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Abstract
Perfect vortex beams can only propagate stably with integer topological charges. Thus, creating perfect fractional vortex beams capable of stable propagation in free space, as perfect integer vortex beams, is crucial. This study proposed perfect vortex beams carrying fractional topological charge of l + 0.5, which are special solutions of the wave equation, and can maintain stable propagation with physical laws same as integer topological charge. Perfect fractional vortex beams were created in free space, which can break the cognition of traditional fractional perfect vortex beams and promote the development of scientific fields such as optical communication, quantum sensing, and optical imaging.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The physical nature of light beam propagation with optical vortices $\textrm{exp}({il\varphi } )$ obeys the well-known paraxial wave equation, wherein the topological charge (TC) l is an integer and exhibits unlimited states. However, the states are generally studied based on theoretical calculations. In practical applications the TC is limited, because a larger l value results in larger beam size and lower energy efficiency. For instance, the simultaneous coupling of multiple vortex beams (VBs) into a single air-core fiber used for multiplexed communications is challenging [1]. To solve this issue, light beams have been proposed to possess unchanged annular intensity profiles and beam size independent of TCs, namely, perfect vortex beams [2]. These have ushered in further research on fundamental and applied research in optics. Consequently, based on the superior characteristics of the perfect vortex beams, a perfect fractional vortex beam such as the perfect integer vortex beam can be created, which is expected to function as novel perfect beam. In essence, the beam can carry arbitrary TC; however, stable propagation is the basis of solving practical engineering problems. The perfect fraction TC beam that can stably propagate in free space can expand the application of perfect beams, such as harmonic generation [3], perfect quantum vortex states [4], parametric down-conversion [5], Quantum secret sharing [6], vectorial perfect vortices [7,8], optical manipulation [9], microscopy imaging [10], and optical communications [11].
VBs are the foundation of perfect VBs, hence they have identical helical phase of $\textrm{exp}({il\varphi } )$ and can be created via varied generation methods [12–16]. Kotlyar et al. studied the best method to produce perfect VBs based on three methods [17], and produced elliptic perfect optical vortices [18]. Kovalev et al. realized a highly efficient element for generating elliptic perfect optical vortices [19]. However, these methods cannot achieve stable propagation of perfect VBs with fractional TCs in free space. In the physical essence, in light beams with fractional TCs, the phase appears as a discontinuity along the phase step. Thus, this fractional phase is not a solution to the wave function. Consequently, such a beam cannot maintain stability in free space and thus changes as it propagates. This phenomenon has become a scientific consensus [20–23]. Therefore, several studies have attempted to improve the stability of fractional VBs in recent years [24–26]. In our previous work, light beam carrying natural non-integer orbital angular momentum in free space was comprehensively studied [27]; however, the physical nature and optical properties of fractional perfect VBs remain unknown. If the fundamental problem of fractional VBs being able to propagate stably and create a perfect fractional VBs like their integer counterpart, the capacity of optical communication can be enhanced. In addition, the degree of freedom for optical manipulation can be improved and thus usher in several opportunities in optics [28].
This study tackled this scientific problem by proposing the creation of a perfect fractional vortex beam as a new type of beam. The proposed perfect beam with TCs ($l + 0.5$) exhibited a unique state of polarization, obeyed the wave equation, and could propagate stably in free space. Moreover, the physical properties of this beam were identical to that of integer TCs. This beam was generated through the successful combination of theoretical and experimental results, and consequently, the electric field stability was analyzed in free space. This study confirmed to the possibility of creating a perfect fractional vortex beam [21], which may change the perception of perfect beams with fractional TCs and result in innovations in the corresponding technology in optics.
2. Principle
In principle, perfect VBs can be obtained via the Fourier transform of Bessel-Gaussian beam. Therefore, they can be created by using an axicon to generate Bessel-Gaussian beam, followed by use of Fourier transforms through thin lens. Figure 1 shows a typical optical system for generating Bessel-Gaussian beams using an axicon. The vortex beam with TC $l$ was generated in the purple box and the perfect VBs was obtained in the yellow box. However, the Bessel-Gaussian beam exists only in a limited range ($z < {z_{max}})\textrm{}$ behind the axicon. Therefore, the front focal plane of the thin lens must be maintained within this range. Further, ${d_z}$ is the distance between the axicon and the thin lens with focal length f. Consequently, perfect VBs can be created under the condition that ${d_z} \le \textrm{}f + {z_{max}}$.
