Partially Coherent Flat-Topped Beam Generated by an Axicon

Abstract

The intensity distribution of a partially coherent beam with a nonconventional correlation function, named the multi-Gaussian Schell-model (MGSM) beam, focused by an axicon was investigated in detail. Our numerical results showed that an optical needle with a flat-topped spatial profile and long focal depth was formed and that we can modulate the focal shift and focal depth of the optical needle by varying the width of the degree of coherence (DOC) and the parameters of the correlation function. The adjustable optical needle can be applied for electron acceleration, particle trapping, fiber coupling and percussion drilling.

1. Introduction

Over the past years, axicon lenses have been studied widely due to their unique properties of producing focal lines of narrow transverse width over extended distances [1,2]. The axicon lens and its combination with other optical elements have been applied in many fields such as laser machining, corneal surgery, and atom alignment and trapping [2,3,4,5]. Since 1986, when Durnin first put forward a non-diffracting beam named a Bessel beam [6], a great deal of attention has been concentrated on the axicons lens as it provides the most energy efficient method of realizing generally diffraction-free beams [7,8,9]. Later, the research was extended from completely coherent beams to partially coherent beams. For a spatially partially coherent light, Friberg et al. proposed a method of designing annular-aperture axicons or apodized annular logarithmic axicons to generate a uniform intensity axial line image [10,11]. Shukri et al. designed diffractive axicons to produce uniform line images in a twisted Gaussian Schell-model (TGSM) illumination [12,13]. Pu et al. generated adjustable partially coherent bottle beams by focusing partially coherent light with an axicon-lens system [14,15]. Alkelly et al. examined the focal depth of partially coherent fields without aperture [16]. To date, the research on focusing partially coherent fields by an axicon lens has been restricted to Gaussian Schell-model beams. The results of the above-mentioned research show that the coherence length of partially coherent light has an intimate relationship with the axial intensity distribution of the focused field formed by an axicon. In an experiment, we can easily obtain different values of the coherence width by varying the focused beam spot on a rotating ground glass disk (RGGD) through adjusting the distance between the focus lens and RGGD [17].

With the rapid development of partially coherent light and its wide applications [17], nonconventional partially coherent light, the correlation function of which does not satisfy a Gaussian distribution, has been proposed [18,19]. It has been confirmed both theoretically and experimentally that the correlation function provides a new effective way to shape the intensity profile [20,21,22,23,24,25,26,27,28,29,30]. What is the effect of the nonconventional correlation function on the intensity distribution of beams focused by an axicon lens? Among a variety of nonconventional partially coherent beams, we are more interested in beams with a flat-topped spatial profile, which have shown great value in lithography, integrated circuit trimming, laser welding, and biomedical engineering. In 2012, a multi-Gaussian Schell-model (MGSM) beam [31], the degree of coherence (DOC) of which was modeled by multi-Gaussian distribution, was proposed by Korotkova et al. The MGSM beam can form flat square or rectangular intensity distributions in the far field [31,32,33]. Later, generalized GMSM beam and elliptical MGSM beam were proposed, which can produce prescribed flat-topped spatial profiles [34,35]. For these nonconventional partially coherent beams, extra parameters of a correlation function can be used to control the flat-topped spatial profiles. In this paper, we will investigate the intensity distribution of a MGSM beam focused by an axicon and explore the effect of a nonconventional correlation function on the focusing properties of a MGSM beam.

2. Analytical Expression of a MGSM Beam Focused by an Axicon Lens

The cross-spectral density (CSD) function of a MGSM beam in the source plane is described as follows [31]:

$$W\left({\rho }_1,{\rho }_2\right)=G_0\exp \left(-\frac{{\rho }_1^2+{\rho }_2^2}{2{\sigma }^2}\right)\mu \left({\rho }_1,{\rho }_2\right),$$

(1)

where $G_0$ is a normalized constant, ${\rho }_1,{\rho }_2$ are two arbitrary points in the source plane, $\sigma$ is the beam waist width of the source, $\mu \left({\rho }_1,{\rho }_2\right)$ is the spectral DOC. The DOC of a MGSM beam is expressed by [31]:

$$\mu \left({\rho }_1,{\rho }_2\right)=\frac{1}{C_0}{\displaystyle \sum _{m=1}^M\frac{{\left(-1\right)}^{m-1}}{m}\left(\begin{array}{c}M\\ m\end{array}\right)}\exp \left[-\frac{{\left({\rho }_2-{\rho }_1\right)}^2}{2m{\delta }^2}\right],$$

(2)

where $C_0={\displaystyle \sum _{m=1}^M\frac{{\left(-1\right)}^{m-1}}{m}\left(\begin{array}{c}M\\ m\end{array}\right)}$ is the normalization factor, M is the beam order and δ is the initial correlation width of the beam. A MGSM beam would reduce to a GSM beam when M = 1 or a coherent Gauss beam when $\delta =\infty$ and M = 1. Figure 1 shows the distribution of the DOC of a MGSM beam for different values of the beam order M. Generally, it displays a more complicated form represented by a sum of positive and negative exponentials.

