Generating double focal spots by focusing a radially polarized double-ring-shaped beam with an annular classical axicon

Ibrahim G. H. Loqman; Abdu A. Alkelly; Hassan T. Al-Ahsab; Hassan T. Al-Ahsab

doi:10.1364/OPTCON.465483

1. Introduction

In recent years, vector double-ring-shaped beams have been used for generating different focal shapes due to their unique characters. Such beams have been used to generate different novel focal shapes, under certain conditions, such as longitudinally polarized needle [1], dark channel with a long depth of focus and sub-wavelength focal hole [2], multiple focal hole [3], super-long dark channel [4], three dimensional dark spot [5,6], ultra-long dark channel [7], different focal patterns dark hollow focus, focal spot and flat-topped beam [8], multiple optical cages [9] and multiple focal hole segments [10]. Furthermore, circularly polarized double-ring-shaped beam has been used for generating multiple focal spots with birefringence [11]. Radially polarized double-ring-shaped beam can be experimentally generated from laser cavity [12,13]. It can produce a strong longitudinal field, a sharp focal spot and a large focal depth [1,5,14–16]. Therefore, double-ring-shaped polarized focused beams have been utilized in different applications such as trapping particles, atom switches, optical data storage, recording system etc. [17–19].

Recently, axicon lenses have been used for generating long-distance propagating Bessel beams [20,21], longitudinally polarized thin needle [22], small focal spot [23,24], partially coherent flat-topped beam [25] and different focal shapes [26–30]. Furthermore, the axicon lens and its combination with other optical elements have been applied in different fields such as atom guiding and trapping, laser machining and corneal surgery [31–34]. Therefore, focusing radially polarized double-ring-shaped beam by annular classical axicon can produce a new focal shapes. In this article, double focal spots and flat-topped beam are generated using a simple optical system: annular classical axicon illuminated by a radially polarized double-ring-shaped beam. Their properties can be altered by changing system and beam parameters.

2. Theoretical model

Radially polarized double-ring-shaped beam form and a schematic of an optical system are shown in Fig. 1. Figure 1(a) demonstrates the $\pi$ phase difference between the inner and the outer rings which cause the destructive interference between the components of the electric field in the focal region.

Fig. 1. (a)Radially polarized double-ring-shaped beam and (b) schematic of optical system.

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Based on the vector Debye theory, the electric field $\textbf {E}(r, \phi, z)$ in the focal region of annular classical axicon for radially polarized focused beam can be written as [35]

(1)$$\begin{aligned} \textbf{E}(r, \varphi, z) =\left[\begin{array}{cc}E_{r}\\ E_{\varphi}\\ E_{z}\end{array}\right]={-} \frac{i A}{\pi} & \int_{\sigma . \alpha}^{\alpha}\int_{0}^{2\pi} \sqrt{\cos\theta}\sin\theta T(\theta) E(\theta, \phi) \left(\begin{array}{cc} \cos\theta \cos(\phi-\varphi)\\ \cos\theta\sin(\phi-\varphi) \\ - \sin\theta \end{array}\right)\\ & \times \exp(i k z \cos\theta+r \sin\theta\cos(\phi-\varphi)) d\theta d\phi, \end{aligned}$$

where $r,\phi,z$ are the cylindrical coordinates in the focal region, A is a constant, $\alpha =\arcsin (NA/n)$ is the convergence angle corresponding to the radius of the incident optical aperture, $NA$ is the numerical aperture and $n$ is the refractive index in the image space, $\sigma$ is the annular obstruction i.e. inner focus angle $\theta _{min}$ to $\alpha$, $k=2\pi /\lambda$ is the wave number. $T(\theta )=\exp (i B \sin \theta /\sin \alpha )$ is the transmittance function of the axicon where $B$ is the axicon parameter [26,27]. $E(\theta, \phi )$ is the incident electric field given by

(2)$$E(\theta, \phi)=\left(\sqrt{2} \frac{\sin \theta}{\sin\alpha}\right) L^{m}_{p}\left[2\left(\beta \frac{\sin \theta}{sin\alpha}\right)^{2}\right] \exp\left[-\left(\beta \frac{\sin \theta}{sin\alpha}\right)^{2}\right]\exp(i m \phi),$$

where $L^{m}_{p}$ is the generalized Laguerre polynomial of degree $p$ and order $m$, in the case of double-ring-shaped beam $m=p=1$, and $\beta$ is the truncation parameter i.e. pupil radius to the beam waist ratio. By substituting Eq. (2) into Eq. (1), the integration over $\phi$ can be evaluated using formulae

