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Abstract
As an important parameter to determine the transmission characteristics of a Gaussian beam, the beam waist holds a huge impact in laser technology and imaging systems. Although it is necessary to clearly measure the specific value of the beam waist, the traditional measurement steps are complex and easily introduce error in the measurement process. In this work, we propose an effective method using the in-plane spin splitting (IPSS) generated by the photonic spin Hall effect (PSHE) to precisely estimate the beam waist. We establish a highly sensitive propagation model to describe the relationship between the IPSS shifts and the beam waist of an arbitrary linearly polarized light and then combine with the quantum weak measurement system to amplify the IPSS shifts. We reveal that the IPSS shifts are sensitive to the variation of beam waists when the beam is reflected near the Brewster angle. With the huge amplified IPSS shifts (maximum of 1500 microns), the variation of beam waist can be accurately detected, even by propagation amplification alone. Prospectively, our scheme may provide an effective method for accurately determining the Gaussian beam waist of arbitrary polarization.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since researchers theoretically proposed the photonic spin Hall effect (PSHE), and considered that the trajectory of light beam would be affected by the Berry geometric phase corresponding to the spin-orbit coupling [1–3], it had attracted much attention [4–15]. The spin-dependent shifts of PSHE are sensitive to changes in physical parameters, and therefore, it has been shown great potentials for precise measurement such as determining the thickness of metallic nano-films [16] and the layer numbers of graphene [17], observing the optical rotation of chiral molecules [18], detecting the Fermi energy levels of Graphene [19], and measuring the optical conductivity of atomically thin crystals [20].
Laser has revolutionized scientific and engineering technology since it was invented. In order to achieve higher energy or finer structure, many applications need to focus the laser beam such as ultrafast laser processing [21], nano-optical tweezers [22], laser surgery [23], and nonlinear optics [24,25]. Hence, it is necessary to obtain the precise value of focused Gaussian beam waist. In past years, many methods have been explored to measure the focused beam waist such as knife-edge scanning [26], pinhole method [27], slit scanning [28], and CCD-based system [29]. However, these traditional methods need to clarify the specific position of the waist, and the accuracy is not enough. In previous works, Qiu et al. revealed that in the case of horizontal incident polarization, the diffraction length has a significant effect on the transverse photonic spin splitting (TPSS) near the Brewster angle [30]. Xie et al. proposed the TPSS to measure the beam waists with only horizontally polarized light [31]. However, the output light is generally an arbitrary linearly polarized light in most laser applications. For such a common case, how to precisely measure their beam waist remains elusive and can be quite challenging. Besides, in addition to the TPSS generated by PSHE, the in-plane spin splitting (IPSS) induced by the gradient of Fresnel coefficient can also be used as a probe in a weak measurement system [32]. Moreover, the IPSS can reach the order of microns, much larger than TPSS. When the incident angle is near the Brewster angle, the IPSS can be significantly enhanced, and it is extremely sensitive to the incident polarization [33]. Therefore, it is reasonable that the IPSS can be used to precisely estimate the Gaussian beam waist with arbitrary linearly polarization. Although the potentiality of IPSS as a probe is huge, a clear relationship between beam waist and the IPSS has never been explored before.
In this paper, we propose an effective method to improve the measurement accuracy of beam waist with the IPSS. Firstly, we theoretically establish a highly sensitive propagation model to describe the relationship between the IPSS and beam waists when an arbitrary linearly polarized beam is reflected from an air-glass interface. By analyzing the expression of the IPSS in this model, we can find that the IPSS is extremely sensitive to the variation of beam waist near the Brewster angle, as well as the polarization states of incident light. Then we combine with a quantum weak measurement system based on the propagation model, the maximum amplified shifts can reach up to about 1500 microns. What’s more, when the polarization angle changing within $0\sim2^\circ$, the measurement accuracy of beam waist can be improved to a certain extent accurately detected by propagation amplification only. The experimental results are match with the theoretical prediction well.
