Propagation of a partially coherent Lorentz-Gauss beam through a paraxial ABCD optical system

Guoquan Zhou

doi:10.1364/OE.18.004637

Optics Express
Vol. 18,
Issue 5,
pp. 4637-4643
(2010)
•https://doi.org/10.1364/OE.18.004637

Propagation of a partially coherent Lorentz-Gauss beam through a paraxial ABCD optical system

Guoquan Zhou

Open Access

Get PDF
Email
Share
Get Citation
Copy Citation Text
Guoquan Zhou, "Propagation of a partially coherent Lorentz-Gauss beam through a paraxial ABCD optical system," Opt. Express 18, 4637-4643 (2010)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

More Like This

Average intensity and spreading of a Lorentz-Gauss beam in turbulent atmosphere
Guoquan Zhou, et al.
Opt. Express 18(2) 726-731 (2010)

Propagation of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical...
Guoquan Zhou, et al.
Opt. Express 20(9) 9897-9910 (2012)

Paraxial propagation of a partially coherent Hermite-Gaussian beam through aligned and misaligned...
Yangjian Cai, et al.
J. Opt. Soc. Am. A 24(8) 2394-2401 (2007)

Related Topics
Table of Contents Category
- Coherence and Statistical Optics
Optics & Photonics Topics
?

The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: November 20, 2009
- Revised Manuscript: January 14, 2010
- Manuscript Accepted: February 5, 2010
- Published: February 22, 2010

Abstract

Based on the generalized Huygens-Fresnel integral and the Hermite-Gaussian expansion of a Lorentz distribution, analytical expressions for the mutual coherence function, the effective beam size, and the spatial complex degree of coherence of a partially coherent Lorentz-Gauss beam through a paraxial and real ABCD optical system are derived, respectively. As a numerical example, the focusing of a partially coherent Lorentz-Gauss beam is considered. The normalized intensity distribution, the effective beam size, and the spatial complex degree of coherence for the focused partially coherent Lorentz-Gauss beam are numerically demonstrated in the focal plane. The influence of the spatial coherence length on the normalized intensity distribution, the effective beam size, and the spatial complex degree of coherence is mainly discussed.

1. Introduction

With the identical spatial extension, the angular spreading of a Lorentz-Gaussian distribution is higher than that of a Gaussian distribution [1]. Therefore, Lorentz-Gauss beams provide appropriate models to describe the radiation emitted by a single mode diode laser [2,3]. The Lorentz beam is a special case of Lorentz-Gauss beams. Within the framework of the paraxial and non-paraxial cases, the properties of Lorentz-Gauss beams have been extensively investigated [4–14]. However, the reported researches were mainly confined to the case of fully coherent Lorentz-Gauss beams. In the practical optical systems, laser beams are almost partially coherent [15], which denotes that fully coherent laser sources are the ideal cases. In the paraxial optics, an arbitrary optical system is described by an ABCD matrix, which is very simple and convenient to the practical applications. To properly design an optical system that includes a single mode diode laser, the analysis of propagation of a Lorentz-Gauss beam passing through an ABCD optical system is prerequisite. In the remainder of this paper, therefore, the propagation of a partially coherent Lorentz-Gauss beam through a paraxial and real ABCD optical system is investigated. Moreover, analytical expressions for the mutual coherence function, the effective beam size, and the spatial complex degree of coherence are derived by means of the mathematical techniques. A numerical example is also demonstrated.

2. Propagation of a partially coherent Lorentz-Gauss beam through a paraxial and real ABCD optical system

In the Cartesian coordinate system, the z-axis is taken to be the propagation axis. The mutual coherence function of a partially coherent Lorentz-Gauss beam in the source plane z=0 is characterized by

Γ (x_{01}, x_{02}; y_{01}, y_{02}; 0) = E (x_{01}, y_{01}, 0) E (x_{02}, y_{02}, 0) g (x_{01}, x_{02}; y_{01}, y_{02}),

with E(x ₀₁,y ₀₁,0), E(x ₀₂,y ₀₂,0), and g(x ₀₁, x ₀₂;y ₀₁, y ₀₂) given by

E (x_{0 p}, y_{0 p}, 0) = \frac{1}{w_{0 x} w_{0 y} [1 + {(x_{0 p} / w_{0 x})}^{2}] [1 + {(y_{0 p} / w_{0 y})}^{2}]} \exp (- \frac{x_{0 p}^{2} + y_{0 p}^{2}}{w_{0}^{2}}),

g (x_{01}, x_{02}; y_{01}, y_{02}) = \exp [- \frac{{(x_{01} - x_{02})}^{2}}{2 σ_{x}^{2}} - \frac{{(y_{01} - y_{02})}^{2}}{2 σ_{y}^{2}}],

where p=1 or 2 (hereafter). w ₀ _x and w ₀ _y are the parameters related to the beam widths of the Lorentz part in the x- and y-directions, respectively. w ₀ is the waist of the Gaussian part.