According to the generalized Huygens–Fresnel principle [36,37], a MGSM beam propagating through an axicon, as shown on Figure 2, can be dealt with the following generalized Collins integral formula:

$$\begin{array}{l}W\left(r_1,r_2\right)=\frac{1}{{\lambda }^2{B}^2}{\displaystyle \iint W\left({\rho }_1,{\rho }_2\right)}{T}^{*}\left({\rho }_1\right)T\left({\rho }_2\right)\times \exp \{\frac{-jk}{2}[\left(\frac{A^{*}}{B^{*}}{\rho }_1^2-\frac{A}{B}{\rho }_2^2\right)\\ -2(\frac{1}{B^{*}}{\rho }_1{r}_1-\frac{1}{B}{\rho }_2{r}_2)+\left(\frac{D^{*}}{B^{*}}{r}_1^2-\frac{D}{B}{r}_2^2\right)]\}{d}^2{\rho }_1{d}^2{\rho }_2,\end{array}$$

(3)

where $r_1,r_2$ are two arbitrary points in the output plane, $\lambda$ is the wavelength, A, B, C and D are the transfer matrix elements of an optical system, $T$ is the transmittance function of the axicon lens, which can be expressed as follows:

$$T\left(\rho \right)=\{\begin{array}{cc}\exp \left[jk\left(n_d\Re -\beta \rho \right)\right]& \rho \le R\\ 0& \rho >R\end{array},$$

(4)

with

$$\beta =\left(n_d-1\right)\tan \frac{\pi -\alpha }{2}.$$

(5)

Here $n_d$ is the relative refractive index of the axicon to the medium around it, $\Re$ is the maxima of its thickness, α is the apex angle of the axicon lens.

One can rewrite Equation (3) in a polar coordinate system with $\rho =\left(\rho \cos \phi ,\rho \sin \phi \right)$, $r=\left(r\cos \theta ,r\sin \theta \right),$ as:

$$\begin{array}{l}W\left(r_1,r_2\right)=\frac{1}{{\lambda }^2{B}^2}\frac{1}{C_0}{\displaystyle \sum _{m=1}^M\left(\begin{array}{c}M\\ m\end{array}\right)}\frac{{\left(-1\right)}^{m-1}}{m}\exp \left[\frac{-jk}{2}\left(\frac{D^{*}}{B^{*}}{r}_1^2-\frac{D}{B}{r}_2^2\right)\right]\\ \times {\displaystyle \underset{0}{\overset{+\infty }{\int }}{\displaystyle \underset{0}{\overset{+\infty }{\int }}{\rho }_1{\rho }_2}}\exp \left[-\frac{1}{2{\Delta }_m^2}\left({\rho }_2^2+{\rho }_1^2\right)\right]\exp \left[jkf\left({\rho }_1,{\rho }_2\right)\right]\\ \times {\displaystyle \underset{0}{\overset{2\pi }{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}\exp \left[\frac{{\rho }_1{\rho }_2\cos \left({\phi }_2-{\phi }_1\right)}{m{\delta }^2}\right]\exp \{[jk(\frac{1}{B^{*}}{\rho }_1{r}_1\cos \left({\phi }_1-{\theta }_1\right)}}\\ -\frac{1}{B}{\rho }_2{r}_2\cos \left({\phi }_2-{\theta }_2\right))]\}d{\phi }_{1}d{\phi }_{2}d{\rho }_{1}d{\rho }_2,\end{array}$$

(6)

where

$$\frac{1}{{\Delta }_m^2}=\frac{1}{{\sigma }^2}+\frac{1}{m{\delta }^2}.$$

(7)

And after tedious integration, the expression of the CSD in the output plane can be obtained as:

$$f\left({\rho }_1,{\rho }_2\right)=\left(\frac{A}{2B}{\rho }_2^2-\beta {\rho }_2\right)-\left(\frac{A^{*}}{2B^{*}}{\rho }_1^2-\beta {\rho }_1\right).$$