2. Experiment setup
The experiment setup is shown in Fig. 1(b) [3,34,35]. A linearly polarized Gaussian beam (632.8 nm) is generated by a He-Ne laser (Thorlabs HNL210L), after that, a half-wave plate is used to adjust the beam intensity. The light beam pass through a lens L1 and a polarizer P1 to obtain a preselected state, and then reflected at the surface of a BK7 glass. The reflected beam splits into left-handed and right-handed components, and generate displacement in both directions perpendicular and parallel to the incident plane, which are the TPSS and the IPSS [32,36,37] [shown in Fig. 1(a)]. The central wave vector of reflected light is rotated by an angle ${{\varphi }_P}$ [shown in Fig. 1(c)], here ${{\varphi }_P} = {\tan ^{ - 1}}\frac{{{r_{s\theta i}}}}{{{r_{p\theta i}}}}\tan {{\varphi }_i}$, ${{\varphi }_i}$ is the polarization angle, ${r_{p\theta i}}$ and ${r_{s\theta i}}$ are the Fresnel reflection coefficients for s and p plane waves at the incident angle of ${\theta _i}$. The polarizer P2 is rotated by an angle $\mathrm{\Delta }$ to post-select the final state. Finally, the beam is collimated through a lens L2 and recorded by a CCD camera (Coherent Laser-Cam HR).
Fig. 1. (a) Schematic diagram of the IPSS at an air-glass interface. Different beam waists of the light beam acquire different IPSS: $\delta _1^{X - }$ and $\delta _1^{X + }$ for ${w_1}$, $\delta _2^{X - }$ and $\delta _2^{X + }$ for ${w_2}$. The symbols “$+$” and “$-$” represent the left- and right-circular polarization components, respectively. $\delta _{}^{X + }$ and $\delta _{}^{X - }$ indicate the IPSS for left- and right-circular polarization components, respectively. (b) Experimental setup: the light source is a He-Ne laser at 632.8 nm; HWP, half-wave plate for adjusting the intensity of the Gaussian light beam; P1 and P2 are polarizers. P1 selects the incident polarization angle, P2 adjusts the post-selection angle; BK7, prism with refractive index n=1.515 (at 632.8 nm) for inducing the spin displacement; L1 and L2, short-focus and long-focus lens; CCD, charge-coupled device; (c) The schema of post-selection. The post-selected angle is $\mathrm{\Delta }$. The central wave vector of reflected light is rotated by an angle ${{\varphi }_P}$.
In our experiment, we chose the Brewster angle as the incident angle and fix the propagation distance ${z_r} = 250mm$ the focused beam waist w can be calculated by the following formula [30,31]:
where ${F_1}$ is the focal length of L1,$R = \frac{{{k_0}w_0^2}}{2}$ is the initial Rayleigh distance, ${w_0} = 0.35mm$ is the initial He-Ne laser beam waist, and the transmission distance $l = 200mm$. In our experiment, F1 is chosen as $30mm$, $35mm$, and $40mm$, and ${F_2}$ is selected as $170mm$, $165mm$, $160mm$. Thus, the corresponding beam waists of the focused beam are $16.63\mu m$, $19.44\mu m$ and $22.26\mu m$, respectively. By measuring the IPSS shifts, we can indirectly determine the focused beam waist with arbitrary polarization.3. Results and discussion
Firstly, we explore the relationship between the IPSS and beam waists by considering different polarized angles and incidence angles, as shown in Fig. 2 (the detail calculation process of IPSS with different beam waist can be found in the Supplement 1). The wavelength of exciting light is chosen at 632.8 nm, the reflecting medium was BK7 glass with refractive index of 1.515 at 632.8 nm (corresponding Brewster angle ${\theta _B} \approx 56.57{^\circ }$), and the beam waists are chosen as $w = 14\mu m$, $18\mu m$, $22\mu m$, and $26\mu m$ respectively. From Fig. 2, with increasing the incident angles ${\theta _i}$ from $55{^\circ }$ to $58{^\circ }$, the IPSS shifts firstly increase until reach the maximum at the Brewster angle ${\theta _B}$, and then decrease gradually. When the incident angle is at the Brewster angle ${\theta _B}$, the IPSS shifts vary significantly as the polarized angle changing within $0 \sim 1.5{^\circ }$. Besides, noting the huge difference of IPSS shifts at different beam waists with the same polarized angle and incidence angle, we can conclude that the IPSS shifts are extremely sensitive to the variation of beam waists at the Brewster angle ${\theta _B}$. Figure 2 also shows that when the incident angle is slightly away from the Brewster angle ${\theta _B}$, the IPSS shifts vary as the polarized angle changing in a certain range. When the incident angle is far away from the ${\theta _B}$, the sensitivity will be weakened.