g (x_{01}, x_{02}; y_{01}, y_{02})

is the complex degree of spatial coherence of beams generated by a Schell-model source. σ_x and σ_y are the spatial coherence length in the x- and y-directions, respectively. The time-dependent factor exp(-iωt) is omitted in the Eq. (1), and ω is the circular frequency. The Lorentz distribution can be expanded into the linear superposition of Hermite-Gaussian functions:

\frac{1}{(x_{0 p}^{2} + w_{0 x}^{2}) (y_{0 p}^{2} + w_{0 y}^{2})} = \frac{π}{2 w_{0 x}^{2} w_{0 y}^{2}} \sum_{m = 0}^{N} \sum_{n = 0}^{N} a_{2 m} a_{2 n} H_{2 m} (\frac{x_{0 p}}{w_{0 x}}) H_{2 n} (\frac{y_{0 p}}{w_{0 y}}) \exp (- \frac{x_{0 p}^{2}}{w_{0 x}^{2}} - \frac{y_{0 p}^{2}}{w_{0 y}^{2}}),

where N is the number of the expansion. a ₂ _m and a ₂ _n are the weight coefficients and can be indexed in [16]. H ₂ _m(.) and H ₂ _n(.) are the 2mth- and 2nth-order Hermite polynomials, respectively. Therefore, Eq. (2) can be rewritten as follows:

E (x_{0 p}, y_{0 p}, 0) = \frac{π}{2 w_{0 x} w_{0 y}} \sum_{m = 0}^{N} \sum_{n = 0}^{N} a_{2 m}^{} a_{2 n}^{} H_{2 m} (\frac{x_{0 p}}{w_{0 x}}) H_{2 n} (\frac{y_{0 p}}{w_{0 y}}) \exp (- \frac{x_{0 p}^{2}}{w_{x}^{2}} - \frac{y_{0 p}^{2}}{w_{y}^{2}}),

where

\frac{1}{w_{j}^{2}} = \frac{1}{w_{0}^{2}} + \frac{1}{2 w_{0 j}^{2}},

and j=x or y (hereafter). The propagation of a partially coherent Lorentz-Gauss beam through a paraxial ABCD optical system is described by the generalized Huygens-Fresnel diffraction integral:

\begin{array}{l} Γ (x_{1}, x_{2}; y_{1}, y_{2}; z) = \frac{- 1}{λ B} \exp (\frac{i k D}{2 B} [(x_{2}^{2} + y_{2}^{2}) - (x_{1}^{2} + y_{1}^{2})]) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} Γ (x_{01}, x_{02}; y_{01}, y_{02}; 0) \\ \times \exp {\frac{i k}{2 B} [A (x_{02}^{2} + y_{02}^{2}) - A (x_{01}^{2} + y_{01}^{2}) - 2 (x_{02} x_{2} + y_{02} y_{2}) + 2 (x_{01} x_{1} + y_{01} y_{1})]} d x_{01} d x_{02} d y_{01} d y_{02}, \end{array}

where

k = 2 π / λ

is the wave number. λ is the wavelength. A, B, C, and D are matrix elements of the optical system between the source and the output planes. Moreover, there is no inherent aperture between the source and the output planes. Therefore, A, B, C, and D are all real-valued. Using the following mathematical formulae [17]:

\int_{- \infty}^{\infty} H_{2 m^{'}} (x) \exp [- {(x - y)}^{2} / u] d x = \sqrt{π / u} {(1 - u)}^{m^{'}} H_{2 m^{'}} [{(1 - u)}^{- 1 / 2} y],

H_{2 m} (x) = \sum_{l = 0}^{m} \frac{{(- 1)}^{l} (2 m)!}{l! (2 m - 2 l)!} {(2 x)}^{2 m - 2 l},

\int_{- \infty}^{\infty} x^{2 n} \exp (- b x^{2} + 2 c x) d x = (2 n)! \sqrt{\frac{π}{b}} {(\frac{c}{b})}^{2 n} \exp (\frac{c^{2}}{b}) \sum_{s = 0}^{n} \frac{1}{s! (2 n - 2 s)!} {(\frac{b}{4 c^{2}})}^{s},

we can obtain the mutual coherence function of a partially coherent Lorentz-Gauss beam in the output plane as