(8)

$$\begin{array}{l}W\left(r_1,r_2\right)=\frac{k^2}{B^2{C}_0}{\displaystyle \sum _{m=1}^M{\displaystyle \sum _{l=-\infty }^{\infty }{e}^{jl\left({\theta }_2-{\theta }_1\right)}}\left(\begin{array}{c}M\\ m\end{array}\right)}\frac{{\left(-1\right)}^{m-1}}{m}\exp \left[\frac{-jk}{2}\left(\frac{D^{*}}{B^{*}}{r}_1^2-\frac{D}{B}{r}_2^2\right)\right]\\ \times {\displaystyle \underset{0}{\overset{+\infty }{\int }}{\displaystyle \underset{0}{\overset{+\infty }{\int }}{\rho }_1{\rho }_2}}\exp \left[-\frac{1}{2{\Delta }_m^2}\left({\rho }_2^2+{\rho }_1^2\right)\right]\exp \left[jkf\left({\rho }_1,{\rho }_2\right)\right]\\ \times C_{lm}\left(B,{\rho }_1,{\rho }_2,r_1,r_2\right)d{\rho }_{1}d{\rho }_2,\end{array}$$

(9)

with

$$C_{lm}\left(B,{\rho }_1,{\rho }_2,r_1,r_2\right)=j^{-l}{J}_l\left(j\frac{{\rho }_1{\rho }_2}{m{\delta }^2}\right)J_l\left(k\frac{{\rho }_1{r}_1}{B^{*}}\right)J_l\left(k\frac{{\rho }_2{r}_2}{B}\right).$$

(10)

where $J_l$ is the first kind Bessel function with order $l$ . In Equation (9), the phase factor $\exp \left[jkf\left({\rho }_1,{\rho }_2\right)\right]$ oscillates around zero intensively within the range of the optical frequencies. Applying the stationary phase method in [12,13,38], the integration result of Equation (9) can be acquired:

$$\begin{array}{l}W\left(r_1,r_2\right)=\frac{2\pi k}{B^2\sqrt{\left|ab\right|}{C}_0}{\rho }_{1s}{\rho }_{2s}\exp \left[\frac{-jk}{2}\left(\frac{D^{*}}{B^{*}}{r}_1^2-\frac{D}{B}{r}_2^2\right)\right]\exp \left[jkf\left({\rho }_{1s},{\rho }_{2s}\right)\right]\\ \times {\displaystyle \sum _{m=1}^M{\displaystyle \sum _{l=-\infty }^{\infty }\left(\begin{array}{c}M\\ m\end{array}\right)}}\frac{{\left(-1\right)}^{m-1}}{m}\exp \left[-\frac{1}{2{\Delta }_m^2}\left({\rho }_{2s}^2+{\rho }_{1s}^2\right)\right]\\ \times C_{lm}\left(B,{\rho }_{1s},{\rho }_{2s},r_1,r_2\right)\exp \left[jl\left({\theta }_2-{\theta }_1\right)\right],\end{array}$$

(11)

with

$$a=\frac{{\partial }^2}{\partial r_1^2}f\left({\rho }_1,{\rho }_2\right)=-\frac{A^{*}}{B^{*}},b=\frac{{\partial }^2}{\partial r_2^2}f\left({\rho }_1,{\rho }_2\right)=\frac{A}{B},{\rho }_{1s}=\frac{\beta B^{*}}{A^{*}},{\rho }_{2s}=\frac{\beta B}{A}.$$

(12)

We call the stationary phase points for ${\rho }_{1s}$ and ${\rho }_{2s}$ are roots of equation:

$$\frac{\partial }{\partial {\rho }_1}f\left({\rho }_1,{\rho }_2\right)=\frac{\partial }{\partial {\rho }_2}f\left({\rho }_1,{\rho }_2\right)=0.$$

(13)

The transfer matrix for a MGSM beam passing through an axicon is given as [36]:

$$\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)=\left(\begin{array}{cc}1& z\\ 0& 1\end{array}\right).$$

(14)

Replacing the elements of the transfer matrix in Equation (11) with the elements in Equation (14), we can obtain the final analytical expression for the average intensity of a MGSM beam focused by an axicon:

$$\begin{array}{l}I\left(r,z\right)=\frac{2\pi k}{z^2\sqrt{\left|ab\right|}{C}_0}{\rho }_{1s}{\rho }_{2s}\exp \left[jkf\left({\rho }_{1s},{\rho }_{2s}\right)\right]\\ \times {\displaystyle \sum _{m=1}^M{\displaystyle \sum _{l=-\infty }^{\infty }\left(\begin{array}{c}M\\ m\end{array}\right)}}\frac{{\left(-1\right)}^{m-1}}{m}\exp \left[-\frac{1}{2{\Delta }_m^2}\left({\rho }_{2s}^2+{\rho }_{1s}^2\right)\right]\\ \times C_{lm}\left(z,{\rho }_{1s},{\rho }_{2s},r,r\right).\end{array}$$

