Generalized Newton&#x2019;s rings with vortex beams

Jiadian Yan; Jun Yao; Yidong Liu; Yidong Liu; Yuanjie Yang

doi:10.1364/OE.476043

1. Introduction

Optical vortices have been widely studied since Allen et al. shown the connection between the spiral phase singularity of the Laguerre-Gaussian (LG) beam and the quantized orbital angular momentum (OAM) [1]. Light beam carrying OAM is associated with an azimuthal phase structure $\exp (il\varphi )$, with $\varphi$ and $l(l\in \mathbb {Z})$ the angular coordinate and azimuthal index, respectively. The azimuthal index is also called as the topological charge (TC) of the OAM beam, which presents an OAM of $l\hbar$ per photon ($\hbar =h/(2\pi )$ is reduced Planck’s constant) [1]. Due to the phase singularity at the center, the intensity profile of a vortex beam is always doughnut-shaped. OAM beams have been widely used in a variety of interesting applications [2–4], such as generation of Pancharatnam–Berry phase [5], optical microscopy [6–8], optical micromanipulation [9–12], quantum information [13,14], free-space and fiber optical communication [15–19], and etc.

To distinguish different OAM states is one of the most fundamental steps towards the utilization of OAM beams. There are some existing methods to achieve this goal, such as interference with a plane wave or a spherical wave [20,21], bifurcation fringes of its self-interference [20], Cartesian to log-polar coordinate transformation [22,23], and diffraction patterns of various apertures or slits [24–29]. Meanwhile, several classic interference experiments adopting OAM beams have been reported [30,31]. It is well known that, interference is a key evidence of the light’s wave nature and the interference pattern presents rich information of intensity and phase which is important in high precision measurement. Sztul and Alfano revealed the interference characteristics of OAM beams in Young’s double-slit interference experiment and pointed out the twist fringes are closely related to the topological charges of incident OAM beams [30].

Besides the double-slit interference, the Newton’s rings experiment is also one of the classic examples in optics. The Newton’s rings pattern shows equal thickness interference fringes and is usually observed in the interferometry with a spherical surface [32]. Due to the rotationally symmetric spherical surface and rotationally symmetric illumination such as the planar illumination and the Gaussian beams, the rotationally symmetric optical path length difference is induced and therefore the Newton’s rings pattern is consisted of a set of concentric circular fringes. The position and the curvature radius of a spherical surface can be measured by analyzing these Newton’s rings patterns. Although the concentric Newton’s rings pattern is similar to the doughnut-shaped intensity pattern of a vortex beam, the underlying physical mechanisms are totally different. What will happen when a vortex beam encounters the spherical surface is not unveiled till now and arises our curiosity.

In this paper we put forward the generalized Newton’s rings interference experiments with optical LG beams as illumination. We find that the generalized Newton’s rings pattern is consisted of the spiral pattern and the fork-shaped pattern which are significantly related to the topological charge of the incident optical vortex beam. This new phenomenon may lead to a better understanding and interpretation of the nature of the OAM of vortex beams. Due to its simplicity, the generalized Newton’s rings can be used for detecting and measuring the topological charges of vortex beams. While, it is noted that the generalized Newton’s rings patterns will become more complicated and difficult to recognized for vortex beams with high topological charges.

2. Methods and discussions

2.1 Theoretical discussions

Without loss of generality, LG beam was chosen to theoretical and experimental demonstrate the generalized Newton’s rings patterns in this work, for it is one of the most typical vortex beams and it has a complex field amplitude shown as followed [12]:

(1)$$\begin{aligned}LG_p^l(\rho,\varphi,z)=&\frac{\omega_0}{\omega(z)}\sqrt{\frac{2p!}{\pi(|l|+p)!}}\bigg(\frac{\sqrt{2}\rho}{\omega(z)}\bigg)^{|l|}L_p^{|l|}\bigg[2\bigg(\frac{\rho}{\omega(z)}\bigg)^2\bigg] \\ &\times\exp[i(2p+|l|+1)\xi(z)] \\ &\times\exp\bigg[-\bigg(\frac{\rho}{\omega(z)}\bigg)^2\bigg]\exp\bigg[-\frac{ik\rho^2}{2R(z)}\bigg]\exp(il\varphi) \end{aligned}$$

where $\omega _0$ is the waist size of the beam, $L_p^{|l|}(2\rho ^2/\omega (z)^2)$ is the associated Laguerre polynomial, $l$ is the azimuthal index, while the $p$ is the radial index of the LG beam, a set of parameters $\omega (z)$, $\xi (z)$, $R(z)$ are defined as: $\omega (z)=\omega _0\sqrt {1+(z/z_R)^2}$, $\xi (z)=\arctan (z/z_R)$, $R(z)=z(1+(z_R/z)^2$, while the $z_R$ here is the Rayleigh range [12,33]. In most cases, the field amplitude of LG beam can always be separated into radial part and azimuthal part as followed:

(2)$$LG_p^l(\rho,\varphi,z)=A_p^l(\rho,z)\exp(il\varphi)$$

for the sake of simplicity.

In a standard Newton’s rings interference experiment, the spherical surface is usually induced by a plano-convex lens (PCL). With plane wave illumination, the interference pattern on the screen is introduced by the reflected beams from each surface of the PCL and the optical path difference between these two beams is resulted from the geometry of the spherical surface. When a plane wave illuminates vertically, the two reflected beams propagate along the same axis with a spherical phase difference caused by the PCL and there is no more additional phase difference. The interference pattern will therefore be a series of concentric circular fringes formed around the axis of the two reflected beams and the PCL. The derivations of the intensity distribution of the standard Newton’s rings pattern are given as:

(3)$$I(\rho,\varphi)=\bigg|E_0\exp\bigg(ik\frac{\lambda}{2}\bigg)+E_0\exp\bigg(ik\frac{\rho^2}{R}\bigg)\bigg|^2\propto\cos^2\bigg(\frac{k\rho^2}{2R}-\frac{\pi}{2}\bigg)$$

Particularly, Eq. (3) shows the intensity distribution of standard Newton’s rings pattern is azimuthally isotropic. And furthermore, it indicates that the classic Newton’s rings interference experiment with vertical plane wave illumination contains rotational symmetry. While the rotational symmetry is one of the intrinsic properties of the optical vortex beams. Therefore, it is predictable that the rotational symmetry would be conserved if the vertical illumination carried OAM, and the interference pattern would be the similar concentric circular fringes as the case with plane wave illumination.

In contrast, the same experiment setup but using oblique illumination will break this rotational symmetry and will exhibit generalized interference patterns with extra details due to this symmetry broken. Two reflected beams are separated due to the refraction caused by the PCL when incident beam obliquely illuminates, especially the two reflected beams propagated in parallel but separated spatially when the incident beam obliquely illuminates at the center of the spherical side of PCL. Therefore, in Newton’s rings experiment with oblique illumination, the interference patterns of these two reflected beams are not only resulted from the spherical phase induced by the PCL but also additionally resulted from the spatial dislocation caused by the refraction in the PCL due to the oblique incidence.

Based on the analysis by geometrical optics, we can find out the spatial dislocation of the two reflected beams resulted from the PCL with an oblique illumination. As shown in Fig. 1(a), when a LG beam passes through the PCL and reflects at the center of the spherical side of the lens, the reflected light and the incident light inside the PCL are symmetric about the axis of the lens due to the law of reflection and the geometry of sphere. Therefore, reflected beam $E_1$ comes from the spherical side of the lens will exit along the propagation direction which is mirrored to the incident beam about the axis of the PCL. While the reflected beam $E_2$ comes from the flatten side of the lens will also emit mirroredly to the incident beam about the normal, which is parallel to the axis of the PCL, these two reflected beams will propagate in parallel as shown in Fig. 1(a). Moreover, the spatial dislocation between these two beams can be derived from the geometric schematic shown in Fig. 1(a) as followed:

(4)$$\Delta=2d\tan\bigg(\arcsin\bigg(\frac{\sin\theta}{n}\bigg)\bigg)\cos\theta$$

where $d$ is the thickness of the lens, $\theta$ is the incident angle and $n$ is the relative refractive index. If the incident angle $\theta$ is small enough, the spatial dislocation can be approximated to:

(5)$$\Delta\approx \frac{2\theta d}{n}$$

Fig. 1. (a) Spatial dislocation $\Delta$ between the two reflect beams $E_1$, $E_2$ induced by the refraction of the illumination through the PCL; (b) cross-section schematic of the two reflect beams and the coordinate setup at the observing screen plane.