Γ (x_{1}, x_{2}; y_{1}, y_{2}; z) = Γ (x_{1}, x_{2}, z) Γ (y_{1}, y_{2}, z),

with

Γ (x_{1}, x_{2}, z)

and

Γ (y_{1}, y_{2}, z)

given by

\begin{array}{l} Γ (j_{1}, j_{2}, z) = \frac{π^{2}}{2} \sqrt{\frac{i}{λ B α_{1 j} α_{2 j}}} \exp (\frac{β_{j}^{2}}{α_{2 j}} - \frac{k^{2} w_{0 j}^{2} j_{2}^{2}}{4 α_{1 j} B^{2}} + \frac{i k D}{2 B} (j_{2}^{2} - j_{1}^{2})) \sum_{m = 0}^{N} \sum_{m^{'} = 0}^{N} a_{2 m}^{} a_{2 m^{'}}^{} {(1 - \frac{1}{α_{1 j}})}^{m^{'}} \\ \begin{matrix} \begin{matrix}  \end{matrix} & \begin{matrix}  \end{matrix} \end{matrix} \times \sum_{l_{1} = 0}^{m} \frac{{(- 1)}^{l_{1}} (2 m)!}{l_{1}! (2 m - 2 l_{1})!} \sum_{l_{2} = 0}^{m^{'}} \frac{{(- 1)}^{l_{2}} (2 m^{'})!}{l_{2}! (2 m^{'} - 2 l_{2})!} \sum_{l_{3} = 0}^{2 m^{'} - 2 l_{2}} (_{\begin{matrix} \begin{matrix} l_{3} \end{matrix} \end{matrix}}^{2 m^{'} - 2 l_{2}}) 2^{2 (m + m^{'}) - 2 l_{1} - 2 l_{2}} γ_{j}^{l_{3}} δ_{j}^{2 m^{'} - 2 l_{2} - l_{3}} \\ \begin{matrix} \begin{matrix}  \end{matrix} & \begin{matrix}  \end{matrix} \end{matrix} \times (2 m + l_{3} - 2 l_{1})! {(\frac{β_{j}^{}}{α_{2 j}})}^{2 m + l_{3} - 2 l_{1}} \sum_{s = 0}^{[m - l_{1} + l_{3} / 2]} \frac{1}{s! (2 m + l_{3} - 2 l_{1} - 2 s)!} {(\frac{α_{2 j}}{4 β_{j}^{2}})}^{s}, \end{array}

where [m-l ₁+l ₃/2] gives the greatest integer less than or equal to (m-l ₁+l ₃/2), and

α_{1 j} = (\frac{1}{w_{j}^{2}} + \frac{1}{2 σ_{j}^{2}} - \frac{i k A}{2 B}) w_{0 j}^{2}, α_{2 j} = (\frac{1}{w_{j}^{2}} + \frac{1}{2 σ_{j}^{2}} + \frac{i k A}{2 B}) w_{0 j}^{2} - \frac{w_{0 j}^{4}}{4 α_{1 j} σ_{j}^{4}},

β_{j} = \frac{i k w_{0 j} j_{1}}{2 B} - \frac{i k w_{0 j}^{3} j_{2}}{4 α_{1 j} B σ_{j}^{2}}, γ_{j} = \frac{w_{0 j}^{2}}{2 {(α_{1 j}^{2} - α_{1 j})}^{1 / 2} σ_{j}^{2}}, δ_{j} = \frac{k w_{0 j}^{} j_{2}}{2 i B {(α_{1 j}^{2} - α_{1 j})}^{1 / 2}} .

The effective beam size of the partially coherent Lorentz-Gauss beam in the x- and y-directions of the output plane is defined as [18]

W_{j z} = \sqrt{\frac{2 \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} j^{2} Γ (x, x; y, y; z) d x d y}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} Γ (x, x; y, y; z) d x d y}} .

Substituting Eq. (11) into Eq. (15), the analytical effective beam size of the partially coherent Lorentz-Gauss beam yields

W_{j z} = \sqrt{\frac{2 Ω_{1 j}}{Ω_{2 j}}},

with Ω₁ _j and Ω₂ _j given by

\begin{array}{l} Ω_{1 j} = \sum_{m = 0}^{N} \sum_{m^{'} = 0}^{N} a_{2 m}^{} a_{2 m^{'}}^{} {(1 - \frac{1}{α_{1 j}})}^{m^{'}} \sum_{l_{1} = 0}^{m} \frac{{(- 1)}^{l_{1}} (2 m)!}{l_{1}! (2 m - 2 l_{1})!} \sum_{l_{2} = 0}^{m^{'}} \frac{{(- 1)}^{l_{2}} (2 m^{'})!}{l_{2}! (2 m^{'} - 2 l_{2})!} \sum_{l_{3} = 0}^{2 m^{'} - 2 l_{2}} (_{\begin{matrix} \begin{matrix} l_{3} \end{matrix} \end{matrix}}^{2 m^{'} - 2 l_{2}}) \\ \begin{matrix}  \end{matrix} \times 2^{2 (m + m^{'}) - 2 l_{1} - 2 l_{2}} γ_{j}^{l_{3}} ξ_{j}^{2 m^{'} - 2 l_{2} - l_{3}} (2 m + l_{3} - 2 l_{1})! \sum_{s = 0}^{[m - l_{1} + l_{3} / 2]} \frac{1}{s! (2 m + l_{3} - 2 l_{1} - 2 s)! 4^{s}} \\ \begin{matrix}  \end{matrix} \times α_{2 j}^{2 l_{1} + s - 2 m - l_{3}} η_{j}^{2 m + l_{3} - 2 l_{1} - 2 s} Γ (m + m^{'} - l_{1} - l_{2} - s + 1.5) ζ_{j}^{l_{1} + l_{2} + s - m - m^{'} - 1.5}, \end{array}