(15)

3. Numerical Simulation

In this section, we will analyze the focused intensity of a MGSM beam shaped by an axicon based on Equation (15). The beam parameter $\lambda$, the refractive index $n_d$, and the apex angle $\alpha$ of the axicon are set as $\lambda =0.633\mathsf{\mu }m,\sigma =2.5\mathsf{\mu }m,n_d=1.51630$ and $\alpha ={170}^{\circ }$.

In order to clarify the axicon’s effect on a MGSM beam, as seen in Figure 3 and Figure 4, we first calculated the normalized intensity distributions of a completely coherent Gaussian beam with $\delta =\infty ,M=1$ and a partially coherent Gaussian Schell-model (GSM) beam with $\delta =2\sigma ,M=1$. One can see in Figure 3 that the focused field of a completely coherent Gaussian beam displays an optical needle and the transverse intensity profile remains unchanged and demonstrates a non-diffraction property. Figure 4 shows that an optical needle is also formed as the coherence length of the incident beam decreases while the transverse profile of the intensity at different propagation distances varies due to beam diffraction. To give a quantitative evaluation to the effect of the coherence length on the diffraction of focused beams, we calculate the full width at half maximum (FWHM) of the intensity along z for different values of $\delta$, as shown in Figure 5. One sees that the FWHM along the z axis remains unchanged for a fully coherent beam, which is consistent with the transverse profiles of the intensity shown in Figure 3. For $\delta =2\sigma$, the FWHM increases slightly with the increase of z, as shown in Figure 5. For the smaller value of $\delta$, the focused field of a GSM beam shaped by an axicon diffracts more quickly. As a result, the length of coherence should not be too small to keep the non-diffraction property.

Next, we concentrated our attention on the focused field of a MGSM beam passing through an axicon. Figure 6 shows the normalized intensity generated by a MGSM beam with M = 4 and δ = 0.2σ. One can see that the focused field of a MGSM beam diffracts more rapidly than that of a GSM beam. The flat-topped profile emerges at z = 18 mm and from z = 18 mm to z = 22 mm the beam profile remains flat-topped. The axicon lens plays the role of extending the longitudinal range of the flat-topped spatial profile. As we increase the beam order M to 12 in Figure 7, the focused field experiences a more complicated variance. The spatial profile varies from a concave top at z = 14 mm to a flat-top at z = 18 mm and then to a cusp top at z = 20 mm. It can be concluded that the focused field of a partially coherent beam is closely related to the characteristic parameters of a correlation function. The FWHM of different beam orders M with $\delta =0.2\sigma$ in Figure 8 clearly demonstrates that as the beam order increases, the focused field diffracts more rapidly.

For a non-conventional partially coherent beam, the structure of a correlation function also induces the effect of focal shift [39]. The relative focal shift is defined as:

$$z_f=-\frac{z_{\max }-z_{G\max }}{z_{G\max }},$$

(16)

where $z_{\max }$ and $z_{G\max }$ are the intensity maxima position of a MGSM beam and the corresponding coherent Gaussian light focused by an axicon respectively. It is obvious from Figure 9 that either the lower value of coherence or the larger beam order leads to the increment of relative focal shift.

From above simulation results, we can generate an optical needle with a flat-topped spatial profile and long focal depth by focusing a MGSM beam with an axicon. The advantages of the formed optical needle are that the focal shift and focal depth can be modulated by varying the width of the degree of coherence (DOC) and the characteristic parameters of a correlation function.

4. Conclusions

Appling the generalized Huygens–Fresnel principle and the stationary phase method, the CSD of a nonconventional partially coherent beam called a MGSM beam passing through an axicon lens was derived in detail. The simulation result, based on the obtained equation, showed that an optical needle with a flat-topped spatial profile and long focal depth can be generated. Compared with the complete coherent light and conventional partially coherent light illumination, a MGSM beam illumination has the advantage not only in the prescribed spatial intensity profile, but also in the convenient control of the focal shift and focal depth. Our results will be useful for electron acceleration [40], particle trapping [41], fiber coupling and percussion drilling [42], where a partially coherent flat-topped beam spot is required.