(3)$$\begin{aligned} \int_{0}^{2\pi} cos(\varphi-\phi)\exp\left[i m \varphi+i t cos((\varphi-\phi)\right]d\varphi =\pi i^{m+1} \exp(i m \phi)\left[J_{m+1}(t)-J_{m-1}(t)\right], \end{aligned}$$

(4)$$\begin{aligned}\int_{0}^{2\pi}sin(\varphi-\phi)\exp\left[i m \varphi+i t cos(\varphi-\phi)\right]d\varphi =\pi i^{m}\exp(i m \phi)\left[J_{m+1}(t)+J_{m-1}(t)\right], \end{aligned}$$

and

(5)$$\int_{0}^{2\pi}\exp\left[i m \varphi+ i t cos(\varphi-\phi)\right]d\varphi =2\pi (i)^{m}\exp(i m \phi)J_{m}(t)$$

where $J$ is the Bessel function of the first kind of order $m$.

Therefore the electric field in the focal region is given by

(6)$$\begin{aligned} \textbf{E}(r, \varphi, z) =\left[\begin{array}{cc}E_{r}\\ E_{\varphi}\\ E_{z}\end{array}\right]= i^{m} A &\exp(i m \varphi) \int_{\sigma . \alpha}^{\alpha}\sqrt{\cos\theta} T(\theta) \left(\sqrt{2} \frac{\sin \theta}{\sin\alpha}\right) L^{m}_{p}\left[2\left(\beta \frac{\sin \theta}{sin\alpha}\right)^{2}\right]\\ & \times \exp\left[-\left(\beta \frac{\sin \theta}{sin\alpha}\right)^{2}\right] \exp\left[i k z \cos\theta\right] \sin\theta Q(r,\theta) d\theta, \end{aligned}$$

where

\begin{align*} Q(r,\theta)=\left(\begin{array}{cc} \cos\theta\left(J_{m+1}[k r \sin\theta]-J_{m-1}[k r \sin\theta]\right)\\ -i \cos\theta \left(J_{m+1}[k r \sin\theta]+J_{m-1}[k r \sin\theta]\right) \\ 2 i \sin\theta J_{m}[k r \sin\theta]\end{array}\right), \end{align*}

3. Results and discussion

Based on Eq. (6) some numerical calculations are accomplished to illustrate the intensity distribution in the focal region of the radially polarized double-ring-shaped beam focused by annular classical axicon. Herein, the intensity distribution in the focal region of annular classical axicon can be obtained by $I=| E_{r}|^{2}+|E_{\varphi }|^{2}+|E_{z}|^{2}$. For simplicity, we assume that $A=1$ and $n=1$ with $\lambda =632.8 nm$ and $B=2$. It is found that the intensity distribution is considerably affected by the other parameters $\beta$, $NA$ and $\sigma$. Figure 2 shows the intensity distribution in the focal region with $\sigma =0.75$, $\sigma =0.50$, $\sigma =0.25$ and $\sigma =0$ where $NA=0.75$ and $\beta =1.1$. As it is shown in Fig. 2(a) the intensity distribution forms uniform double focal spots that are joined with decreasing the value of $\sigma$ to form an expanded focal spot Fig. 2(c) with depth of focus (DOF) 5.76$\lambda$ and full width at half maximum (FWHM) 1.12$\lambda$ at $z=0$. By decreasing the value of $\sigma$ the focal spot becomes more tight. In the case of classical axicon without annular obstruction $\sigma =0$ tightly focused spot is obtained with DOF 4.97$\lambda$ and FWHM 1.12$\lambda$ Fig. 2(d).

Fig. 2. Intensity distribution in the r-z plane of the double-ring-shaped beam focused by annular classical axicon with $NA=0.75$ and $\beta =1.1$ for (a) $\sigma =0.75$, (b) $\sigma =0.50$, (c) $\sigma =0.25$ and (d) $\sigma =0$.