\begin{array}{l} Ω_{2 j} = \sum_{m = 0}^{N} \sum_{m^{'} = 0}^{N} a_{2 m}^{} a_{2 m^{'}}^{} {(1 - \frac{1}{α_{1 j}})}^{m^{'}} \sum_{l_{1} = 0}^{m} \frac{{(- 1)}^{l_{1}} (2 m)!}{l_{1}! (2 m - 2 l_{1})!} \sum_{l_{2} = 0}^{m^{'}} \frac{{(- 1)}^{l_{2}} (2 m^{'})!}{l_{2}! (2 m^{'} - 2 l_{2})!} \sum_{l_{3} = 0}^{2 m^{'} - 2 l_{2}} (_{\begin{matrix} \begin{matrix} l_{3} \end{matrix} \end{matrix}}^{2 m^{'} - 2 l_{2}}) \\ \begin{matrix}  \end{matrix} \times 2^{2 (m + m^{'}) - 2 l_{1} - 2 l_{2}} γ_{j}^{l_{3}} ξ_{j}^{2 m^{'} - 2 l_{2} - l_{3}} (2 m + l_{3} - 2 l_{1})! \sum_{s = 0}^{[m - l_{1} + l_{3} / 2]} \frac{1}{s! (2 m + l_{3} - 2 l_{1} - 2 s)! 4^{s}} \\ \begin{matrix}  \end{matrix} \times α_{2 j}^{2 l_{1} + s - 2 m - l_{3}} η_{j}^{2 m + l_{3} - 2 l_{1} - 2 s} Γ (m + m^{'} - l_{1} - l_{2} - s + 0.5) ζ_{j}^{l_{1} + l_{2} + s - m - m^{'} - 0.5}, \end{array}

where

ξ_{j} = \frac{k w_{0 j}^{}}{2 i B {(α_{1 j}^{2} - α_{1 j})}^{1 / 2}}, η_{j} = \frac{i k w_{0 j}}{2 B} - \frac{i k w_{0 j}^{3}}{4 α_{1 j} B σ_{j}^{2}}, ζ_{j} = \frac{k^{2} w_{0 j}^{2}}{4 α_{1 j} B^{2}} - \frac{η_{j}^{2}}{α_{2 j}},

and Γ(.) is a Gamma function.

The spatial complex degree of coherence of the partially coherent Lorentz-Gauss beam at two points (x ₁, y ₁, z) and (x ₂, y ₂, z) turns out to [19,20]

μ (x_{1}, x_{2}; y_{1}, y_{2}; z) = μ (x_{1}, x_{2}, z) μ (y_{1}, y_{2}, z),

with

μ (x_{1}, x_{2}, z)

and

μ (y_{1}, y_{2}, z)

given by

μ (j_{1}, j_{2}, z) = \frac{Γ (j_{1}, j_{2}, z)}{{[Γ (j_{1}, j_{1}, z) Γ (j_{2}, j_{2}, z)]}^{1 / 2}} .

Inserting Eq. (12) into Eq. (21), one can calculate the spatial complex degree of coherence.