Fig. 2. The IPSS shifts with different polarized angles and incidence angles. The refractive index of glass is chosen as n=1.515, and the corresponding Brewster angle is 56.57°. The beam waists are selected as $w = 14\mu m$, $18\mu m$, $22\mu m$, and $26\mu m$ respectively.
The analytical derivation for the IPSS for an arbitrary linearly polarized light can explain this phenomenon. In general, the influence of the beam waist w on the IPSS can almost be offset. However, ${r_{p\theta i}} \to 0$ when the incident angle is approach to the Brewster angle, the term $\chi = \frac{{\partial {r_p}}}{{\partial {\theta _i}}}$ is dominant (see Supplement 1), resulting that the IPSS shifts are larger and more sensitive nearby the Brewster angle ${\theta _B}$. Based on these cases, we can use the large IPSS shifts near the Brewster angle ${\theta _B}$ to detect the variation of an arbitrary linearly polarized beam waist.
In order to achieve higher measurement accuracy, a quantum weak measurement system is used to amplifying the IPSS shifts (the detail calculation process of amplified IPSS can be found in Supplement 1). The accuracy of the measurement depends on the sensitivity of the amplified shift $\frac{{\partial A_w^{\bmod }{\delta ^{X \pm }}}}{{\partial w}}$ ($A_w^{\bmod }{\delta ^{X \pm }}$ is the amplified shift) and the experimental measurement accuracy [31]. To investigate $\frac{{\partial A_w^{\bmod }{\delta ^{X \pm }}}}{{\partial w}}$ with different incident angle and post-selected angle, we fix the incident angle as the Brewster angle and the free propagation length ${z_r} = 250mm$, as shown in Fig. 3. It is found that the sensitivity factor is huge when polarization angle is more than $5{^\circ }$, and the optimal sensitivity of the amplified shift is about 81.6. In addition, the measurement error of our experiment is less than $3\mu m$ (the error caused by the misalignment of the experimental setup is not considered here). Therefore, the measurement accuracy of our method can reach about $0.03\mu m$ via weak measurement. Compared with the measurement of beam waist with TPSS [31], the measurement accuracy of beam waist with IPSS is improved.
Fig. 3. The sensitivity factor (the sensitivity of the amplified shifts to the beam waist) with different incident angle and post-selected angle. Here the incident angle is chosen as the Brewster angle ${\theta _B}$ and the beam waist is selected as $20\mu m$.