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Therefore, if we set a coordinate at the observing screen and set the origin point at the center of beam $E_2$ the one reflected by the flatten side of the PCL as shown in Fig. 1(b), the intensity distribution of the generalized Newton’s rings introduced by LG beam illumination carrying topological charge $l$ is given as:

(6)$$\begin{aligned}I(x,y)\propto&\bigg|LG_{p,-l}(x,y)\exp\bigg(ik\frac{\lambda}{2}\bigg)+LG_{p,-l}(x,y+\Delta)\exp\bigg(ik\frac{x^2+(y+\Delta)^2}{R^2}\bigg)\bigg|^2 \\ \propto&\bigg|A_{p,-l}(x,y)\exp({-}il\varphi)\exp\bigg(ik\frac{\lambda}{2}\bigg) \\ &+A_{p,-l}(x,y+\Delta)\exp({-}il\varphi')\exp\bigg(ik\frac{x^2+(y+\Delta)^2}{R^2}\bigg)\bigg|^2 \end{aligned}$$

where $\Delta$ is the spatial dislocation between the two reflect beams as Eq. (4)(5), and the change of the topological charge from $l$ of illumination to $-l$ of reflection is the fact that reflection of vortex beams will invert their topological charges as discussed by previous researches [34–36].

From Eq. (4–6), it is obvious that the intensity distribution of generalized Newton’s rings generated by LG beam is connected with the topological charge of the illumination when its incident angle $\theta$ is non-zero. The interference patterns calculated from Eq. (6) can be graphically expressed as Figs. 2(a), (c) and (e), where the topological charges of LG beams are modeling from $l=+1$ to $l=+3$. In this study, we ignore the radial index $p$ of LG beams, since for the cases $p\neq 0$, the interference patterns will be very complicated and it is not easy to find the regularity any longer. The numerical results shown in Fig. 2 indicate that the generalized Newton’s rings are the combinations of the spiral patterns and the fork-shaped patterns instead of the concentric circular fringes in classic Newton’s rings experiment. The fork-shaped dislocation clings to the arm of the spiral pattern, and the numbers of the spiral arms and the order of fork-shaped patterns are equal to the topological charge of the corresponding incident LG beam. While the topological charges of the illuminations invert into $l=-1, \dots, -3$, as shown in Figs. 2(b), (d) and (f), the chirality of the spiral arms and the direction of the fork patterns invert simultaneously.

Fig. 2. Numerical calculations of the generalized Newton’s rings obtained from vortex beams with different topological charges. (a) $l=+1$, (b) $l=-1$, (c) $l=+2$, (d) $l=-2$, (e) $l=+3$, (f) $l=-3$. The number of spiral arms in the interference pattern indicates the topological charge $l$, and the fork-shaped dislocation (marked by the dashed green circles) indicates the topological charge $l$ as well, namely, one interference fringe splits into $|l|+1$ fringes.

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2.2 Experimental method

With aforementioned theoretical analyses, the experimental setup to investigate the generalized Newton’s rings patterns generated by LG illuminations is designed as shown in Fig. 3. In our experiment we used a PCL, where the difference between the curvature on each side of the lens can introduce Newton’s rings in its reflected beams. The thickness $d$ of the PCL is $3.5 mm$ and its relative refractive index $n$ is $1.517$. A He-Ne laser ($\lambda = 632.8 nm$) beam properly expanded and collimated by a couple of lenses, i.e. Lens 1 and Lens 2, illuminates a reflective spatial light modulator (SLM) parallel with the PCL. Spiral phases and proper blazed phase are displayed on SLM to generate the illuminating LG beams with topological charges $l=1,2,\dots,4$ and incident angle $\theta$ about $36^{\circ }$. Since the SLM only modulates specifically polarized light, a polarizer is placed before the SLM to match its polarization. The illuminating LG beam is then selected by an aperture slot (AS 2) placing in front of the PCL, allowing only one mode of LG beam to pass through. Due to the LG beams carrying spherical phases naturally, an additional lens (Lens 3) is chosen to reshape the wavefronts and relocate the waist positions of the illuminating LG beams in order to optimize the interference results. The two reflected beams and their interference patterns from the PCL are recorded and measured by a CCD camera.