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We found that identical separated double focal spots can be obtained in the focal region by choosing appropriate values of parameters. Figure 3 illustrates that identically separated double focal spots are obtained when $\sigma =0.78$ and the other parameters are same as those in Fig. 2. The DOF of both spots is around $5.31 \lambda$ with FWHM $1.37 \lambda$. Double focal spots properties and shapes are changed remarkably for different values of NA when $\sigma$ is constant. For instance, DOF and FWHM of double focal spots are $7.58 \lambda$ and $1.137 \lambda$, respectively, when $NA=0.56$ which means that double spots are extended by decreasing the value of $NA$. On the other hand, the double focal spots are squeezed by increasing the $NA$. It can be seen that using annular axicon is useful for generating entirely separated double focal spots better than generated using cosine phase plate with high NA lens for focusing double ring shaped radially polarized multi Gaussian beam [16] and a uniaxial birefringent crystal for focusing double ring shaped radially polarized beam [11,36]. These double focal spots are joined either by decreasing the value of $\sigma$ or increasing $\beta$.

Fig. 3. Intensity distribution in the r-z plane of the double-ring-shaped beam focused by annular classical axicon with $\sigma =0.78$, NA=0.75 and $\beta =1.1$.

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Figure 4 illustrates the intensity distribution of radially polarized double-ring-shaped beam focused by annular classical axicon for $\beta =1.2$ and the other parameters are same as those in Fig. 2. It is clearly shown that the focal shape in the focal region remarkably changes depending on annular obstruction $\sigma$. Figure 4(a) shows a long flat-topped in the focal region is generated with DOF $5.33 \lambda$ when $\sigma =0.75$. By decreasing the annular obstruction value unseparated two focal spots is formed Fig. 4(b) when $\sigma =0.50$. Figures 4(c) and 4(d) show nonuniform focal spots in the focal region is formed for small values of $\sigma$ and $\sigma =0$.

Fig. 4. Intensity distribution in the r-z plane of the double-ring-shaped beam focused by annular classical axicon with $\beta =1.2$ the other parameters are same as those in Fig. 2.

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Figure 5 shows that a long flat-topped beam can be generated in the focal region of classical axicon with double-ring-shaped focused beam for $\sigma =0.65$, $NA=0.75$ and $\beta =1.3$. The DOF of the generated flat-topped is $8.80\lambda$ with FWHM$=1.43 \lambda$ at $z=0$. By increasing the value of truncation parameter $\beta$, $\sigma$ should adjust carefully in order to obtain a flat-topped beam in the focal region. For instance, with $\beta =1.5$ to $1.9$ flat-topped beam can be obtained when $\sigma =0.85$ with FWHM and DOF around $1.4 \lambda$ and $10 \lambda$, respectively. These results indicated that the annular axicons can produce a flat-topped beam with long focal depth and small FWHM better than those of cosine phase plate [16]. This flat-topped may find applications in material processing, optical data processing and inertial confinement fusion.

Fig. 5. Intensity distribution in the r-z plane of the radially polarized double-ring-shaped beam focused by annular classical axicon when $\sigma =0.65$, $NA=0.75$ and $\beta =1.3$

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Figure 6 illustrates the dependence of both DOF and the distance between double focal spots for different values of NA when $\sigma =0.75$ and $\sigma =0.80$. It can be found that the DOF and the distance between double spots are considerably affected by NA and $\sigma$. DOF and the distance between double focal spots are increasing by decreasing the value of numerical aperture. The effect of NA and $\sigma$ on DOF in the case of annular classical axicon are similar to those of high NA lens [37]. Furthermore, increase of annular obstruction cause an increase for the DOF and the distance between double spots. Long identical double spots separated by $35.12\lambda$ and $44.52\lambda$ with DOF $21.05\lambda$ and $23.20\lambda$ for $\sigma =0.75$ and $\sigma =0.80$, respectively. It is found that for high values of $\sigma \geq 0.85$ one spot is generated in the focal region of annular axicon because the inner ring is obstructed by annular obstruction.

Fig. 6. Depth of focus (dotted line) and distance between double spots (triangular line) of the radially polarized double-ring-shaped beam with classical axicon when $B=2$ and $\beta =1.1$ for $\sigma =0.75$ (red line) and $\sigma =0.80$ (blue line).