Figure 4 shows the amplified shift changing with polarization angle for different beam waists. Here, the post-selected angle ${\Delta = 2^\circ }$ (the weak value and propagation amplification mechanism work together). The difference from Fig. 2 is that when the beam waist w increases from $14\mu m$ to $26\mu m$, the amplified shift decreases. This phenomenon is attributed to the propagation amplification factor $F = \frac{{\lambda {z_r}}}{{\pi {w^2}}}$, which leads to $\frac{{\partial A_w^{\bmod }}}{{\partial w}}\textrm{ < }0$. Meanwhile, even the incident angles are far away from ${\theta _B}$, the IPSS shifts are still sensitive to the variation of beam waists after amplifying [shown in Figs. 4(b) and 4(c)]. The similarity to Fig. 2 is that Fig. 4(a) shows the amplified shifts corresponding to different beam waists reach the maximum when the incident angle is at Brewster angle. In this case, Figs. 4(d) and 4(e) show that the experimental results are in good agreement with our theory when the polarization angle in a range of $5{^\circ } \sim 25{^\circ }$(dots, circles, and triangles denote experimental data and solid curves, dash curves, and point curves represent theoretical data). Compared with previous work which used the transverse shifts [31], the experimental data are closer to the theoretical curve, it further confirms that our scheme has a higher precision.
Fig. 4. The amplified IPSS shifts with different incident and polarization angles under weak value and propagation amplification mechanism together (${\Delta = 2^\circ }$). Here the propagation distance of the light beam is selected as ${z_r} = 250mm$. (a) represents the amplified shift when the incident angle is chosen as the Brewster angle ${\theta _B}$; (b) and (c) respectively show the amplified shift when the incident angle is less than and greater than the Brewster angle ${\theta _B}$; (d) shows theoretical and experimental results of the amplified shifts measured by the CCD when the incident angle is selected as the Brewster angle ${\theta _B}$. The solid curves, dash curves, and point curves are theoretical result, and the dots, circles, and triangles represent the experimental data of three different beam waists: $16.63\mu m$, $19.44\mu m$ and $22.26\mu m$, respectively; (e) shows the theoretical and experimental intensity profile of the reflected light beam after L2.
It can be seen from Fig. 3, when the polarization angle is $0 \sim 2{^\circ }$, the amplified factor is still small. In this case, the weak value amplification factor $A_w^{}$ is relatively small. To obtain larger shift, we try to reduce the weak value amplification and only use propagation amplification with polarization angle in the range of $0 \sim 2{^\circ }$. When the post-selected angle is ${\Delta = 45^\circ }$, the weak-value amplification does not work, and then the final amplification factor is constituted by the propagation amplification, results shown in Fig. 5. When the incident angle is the Brewster angle ${\theta _B}$, it is found that the amplified IPSS shift can still reach up to 1500 microns and very sensitive to polarization angle ${\; }$ in range $0 \sim 2{^\circ }$. It should be noted that, in this range of polarization angle, the amplified shifts are not sensitive to the beam waists in Fig. 4(a). Further, we also experimentally demonstrate that the beam waists can still be precisely measured only by propagation amplification with small incident polarized angle, as shown in Figs. 5(b) and 5(c). The experimental results are as good as we expected, which proves that we can precisely measure the beam waist of an arbitrary linearly polarized light.
Fig. 5. The amplified IPSS shifts with different incident and polarization angles under the case of only propagation amplification mechanism (${\Delta = 45^\circ }$). Here the incident angle is chosen as the Brewster angle ${\theta _B}$. (a) represents the amplified shift as the polarized angle changing within $0 \sim 2{^\circ }$; (b) shows theoretical and experimental results of the amplified shifts measured by the CCD. The solid curves, dash curves, and point curves are theoretical results, and the dots, circles, and triangles represent experimental data of three different beam waists: $16.63\mu m$, $19.44\mu m$ and $22.26\mu m$, respectively; (c) is the theoretical and experimental intensity profile of the reflected light beam after L2.
4. Conclusion
In summary, we have investigated the relationship between Gaussian beam waist and the in-plane spin splitting (IPSS) when an arbitrary linearly polarized beam is reflected from an air-glass interface. Considering the large IPSS shifts when the incident angle is the Brewster angle, we have experimentally measured different beam waists through an IPSS-based system. The experimental data agree well with our theoretical calculation. Compared with the traditional method, even the transverse photonic spin splitting (TPSS) of the photonic spin Hall effect (PSHE), due to the high sensitivity of the IPSS to beam waists and the huge amplification factor, the IPSS-based indirect method using weak measurement has much higher sensitivity and accuracy. It is believed that our scheme can provide an effective method to precisely measure an arbitrary linearly polarized Gaussian beam waist.