3. Numerical calculations and analyses

Now, we consider the focusing of the partially coherent Lorentz-Gauss beam. A thin lens with the focal length f is placed in front of the single mode diode laser, so that the partially coherent Lorentz-Gauss beam is transformed into a converging beam. In the case of diode-fiber coupling, a fiber end is placed in the focal region. The matrix elements of this optical arrangement are A=0, B=f, C=−1/f, and D=1. The normalized light intensity, the effective beam size, and the spatial complex degree of coherence for the partially coherent Lorentz-Gauss beam in the focal plane are calculated by using the formulae derived above. As the x- and y-directions are separable, only the x-direction is considered in the following calculations. Moreover, we mainly concentrate on the effect of the spatial coherence length. Figure 1 represents the normalized intensity distribution in the focal plane. The parameters used are chosen as follow: λ=0.8μm, f=1m and w ₀ _x=1mm. w ₀=2mm in Fig. 1(a) and w ₀=∞ in Fig. 1(b). The normalized intensity distribution of the partially coherent Lorentz-Gauss beam spreads with decreasing the spatial coherence length σ_x. The effective beam size of partially coherent Lorentz-Gauss beams in the focal plane versus the coherence length σ_x is depicted in Fig. 2 . The inset shows the detail change of W_xz within the range of 0.2mm≤σ_x≤1mm. With increasing the spatial coherence length σ_x, the effective beam size first quickly decreases and then tends to a minimum value. The better coherence the partially coherent Lorentz-Gauss beam has, the smaller effective beam size it has. If the spatial coherence length σ_x keeps invariant, the partially coherent Lorentz-Gauss beam with smaller w ₀ and w ₀ _x has the larger effective beam size. Figure 3 shows the spatial complex degree of coherence in the focal plane. w ₀=2mm and w ₀ _x=1mm. σ_x=1mm, 2mm, and $2 \sqrt{10}$ mm in Figs. 3(a)–3(c), respectively. With given two given points (x ₁, f) and (x ₂, f), their spatial complex degree of coherence increases with increasing the spatial coherence length σ_x. To quantitatively evaluate the influence of the spatial coherence length on the spatial complex degree of coherence, we calculate the spatial complex degree of coherence at two points (0.1mm, f)and (0.4mm, f) by altering the spatial coherence length σ_x, which is shown in Fig. 4 . With increasing the spatial coherence length σ_x, the spatial complex degree of coherence first quickly increases and then tends to the saturated value 1. With a given σ_x, the partially coherent Lorentz-Gauss beam with the smaller w ₀ and w ₀ _x has the larger spatial complex degree of coherence.

Fig. 1 Normalized intensity distribution in the x-direction of partially coherent Lorentz-Gauss beams with different σ_x in the focal plane. w ₀ _x=1mm. (a) w ₀=2mm. (b) w ₀=∞.

Download Full Size | PDF

Fig. 2 The effective beam size in the x-direction of partially coherent Lorentz-Gauss beams in the focal plane versus the coherence length σ_x. (a) w ₀=2mm. (b) w ₀ _x=2mm.

Download Full Size | PDF

Fig. 3 The spatial complex degree of coherence in the x-direction of partially coherent Lorentz-Gauss beams in the focal plane. w ₀=2mm and w ₀ _x=1mm. (a) σ_x=1mm. (b) σ_x=2mm. (c) σ_x= $2 \sqrt{10}$ mm.

Download Full Size | PDF

Fig. 4 The spatial complex degree of coherence in the x-direction of partially coherent Lorentz-Gauss beams in the focal plane versus the coherence length σ_x. x ₁=0.1mm and x ₂=0.4mm. (a) w ₀=2mm. (b) w ₀ _x=2mm.

Download Full Size | PDF

It should be pointed out that the presented formulae are valid for the case of the three beam parameters being far larger than the wavelength. In the paraxial case, the beam propagation fa-ctor of a partially coherent Lorentz-Gauss beam has been presented as Eq. (22) in [21]. By expanding the complementary error function, the beam propagation factor of a partially coherent Lorentz-Gauss beam can be verified to be larger than that of a corresponding partially coherent Gaussian beam. Therefore, the angular spreading of a partially coherent Lorentz-Gauss beam still preserves high within the paraxial propagation. When the three beam parameters are of the order of the wavelength, an efficaciously non-paraxial method is to be sought to treat the propagation of partially coherent Lorentz-Gauss beams.

4. Conclusions

Based on the generalized Huygens-Fresnel integral and the expansion of Lorentz distribution, the analytical propagation equation of a partially coherent Lorentz-Gauss beam through a paraxial and real ABCD optical system is derived. Moreover, the analytical formulae for the effective beam size and the spatial complex degree of coherence are also presented. As a numerical example, the normalized light intensity, the effective beam size, and the spatial complex degree of coherence for the partially coherent Lorentz-Gauss beam focused by a thin lens are calculated in the focal plane. The effect of the spatial coherence length is mainly discussed. With increasing the spatial coherence length, the partially coherent Lorentz-Gauss beam has the smaller effective beam size and the higher spatial complex degree of coherence. As apertures usually exist in the practical optical system, the propagation of a partially coherent Lorentz-Gauss beam through a paraxial and complex ABCD optical system also deserves to be investigated. This research is useful to the optical designs that are involved in the single mode diode laser.

Acknowledgements

This research was supported by National Natural Science Foundation of China under Grant No. 10974179 and Zhejiang Provincial Natural Science Foundation of China under Grant No. Y1090073.