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4. Conclusion

In conclusion, based on vector Debye theory focusing properties of double-ring-shaped beam by annular classical axicon have been studied theoretically. Simulation results illustrate different focal shapes such as double focal spots and flat-topped beam are obtained in the focal region of the annular classical axicon. The DOF of double spots and the distance between them increase by decreasing the NA. The focused beam may have applications in multi-particle trapping and manipulation, multi-focal microscopy and laser printing.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

References

1. K. Rajesh, N. V. Suresh, P. Anbarasan, K. Gokulakrishnan, and G. Mahadevan, “Tight focusing of double ring shaped radially polarized beam with high na lens axicon,” Opt. Laser Tech. 43(7), 1037–1040 (2011). [CrossRef]

2. B. Tian and J. Pu, “Tight focusing of a double-ring-shaped, azimuthally polarized beam,” Opt. Lett. 36(11), 2014–2016 (2011). [CrossRef]

3. P. Suresh, C. Mariyal, T. S. Pillai, K. Rajesh, and Z. Jaroszewicz, “Study on polarization effect of azimuthally polarized lg beam in high na lens system,” Optik 124(21), 5099–5102 (2013). [CrossRef]

4. K. Lalithambigai, P. Suresh, V. Ravi, K. Prabakaran, Z. Jaroszewicz, K. Rajesh, P. Anbarasan, and T. Pillai, “Generation of sub wavelength super-long dark channel using high na lens axicon,” Opt. Lett. 37(6), 999–1001 (2012). [CrossRef]

5. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31(6), 820–822 (2006). [CrossRef]

6. Y. Zhang, “Generation of three-dimensional dark spots with a perfect light shell with a radially polarized laguerre–gaussian beam,” Appl. Opt. 49(32), 6217–6223 (2010). [CrossRef]

7. K. Lalithambigai, P. Anbarasan, and K. Rajesh, “Generation of ultra-long focal depth by tight focusing of double-ring-shaped azimuthally polarized beam,” J. Opt. 43(4), 278–283 (2014). [CrossRef]

8. N. Umamageswari, K. Rajesh, M. Udhayakumar, K. Prabakaran, and Z. Jaroszewicz, “Tight focusing properties of spirally polarized lg (1, 1)* beam with high na parabolic mirror,” Opt. Quantum Electron. 50(2), 77 (2018). [CrossRef]

9. T. Zeng and J. Ding, “Three-dimensional multiple optical cages formed by focusing double-ring shaped radially and azimuthally polarized beams,” Chin. Opt. Lett. 16(3), 031405 (2018). [CrossRef]

10. K. Lalithambigai, R. Saraswathi, P. Anbarasan, K. B. Rajesh, and Z. Jaroszewicz, “Generation of multiple focal hole segments using double-ring shaped azimuthally polarized beam,” J. At. Mol. Phys. 2013, 1–4 (2013). [CrossRef]

11. R. Murugesan, N. Pasupathy, D. ThiruArul, K. Rajesh, and V. Dhayalan, “Generating multiple focal structures with circularly polarized double ring shaped beam and axial birefringence,” J. Opt. 46(4), 403–409 (2017). [CrossRef]

12. K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut nd: Yvo4 crystal,” Opt. Lett. 31(14), 2151–2153 (2006). [CrossRef]

13. Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical brewster prism,” Opt. Lett. 30(22), 3063–3065 (2005). [CrossRef]

14. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007). [CrossRef]

15. Y. Kozawa and S. Sato, “Small focal spot formation by vector beams,” Prog. Opt. 66, 35–90 (2021). [CrossRef]

16. K. Prabakaran, K. Rajesh, V. Hariharan, A. Musthafa, M. Velu, and V. Aroulmoji, “Focal shifting of double ring shaped radially polarized multi gaussian beam,” Int. journal advanced Sci. Eng. 6(1), 1193–1199 (2019). [CrossRef]

17. Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81(2), 023831 (2010). [CrossRef]

18. Y. Zhang, T. Suyama, and B. Ding, “Longer axial trap distance and larger radial trap stiffness using a double-ring radially polarized beam,” Opt. Lett. 35(8), 1281–1283 (2010). [CrossRef]

19. Y. Zhang, X. Xu, and Y. Okuno, “Theoretical study of optical recording with a solid immersion lens illuminated by focused double-ring-shaped radially-polarized beam,” Opt. Commun. 282(23), 4481–4485 (2009). [CrossRef]