Funding
National Natural Science Foundation of China (11604095); Shenzhen Government’s Plan of Science and Technology (JCYJ20180305124927623, JCYJ20190808150205481); Science Foundation of Civil Aviation Flight University of China (JG2019-19).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
References
1. K. Y. Bliokh and Y. P. Bliokh, “Topological spin transport of photons: the optical Magnus effect and Berry phase,” Phys. Rev. A 333(3-4), 181–186 (2004). [CrossRef]
2. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93(8), 083901 (2004). [CrossRef]
3. O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319(5864), 787–790 (2008). [CrossRef]
4. A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009). [CrossRef]
5. Y. Qin, Y. Li, H. He, and Q. Gong, “Measurement of spin Hall effect of reflected light,” Opt. Lett. 34(17), 2551–2553 (2009). [CrossRef]
6. M. Merano, N. Hermosa, J. P. Woerdman, and A. Aiello, “How orbital angular momentum affects beam shifts in optical reflection,” Phys. Rev. A 82(2), 023817 (2010). [CrossRef]
7. N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett. 36(16), 3200–3202 (2011). [CrossRef]
8. Y. Gorodetski, K. Y. Bliokh, B. Stein, C. Genet, N. Shitrit, V. Kleiner, E. Hasman, and T. W. Ebbesen, “Weak Measurements of Light Chirality with a Plasmonic Slit,” Phys. Rev. Lett. 109(1), 013901 (2012). [CrossRef]
9. N. Shitrit, I. Yulevich, E. Maguid, D. Ozeri, D. Veksler, V. Kleiner, and E. Hasman, “Spin-optical metamaterial route to spin-controlled photonics,” Science 340(6133), 724–726 (2013). [CrossRef]
10. X. Yin, Z. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339(6126), 1405–1407 (2013). [CrossRef]
11. P. V. Kapitanova, P. Ginzburg, F. J. Rodríguez-Fortuño, D. S. Filonov, P. M. Voroshilov, P. A. Belov, A. N. Poddubny, Y. S. Kivshar, G. A. Wurtz, and A. V. Zayats, “Photonic spin Hall effect in hyperbolic metamaterials for polarization-controlled routing of subwavelength modes,” Nat. Commun. 5(1), 3226 (2014). [CrossRef]
12. W. J. M. Kort-Kamp, “Topological phase transitions in the photonic spin Hall effect,” Phys. Rev. Lett. 119(14), 147401 (2017). [CrossRef]
13. Y. Xiang, X. Jiang, Q. You, J. Guo, and X. Dai, “Enhanced spin Hall effect of reflected light with guided-wave surface plasmon resonance,” Photonics Res. 5(5), 467–472 (2017). [CrossRef]
14. X. Ling, X. Zhou, K. Huang, Y. Liu, C. Qiu, H. Luo, and S. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80(6), 066401 (2017). [CrossRef]
15. O. Takayama, J. Sukham, R. Malureanu, A. V. Lavrinenko, and G. Puentes, “Photonic spin Hall effect in hyperbolic metamaterials at visible wavelengths,” Opt. Lett. 43(19), 4602–4605 (2018). [CrossRef]
16. X. Zhou, Z. Xiao, H. Luo, and S. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85(4), 043809 (2012). [CrossRef]
17. X. Zhou, X. Ling, H. Luo, and S. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101(25), 251602 (2012). [CrossRef]
18. X. Qiu, L. Xie, X. Liu, L. Luo, Z. Zhang, and J. Du, “Estimation of optical rotation of chiral molecules with weak measurements,” Opt. Lett. 41(17), 4032–4035 (2016). [CrossRef]