References and links

1. W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11(7), 400–402 (1975). [CrossRef]

2. A. Naqwi and F. Durst, “Focus of diode laser beams: a simple mathematical model,” Appl. Opt. 29(12), 1780–1785 (1990). [CrossRef] [PubMed]

3. J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE 6824, 68240A (2008). [CrossRef]

4. O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. 8(5), 409–414 (2006). [CrossRef]

5. O. Elgawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. 269(2), 274–284 (2007). [CrossRef]

6. A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. 10(11), 115007 (2008). [CrossRef]

7. G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. 55(6), 993–1002 (2008). [CrossRef]

8. G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93(4), 891–899 (2008). [CrossRef]

9. G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55(21), 3573–3577 (2008). [CrossRef]

10. G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25(10), 2594–2599 (2008). [CrossRef]

11. G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26(1), 141–147 (2009). [CrossRef]

12. G. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A 26(2), 350–355 (2009). [CrossRef]

13. G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. 41(8), 953–955 (2009). [CrossRef]

14. G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96(1), 149–153 (2009). [CrossRef]

15. L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).

16. P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976). [CrossRef]

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

18. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express 17(13), 11130–11139 (2009). [CrossRef] [PubMed]

19. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5(5), 713–720 (1988). [CrossRef]

20. M. Born, and E. Wolf, Principles of Optics 7^th ed. (Cambridge University Press, Cambridge, UK, 1999).

21. G. Zhou, “Generalized M² factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. 12(1), 015701 (2010). [CrossRef]

Previous Article Next Article

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.

Fig. 1 Normalized intensity distribution in the x-direction of partially coherent Lorentz-Gauss beams with different σ_x in the focal plane. w ₀ _x =1mm. (a) w ₀=2mm. (b) w ₀=∞.

View in Article | Download Full Size | PDF

Fig. 2 The effective beam size in the x-direction of partially coherent Lorentz-Gauss beams in the focal plane versus the coherence length σ_x . (a) w ₀=2mm. (b) w ₀ _x =2mm.

View in Article | Download Full Size | PDF

Fig. 3 The spatial complex degree of coherence in the x-direction of partially coherent Lorentz-Gauss beams in the focal plane. w ₀=2mm and w ₀ _x =1mm. (a) σ_x =1mm. (b) σ_x =2mm. (c) σ_x =

2 \sqrt{10}

mm.

View in Article | Download Full Size | PDF

Fig. 4 The spatial complex degree of coherence in the x-direction of partially coherent Lorentz-Gauss beams in the focal plane versus the coherence length σ_x . x ₁=0.1mm and x ₂=0.4mm. (a) w ₀=2mm. (b) w ₀ _x =2mm.

View in Article | Download Full Size | PDF

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

Γ (x_{01}, x_{02}; y_{01}, y_{02}; 0) = E (x_{01}, y_{01}, 0) E (x_{02}, y_{02}, 0) g (x_{01}, x_{02}; y_{01}, y_{02}),

E (x_{0 p}, y_{0 p}, 0) = \frac{1}{w_{0 x} w_{0 y} [1 + {(x_{0 p} / w_{0 x})}^{2}] [1 + {(y_{0 p} / w_{0 y})}^{2}]} \exp (- \frac{x_{0 p}^{2} + y_{0 p}^{2}}{w_{0}^{2}}),

g (x_{01}, x_{02}; y_{01}, y_{02}) = \exp [- \frac{{(x_{01} - x_{02})}^{2}}{2 σ_{x}^{2}} - \frac{{(y_{01} - y_{02})}^{2}}{2 σ_{y}^{2}}],

\frac{1}{(x_{0 p}^{2} + w_{0 x}^{2}) (y_{0 p}^{2} + w_{0 y}^{2})} = \frac{π}{2 w_{0 x}^{2} w_{0 y}^{2}} \sum_{m = 0}^{N} \sum_{n = 0}^{N} a_{2 m} a_{2 n} H_{2 m} (\frac{x_{0 p}}{w_{0 x}}) H_{2 n} (\frac{y_{0 p}}{w_{0 y}}) \exp (- \frac{x_{0 p}^{2}}{w_{0 x}^{2}} - \frac{y_{0 p}^{2}}{w_{0 y}^{2}}),

E (x_{0 p}, y_{0 p}, 0) = \frac{π}{2 w_{0 x} w_{0 y}} \sum_{m = 0}^{N} \sum_{n = 0}^{N} a_{2 m}^{} a_{2 n}^{} H_{2 m} (\frac{x_{0 p}}{w_{0 x}}) H_{2 n} (\frac{y_{0 p}}{w_{0 y}}) \exp (- \frac{x_{0 p}^{2}}{w_{x}^{2}} - \frac{y_{0 p}^{2}}{w_{y}^{2}}),