Fig. 1. Schematic of a typical optical system for generating perfect VBs. The vortex beam with TC $l\textrm{}$ is generated in the purple box and the perfect VBs can be obtained in the yellow box. P1 and P2 represent polarization and phase, respectively. The vortex beam is transformed into a Bessel-Gaussian beam after passing through the axicon. In a limited range, the perfect VBs are obtained by the Fourier transform of the lens (the yellow box in Fig. 1).
In our previous work, we have shown that a propagable vortex beam with nature non-integer TC can be expressed as [27]
Equation (2) and (3) present two different vortex beams with fractional TCs vortex phases in free space. (R1-Q1) Note: in our paper the fractional TC is a half-integer TC, l and m represent the TC of scalar vortex beam and the order of VVB, they are independent of each other. l is measured by interference experiments in linearly polarized states (Fig.S3 in Supplement 1), and m is obtained by a polarizer.
To verify the propagation invariance of perfect VBs with fractional TCs, based on the Debye vectorial diffraction theory, the electric field in the focal region of lens in Fig. 1 can be expressed as [29]
According to Eq. (1), the polarization of the $m + 0.5$ order VVB can be considered a combination of x and y linear polarization modes. Therefore, the propagation unit vector of the incident beam after having passed through the lens is ${\boldsymbol V} = \cos [{({m + 0.5} )\varphi } ]{{\boldsymbol V}_{\boldsymbol x}} + \textrm{sin}[{({m + 0.5} )\varphi } ]{{\boldsymbol V}_{\boldsymbol y}}$, where ${{\boldsymbol V}_{\boldsymbol x}}$ and ${{\boldsymbol V}_{\boldsymbol y}}$ are the electric vectors of $|x $ and $|y $, respectively, and can be expressed as [29]
Consequently, the focal light intensity of the perfect VBs can be obtained using $I = |{{E^2}} |$.
3. Experiment
According to Eqs. (1) and (4), the perfect VBs with fractional TCs can maintain stable amplitudes and phases if two conditions are satisfied: the optical vortex with TC $l + 0.5$ and the $m + 0.5$ order VVB. Moreover, the simultaneous modulation of phase and polarization in a single beam has been challenging in the past. In our previous work, the invention of the polarized spatial light modulator (P-SLM) effectively solved this technical limitation [31]. Figure 2 shows the experimental setup comprising three optical systems. The first is the parallel light creation system in the purple box, which can produce high quality beam and provide x linearly polarized beam for SLM. The second is the phase control system, indicated by the blue box. A x linearly polarized parallel beam with the wavelength of 780 $nm$ passes through a SLM and the first 4f system comprising L3 (f = 150 $mm$) and L4 (f = 150 $mm$), and then is transformed into an $m = 30$ order VVB via a vortex polarizer (VP). The VP (VP is realized by the Q-plate technique) was conjugated with the SLM by the first 4f system; The last is the filter system comprising L5 (f = 150 $mm$) and L6 (f = 150 $mm$) in the yellow box. Thus, the arbitrary phase and the polarization can be obtained using this system. These three optical systems constitute P-SLM and can provide the required phase and polarization for fractional VBs in Eq. (1). Further, the fractional perfect VBs can be created using the axicon (the base angle is 2°) and L7 (f = 200 $mm$) after P-SLM and are recorded using a CCD.
Fig. 2. Experimental setup for generating fractional perfect VBs. The P-SLM comprises three optical systems: the parallel light creation (the purple box), the phase control (the blue box) and the filter (the yellow box) systems. A collimated incident x linearly polarized beam with a wavelength of 780 $nm$ can be converted into an m = 30 order VVB using the VP. The phase-only SLM is conjugated with the VP by the 4f systems with L3 (f = 150 $mm$) and L4 (f = 150 $mm$). We can obtain the phase and the polarization through the filter system consisting of L5 (f = 150 $mm$) and L6 (f = 150 $mm$). The fractional perfect VBs can be created by axicon (the base angle is 2°) and L7 (f = 200 $mm$) after P-SLM and are recorded using a CCD. Here, subfigures (a) and (b) are the phase for the phase-only SLM and the experiment for fractional perfect VBs, respectively.