19. Y. Wu, L. Sheng, L. Xie, S. Li, P. Nie, Y. Chen, X. Zhou, and X. Ling, “Actively manipulating asymmetric photonic spin Hall effect with graphene,” Carbon 166, 396–404 (2020). [CrossRef]
20. S. Chen, X. Ling, W. Shu, H. Luo, and S. Wen, “Precision measurement of the optical conductivity of atomically thin crystals via the photonic spin hall effect,” Phys. Rev. Appl. 13(1), 014057 (2020). [CrossRef]
21. M. Malinauskas, A. Zukauskas, S. Hasegawa, Y. Hayasaki, V. Mizeikis, R. Buividas, and S. Juodkazis, “Ultrafast laser processing of materials: from science to industry,” Light: Sci. Appl. 5(8), e16133 (2016). [CrossRef]
22. M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics 5(6), 349–356 (2011). [CrossRef]
23. C. A. Nanni and T. S. Alster, “Complications of Carbon Dioxide Laser Resurfacing. An Evaluation of 500 Patients,” Dermatol. Surg. 24(3), 315–320 (1998). [CrossRef]
24. E. Goulielmakis, M. Schultze, M. Hofstetter, V. S. Yakovlev, J. Gagnon, M. Uiberacker, A. L. Aquila, E. M. Gullikson, D. T. Attwood, R. Kienberger, F. Krausz, and U. Kleineberg, “Single-cycle nonlinear optics,” Science 320(5883), 1614–1617 (2008). [CrossRef]
25. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997). [CrossRef]
26. A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focused light beams,” Appl. Opt. 16(7), 1971–1974 (1977). [CrossRef]
27. P. J. Shayler, “Laser beam distribution in the focal region,” Appl. Opt. 17(17), 2673–2674 (1978). [CrossRef]
28. P. B. Chapple, “Beam waist and M2 measurement using a finite slit,” Opt. Eng. 33(7), 2461–2466 (1994). [CrossRef]
29. J. A. Ruff and A. E. Siegman, “Single-pulse laser beam quality measurements using a CCD camera system,” Appl. Opt. 31(24), 4907–4909 (1992). [CrossRef]
30. X. Qiu, L. Xie, J. Qiu, Z. Zhang, J. Du, and F. Gao, “Diffraction-dependent spin splitting in spin Hall effect of light on reflection,” Opt. Express 23(15), 18823–18831 (2015). [CrossRef]
31. L. Xie, X. Qiu, J. Qiu, Z. Zhang, J. Du, and F. Gao, “Determination of the focused beam waist of lasers with weak measurements,” Chin. Opt. Lett. 13(11), 112401–112404 (2015). [CrossRef]
32. Y. Qin, Y. Li, X. Feng, Y. Xiao, H. Yang, and Q. Gong, “Observation of the in-plane spin separation of light,” Opt. Express 19(10), 9636–9645 (2011). [CrossRef]
33. X. Qiu, Z. Zhang, L. Xie, J. Qiu, F. Gao, and J. Du, “Incident-polarization-sensitive and large in-plane-photonic-spin-splitting at the Brewster angle,” Opt. Lett. 40(6), 1018–1021 (2015). [CrossRef]
34. Y. Aharonov, D. Z. Albert, and L. Vaidman, “How the result of a measurement of a component of the spin of a spin-1/2 particle can turn out to be 100,” Phys. Rev. Lett. 60(14), 1351–1354 (1988). [CrossRef]
35. Y. Qin, Y. Li, X. Feng, Z. Liu, H. He, Y. Xiao, and Q. Gong, “Spin Hall effect of reflected light at the air-uniaxial crystal interface,” Opt. Express 18(16), 16832–16839 (2010). [CrossRef]
36. K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96(7), 073903 (2006). [CrossRef]
37. J. Ren, B. Wang, Y. Xiao, Q. Gong, and Y. Li, “Direct observation of a resolvable spin separation in the spin Hall effect of light at an air-glass interface,” Appl. Phys. Lett. 107(11), 111105 (2015). [CrossRef]