Fig. 3. Schematic of the experimental setup. AS, aperture slot; PCL, plano-convex lens; SLM, reflective spatial light modulator; CCD, charge-coupled device.

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3. Experimental results

Figures 4(a)-(d) show the generalized Newton’s rings interference fringes experimentally for $l =+1,+2,\dots,+4$. As we predicted in theory, the interference pattern in Fig. 4(a) is consisted of spiral pattern with one spiral arm at right and fold-1 fork-shaped pattern at left of the figure, which is corresponding to illumination LG beam with topological charge $l=+1$. Figures 4(b)-(d) show the fringe patterns generated by LG beams with high order topological charges, i.e., $l=+2,\dots,+4$. These interference patterns show the spiral patterns, which carry the same orders to the corresponding topological charges, located at right of the generalized Newton’s rings patterns.

Fig. 4. Experimental generalized Newton’s rings interference patterns obtained from (a) the $l=+1$ beam, (b) $l=+2$ beam, (c) $l=+3$ beam, (d) $l=+4$ beam.

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However, instead of the fork-shaped patterns with the same orders to incident LG beams shown in numerical results, the experimental results show that the fork-shaped patterns of generalized Newton’s rings generated by high-order LG beams, i.e., $l>1$, are split and separated into the corresponding numbers of fold-1 fork-shaped patterns. For example, with the LG beam carried $l=+3$ illuminating, the generalized Newton’s rings shown in Fig. 4(c) is consisted of a triple of fold-1 fork-shaped patterns instead of an order-3 fork-shaped pattern. The separation of the high-order fork-shaped dislocation in the generalized Newton’s rings in our experiment may be explained by the separating of the interference patterns due to the instability of high-order fork-shaped interference patterns discussed by Ricci et al. [37], or maybe it is because the high-order incident vortex beam doesn’t actually exist of perfect mode with high order topological charge.

4. Conclusions

As the example of interference fringes of equal thickness, the Newton’s rings pattern is one of the most classic interference patterns due to its concentric rings pattern induced by a spherical surface as the plane wave illuminating. Our demonstration shows that the generalized Newton’s rings pattern can be generated from the classic Newton’s rings interference experiment setup with oblique Laguerre-Gaussian beam illumination, and the generalized Newton’s rings are interference patterns consisted of spiral patterns and fork-shaped patterns instead of the simple concentric rings. Moreover, our further theoretical analyses and experiments show that both the number of spiral arms and the order of fork-shaped dislocations are equal to the topological charge of the incident LG beam. Considering these novel interference patterns are induced by the helical phase structures of the incident LG beams. Therefore, the generalized Newton’s rings experiment we revealed may be a new approach to determine the topological charges of optical vortex beams. Moreover, the generalized Newton’s rings of the fractional vortex beams [38–41] is also an interesting topic and should be studied in the future. Although this is not the first candidate to measure a given optical vortex beam, this generalized Newton’s rings interference experiment provides an interesting and novel attempt to make a connection between the classic Newton’s rings interference experiment and the novel optical vortex beams.

Funding

National Natural Science Foundation of China (11874102, 12174047); Sichuan Province Science and Technology Support Program (2020JDRC0006); Fundamental Research Funds for the Central Universities (ZYGX2019J102).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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.

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Generalized Newton’s rings with vortex beams

Abstract

1. Introduction

2. Methods and discussions

2.1 Theoretical discussions

2.2 Experimental method

3. Experimental results

4. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (4)

Equations (6)

Optics Express