\frac{1}{w_{j}^{2}} = \frac{1}{w_{0}^{2}} + \frac{1}{2 w_{0 j}^{2}},

\begin{array}{l} Γ (x_{1}, x_{2}; y_{1}, y_{2}; z) = \frac{- 1}{λ B} \exp (\frac{i k D}{2 B} [(x_{2}^{2} + y_{2}^{2}) - (x_{1}^{2} + y_{1}^{2})]) \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} Γ (x_{01}, x_{02}; y_{01}, y_{02}; 0) \\ \times \exp {\frac{i k}{2 B} [A (x_{02}^{2} + y_{02}^{2}) - A (x_{01}^{2} + y_{01}^{2}) - 2 (x_{02} x_{2} + y_{02} y_{2}) + 2 (x_{01} x_{1} + y_{01} y_{1})]} d x_{01} d x_{02} d y_{01} d y_{02}, \end{array}

\int_{- \infty}^{\infty} H_{2 m^{'}} (x) \exp [- {(x - y)}^{2} / u] d x = \sqrt{π / u} {(1 - u)}^{m^{'}} H_{2 m^{'}} [{(1 - u)}^{- 1 / 2} y],

H_{2 m} (x) = \sum_{l = 0}^{m} \frac{{(- 1)}^{l} (2 m)!}{l! (2 m - 2 l)!} {(2 x)}^{2 m - 2 l},

\int_{- \infty}^{\infty} x^{2 n} \exp (- b x^{2} + 2 c x) d x = (2 n)! \sqrt{\frac{π}{b}} {(\frac{c}{b})}^{2 n} \exp (\frac{c^{2}}{b}) \sum_{s = 0}^{n} \frac{1}{s! (2 n - 2 s)!} {(\frac{b}{4 c^{2}})}^{s},

Γ (x_{1}, x_{2}; y_{1}, y_{2}; z) = Γ (x_{1}, x_{2}, z) Γ (y_{1}, y_{2}, z),

\begin{array}{l} Γ (j_{1}, j_{2}, z) = \frac{π^{2}}{2} \sqrt{\frac{i}{λ B α_{1 j} α_{2 j}}} \exp (\frac{β_{j}^{2}}{α_{2 j}} - \frac{k^{2} w_{0 j}^{2} j_{2}^{2}}{4 α_{1 j} B^{2}} + \frac{i k D}{2 B} (j_{2}^{2} - j_{1}^{2})) \sum_{m = 0}^{N} \sum_{m^{'} = 0}^{N} a_{2 m}^{} a_{2 m^{'}}^{} {(1 - \frac{1}{α_{1 j}})}^{m^{'}} \\ \begin{matrix} \begin{matrix}  \end{matrix} & \begin{matrix}  \end{matrix} \end{matrix} \times \sum_{l_{1} = 0}^{m} \frac{{(- 1)}^{l_{1}} (2 m)!}{l_{1}! (2 m - 2 l_{1})!} \sum_{l_{2} = 0}^{m^{'}} \frac{{(- 1)}^{l_{2}} (2 m^{'})!}{l_{2}! (2 m^{'} - 2 l_{2})!} \sum_{l_{3} = 0}^{2 m^{'} - 2 l_{2}} (_{\begin{matrix} \begin{matrix} l_{3} \end{matrix} \end{matrix}}^{2 m^{'} - 2 l_{2}}) 2^{2 (m + m^{'}) - 2 l_{1} - 2 l_{2}} γ_{j}^{l_{3}} δ_{j}^{2 m^{'} - 2 l_{2} - l_{3}} \\ \begin{matrix} \begin{matrix}  \end{matrix} & \begin{matrix}  \end{matrix} \end{matrix} \times (2 m + l_{3} - 2 l_{1})! {(\frac{β_{j}^{}}{α_{2 j}})}^{2 m + l_{3} - 2 l_{1}} \sum_{s = 0}^{[m - l_{1} + l_{3} / 2]} \frac{1}{s! (2 m + l_{3} - 2 l_{1} - 2 s)!} {(\frac{α_{2 j}}{4 β_{j}^{2}})}^{s}, \end{array}

α_{1 j} = (\frac{1}{w_{j}^{2}} + \frac{1}{2 σ_{j}^{2}} - \frac{i k A}{2 B}) w_{0 j}^{2}, α_{2 j} = (\frac{1}{w_{j}^{2}} + \frac{1}{2 σ_{j}^{2}} + \frac{i k A}{2 B}) w_{0 j}^{2} - \frac{w_{0 j}^{4}}{4 α_{1 j} σ_{j}^{4}},

β_{j} = \frac{i k w_{0 j} j_{1}}{2 B} - \frac{i k w_{0 j}^{3} j_{2}}{4 α_{1 j} B σ_{j}^{2}}, γ_{j} = \frac{w_{0 j}^{2}}{2 {(α_{1 j}^{2} - α_{1 j})}^{1 / 2} σ_{j}^{2}}, δ_{j} = \frac{k w_{0 j}^{} j_{2}}{2 i B {(α_{1 j}^{2} - α_{1 j})}^{1 / 2}} .

W_{j z} = \sqrt{\frac{2 \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} j^{2} Γ (x, x; y, y; z) d x d y}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} Γ (x, x; y, y; z) d x d y}} .

