence of the circular aperture on the change of the paraxiality is also discussed in detail. The change for the degree of paraxiality depends on the Gaussian-beam’s self and the circular aperture.

Fig. 1. The degree of paraxiality, the absolute change for the degree of paraxiality, and the relative change for the degree of paraxiality of different Gaussian beams diffracted by a circular aperture as a function of the ratio R/λ, respectively.

Download Full Size | PDF

Fig. 2. The degree of paraxiality, the absolute change for the degree of paraxiality, and the relative change for the degree of paraxiality of different Gaussian beams diffracted by a circular aperture as a function of the ratio R/w ₀, respectively.

Download Full Size | PDF

Acknowledgments

The author is indebted to the reviewers for valuable comments. This work was supported by Scientific Research Fund of Zhejiang Provincial Education Department.

References and links

1. D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun . 224, 5–12 (2003). [CrossRef]

2. D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun . 236, 225–235 (2004). [CrossRef]

3. D. Zhao, H. Mao, and H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A: Pure Appl. Opt . 6, 77–83 (2004). [CrossRef]

4. G. Zhou, “Far-field structure of a linearly polarized plane wave diffracted by a rectangular aperture,” Opt. Laser Technol . 41, 504–508 (2009). [CrossRef]

5. T. Takenaka, M. Kakeya, and O. Fukumitsu, “Asymptotic representation of the boundary diffraction wave for a Gaussian beam incident on a circular aperture,” J. Opt. Soc. Am . 70, 1323–1328 (1980). [CrossRef]

6. K. Tanaka, N. Saga, and H. Mizokami, “Field spread of a diffracted Gaussian beam through a circular aperture,” Appl. Opt . 24, 1102–1106 (1985). [CrossRef] [PubMed]

7. P. Belland and J. P. Crenn, “Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture,” Appl. Opt . 21, 522–527 (1982). [CrossRef] [PubMed]

8. S. Wang, E. Bernabeu, and J. Alda, “ABCD matrix for weakly apertured Gaussian beams in the far field,” Appl. Opt . 30, 1584–1585 (1991). [CrossRef] [PubMed]

9. Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt . 36, 772–778 (1997). [CrossRef] [PubMed]

10. B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett . 28, 2440–2442 (2003). [CrossRef] [PubMed]

11. C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol . 39, 598–604 (2007). [CrossRef]

12. K. Duan and B. Lü, “Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture,” Opt. Express 11, 1474–1480 (2003). [CrossRef] [PubMed]

13. Z. L. Horväth and Z. Bor, “Focusing of truncated Gaussian beams,” Opt. Commun . 222, 51–68 (2003). [CrossRef]

14. H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22, 647–653 (2005). [CrossRef]

15. S. Teng, T. Zhou, and C. Cheng, “Fresnel diffraction of truncated Gaussian beam,” Optik 118, 435–439 (2007). [CrossRef]

16. H. Liu, R. Liang, and X. Huang, “Study on the approximative formula for the far-field of a Gaussian beam under circular aperture diffraction and its divergence,” Chin. Opt. Lett . 5, 4–7 (2007).

17. G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun . 281, 1929–1934 (2008). [CrossRef]

18. X. Du and D. Zhao, “Propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems,” J. Opt. Soc. Am. A 23, 625–631 (2006). [CrossRef]

19. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975). [CrossRef]

20. G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am . 69, 575–578 (1979). [CrossRef]

21. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation for light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985). [CrossRef]

22. S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt . 29, 1940–1946 (1990). [CrossRef] [PubMed]

23. O. E. Gawhary and S. Severini, “Degree of paraxiality for monochromatic light beams,” Opt. Lett . 33, 1360–1362 (2008). [CrossRef] [PubMed]

24. O. E. Gawhary and S. Severini, “Dependence of the degree of paraxiality on field correlations,” Opt. Lett . 33, 1866–1868 (2008). [CrossRef] [PubMed]

25. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, New York, 1980).

Change of the paraxiality of a Gaussian beam diffracted by a circular aperture

Abstract

1. Introduction

2. Change of the paraxiality of a Gaussian beam diffracted by a circular aperture

3. Numerical calculations and discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (2)

Equations (19)

Optics Express