20. N. Zhang, J.-S. Ye, S.-F. Feng, X.-K. Wang, P. Han, W.-F. Sun, Y. Zhang, and X.-C. Zhang, “Generation of long-distance stably propagating bessel beams,” OSA Continuum 4(4), 1223–1233 (2021). [CrossRef]

21. B. K. Gutierrez, J. A. Davis, M. M. Sánchez-López, I. Moreno, and D. M. Cottrell, “Dynamic control of bessel beams through high-phase diffractive axicons,” OSA Continuum 3(5), 1314–1321 (2020). [CrossRef]

22. S. Khonina and S. Degtyarev, “Analysis of the formation of a longitudinally polarized optical needle by a lens and axicon under tightly focused conditions,” J. Opt. Technol. 83(4), 197–205 (2016). [CrossRef]

23. S. Khonina and S. Volotovsky, “Application axicons in a large-aperture focusing system,” Opt.Mem. Neural Networks 23(4), 201–217 (2014). [CrossRef]

24. D. A. Savelyev, “The investigation of the features of focusing vortex super-gaussian beams with a variable-height diffractive axicon,” Comput. Opt. 45(2), 214–221 (2021). [CrossRef]

25. M. Zhang, X. Liu, L. Guo, L. Liu, and Y. Cai, “Partially coherent flat-topped beam generated by an axicon,” Appl. Sci. 9(7), 1499 (2019). [CrossRef]

26. G. Wang, Y. Miao, Q. Zhan, G. Sui, and R. Zhang, “Focusing of linearly polarized lorentz-gaussian beam with helical axicon,” Optik 130, 266–272 (2017). [CrossRef]

27. A. AlKelly, I. G. Loqman, and H. T. Al-Ahsab, “Focus shaping of cylindrically polarized vortex beams by a linear axicon,” Electron. J. Univ. Aden for Basic Appl. Sci. 2(4), 145–150 (2021). [CrossRef]

28. S. Saghafi, K. Becker, F. Gori, M. Foroughipour, C. Bollwein, M. Foroughipour, K. Steiger, W. Weichert, and H.-U. Dodt, “Engineering a better light sheet in an axicon-based system using a flattened gaussian beam of low order,” J. Biophotonics 15(6), e202100342 (2022). [CrossRef]

29. S. N. Khonina, N. L. Kazanskiy, P. A. Khorin, and M. A. Butt, “Modern types of axicons: new functions and applications,” Sensors 21(19), 6690 (2021). [CrossRef]

30. I. G. H. Loqman, A. A. Alkelly, and H. T. Alahsab, “Propagation of azimuthally polarized bessel-gaussian beam through helical axicon,” in Fifth International Conference on Optics, Photonics and Lasers (OPAL 2022) 18-20 May 2022, S. Y. Yurish, ed. (2022), pp. 74–78.

31. M. Rioux, R. Tremblay, and P.-A. Belanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17(10), 1532–1536 (1978). [CrossRef]

32. O. Ren and R. Birngruber, “Axicon: a new laser beam delivery system for corneal surgery,” IEEE J. Quantum Electron. 26(12), 2305–2308 (1990). [CrossRef]

33. S. Muniz, S. Jenkins, T. Kennedy, D. Naik, and C. Raman, “Axicon lens for coherent matter waves,” Opt. Express 14(20), 8947–8957 (2006). [CrossRef]

34. I. Manek, Y. B. Ovchinnikov, and R. Grimm, “Generation of a hollow laser beam for atom trapping using an axicon,” Opt. Commun. 147(1-3), 67–70 (1998). [CrossRef]

35. S. Sato and Y. Kozawa, “Hollow vortex beams,” J. Opt. Soc. Am. A 26(1), 142–146 (2009). [CrossRef]

36. R. Murugesan, D. ThiruArul, and K. Rajesh, “Tightly focusing properties of radially polarized double ring shaped beam through a uniaxial birefringent crystal,” J. Environ. Nanotechnol. 7(2), 18–25 (2018). [CrossRef]

37. P. Suresh, K. Rajesh, T. Sivasubramonia Pillai, and Z. Jaroszewicz, “Effect of annular obstruction and numerical aperture in the focal region of high na objective lens,” Opt. Commun. 318, 137–141 (2014). [CrossRef]

Generating double focal spots by focusing a radially polarized double-ring-shaped beam with an annular classical axicon

Abstract

1. Introduction

2. Theoretical model

3. Results and discussion

4. Conclusion

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (7)

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