W_{j z} = \sqrt{\frac{2 Ω_{1 j}}{Ω_{2 j}}},

\begin{array}{l} Ω_{1 j} = \sum_{m = 0}^{N} \sum_{m^{'} = 0}^{N} a_{2 m}^{} a_{2 m^{'}}^{} {(1 - \frac{1}{α_{1 j}})}^{m^{'}} \sum_{l_{1} = 0}^{m} \frac{{(- 1)}^{l_{1}} (2 m)!}{l_{1}! (2 m - 2 l_{1})!} \sum_{l_{2} = 0}^{m^{'}} \frac{{(- 1)}^{l_{2}} (2 m^{'})!}{l_{2}! (2 m^{'} - 2 l_{2})!} \sum_{l_{3} = 0}^{2 m^{'} - 2 l_{2}} (_{\begin{matrix} \begin{matrix} l_{3} \end{matrix} \end{matrix}}^{2 m^{'} - 2 l_{2}}) \\ \begin{matrix}  \end{matrix} \times 2^{2 (m + m^{'}) - 2 l_{1} - 2 l_{2}} γ_{j}^{l_{3}} ξ_{j}^{2 m^{'} - 2 l_{2} - l_{3}} (2 m + l_{3} - 2 l_{1})! \sum_{s = 0}^{[m - l_{1} + l_{3} / 2]} \frac{1}{s! (2 m + l_{3} - 2 l_{1} - 2 s)! 4^{s}} \\ \begin{matrix}  \end{matrix} \times α_{2 j}^{2 l_{1} + s - 2 m - l_{3}} η_{j}^{2 m + l_{3} - 2 l_{1} - 2 s} Γ (m + m^{'} - l_{1} - l_{2} - s + 1.5) ζ_{j}^{l_{1} + l_{2} + s - m - m^{'} - 1.5}, \end{array}

\begin{array}{l} Ω_{2 j} = \sum_{m = 0}^{N} \sum_{m^{'} = 0}^{N} a_{2 m}^{} a_{2 m^{'}}^{} {(1 - \frac{1}{α_{1 j}})}^{m^{'}} \sum_{l_{1} = 0}^{m} \frac{{(- 1)}^{l_{1}} (2 m)!}{l_{1}! (2 m - 2 l_{1})!} \sum_{l_{2} = 0}^{m^{'}} \frac{{(- 1)}^{l_{2}} (2 m^{'})!}{l_{2}! (2 m^{'} - 2 l_{2})!} \sum_{l_{3} = 0}^{2 m^{'} - 2 l_{2}} (_{\begin{matrix} \begin{matrix} l_{3} \end{matrix} \end{matrix}}^{2 m^{'} - 2 l_{2}}) \\ \begin{matrix}  \end{matrix} \times 2^{2 (m + m^{'}) - 2 l_{1} - 2 l_{2}} γ_{j}^{l_{3}} ξ_{j}^{2 m^{'} - 2 l_{2} - l_{3}} (2 m + l_{3} - 2 l_{1})! \sum_{s = 0}^{[m - l_{1} + l_{3} / 2]} \frac{1}{s! (2 m + l_{3} - 2 l_{1} - 2 s)! 4^{s}} \\ \begin{matrix}  \end{matrix} \times α_{2 j}^{2 l_{1} + s - 2 m - l_{3}} η_{j}^{2 m + l_{3} - 2 l_{1} - 2 s} Γ (m + m^{'} - l_{1} - l_{2} - s + 0.5) ζ_{j}^{l_{1} + l_{2} + s - m - m^{'} - 0.5}, \end{array}

ξ_{j} = \frac{k w_{0 j}^{}}{2 i B {(α_{1 j}^{2} - α_{1 j})}^{1 / 2}}, η_{j} = \frac{i k w_{0 j}}{2 B} - \frac{i k w_{0 j}^{3}}{4 α_{1 j} B σ_{j}^{2}}, ζ_{j} = \frac{k^{2} w_{0 j}^{2}}{4 α_{1 j} B^{2}} - \frac{η_{j}^{2}}{α_{2 j}},

μ (x_{1}, x_{2}; y_{1}, y_{2}; z) = μ (x_{1}, x_{2}, z) μ (y_{1}, y_{2}, z),

μ (j_{1}, j_{2}, z) = \frac{Γ (j_{1}, j_{2}, z)}{{[Γ (j_{1}, j_{1}, z) Γ (j_{2}, j_{2}, z)]}^{1 / 2}} .

Abstract

1. Introduction

2. Propagation of a partially coherent Lorentz-Gauss beam through a paraxial and real ABCD optical system

3. Numerical calculations and analyses

4. Conclusions

Acknowledgements

References and links

Cited By

Figures (4)

Equations (21)

Optics Express