4. Result
4.1 Experiment for perfect VBs of $m + 0.5$ order
According to Eq. (2), perfect VBs comprise an optical vortex with TC of 0.5 and an $m + 0.5$ order VVB. This perfect VB of $m + 0.5$ corresponds to the integer-order VVB, which is a special VVB and a supplement to integer-order VVB. Further, they exhibit amplitude distributions with the same physical laws as in case of integer-order VVBs. To create this light beam in free space, the polarization and phase of the incident light beam (Eq. 2) must be controlled using P-SLM. Here, a new light-field generation method based on the principle of the P-SLM [31] can facilitate independent control of the polarization and phase; that is, the extraction of inherent polarization modes, which has been fulfilled in our previous work [32].
Figure 3 (a, e, i, m q) presents the experimental results of the $m + 0.5$ order Perfect vector VBs, with (a) $m = 0$, (e) $m = 1$, (i) $m = 2$, (m) $m = 3$, (q) $m = 4$. Subfigures (A, B, C, D, E) were coded in the P-SLM to create the corresponding perfect VBs. The theoretical results presented in supplementary Fig. 1 (A, B, C, D, E) are the phases containing two parts: the phase of optical vortex that can carry TC 0.5 and a phase of the axicon. After passing through a polarizer, Fig. 1 (b-d, f-h, j-l, n-p, r-t) shows the $2m + 1$ number of the petal-like patterns and the same physical laws as in integer-order VVBs were followed.
Fig. 3. Experimental results for perfect VBs of $m + 0.5$ order. Perfect VBs with (a) $m = 0$, (e) $m = 1$, (i) $m = 2$, (m) $m = 3$, (q) $m = 4$ are created by phase (A, B, C, D, E) and polarization (F, G, H, I, J), respectively. Subfigures (b-d, f-h, j-l, n-p, r-t) indicate the light intensities of perfect VBs (a, e, i, m, q) passing through the polarizer by the purple arrow. The light intensities are normalized to a unit value, and the phase scale is $0\sim 2\pi $. The theoretical results are shown in Supplementary Fig. 1.
4.2 Experiment for perfect VBs with topological charge $l + 0.5$
According to Eq. (3), perfect VBs theoretically comprise a 0.5-order VVB and an optical vortex with TCs of $l + 0.5$. The P-SLM in Fig. 2 can control the arbitrary polarization and phase simultaneously. Moreover, the polarization was a fixed 0.5-order VVB and the phase was a vortex beam with TCs $l + 0.5$ exhibiting unlimited states.
Figure 4 (a, e, i, m q) shows the experimental results of perfect VBs with topological charge $l + 0.5$ with (a) $l = 0$, (e) $l = 1$, (i) $l = 2$, (m) $l = 3$, (q) $l = 4$. The phase containing 0.5-order VVB and VBs with TCs $l + 0.5\textrm{}$ coded in the P-SLM are shown in Fig. 4 (A, B, C, D, E). Theoretically, the phase comprises optical vortex with TCs $l + 0.5$ and axicon as shown in Supplementary Fig. 2. The different phases obtained in theory and experiments were as follows. One was a 0.5-order VVB created via the P-SLM in experiment; however, only changing ${\boldsymbol V}$ in Eq. (4) can ensure its realization in theory. The other is that the axicon is an optical element in the experiment; however, the phase of axicon was replaced in theory. Owing to the invariable polarization of the 0.5-order VVB, only one petal could be observed when passing through a polarizer, indicated by the purple arrow, and was rotated along with the polarizer, as in Fig. 4 (b-d, f-h, j-l, n-p, r-t). Although all light intensities were approximately identical, as in Fig. 4, the VBs exhibited different TCs. The different TCs were shown using a common interferometric system. In Supplementary Fig. 3, an incident x linearly polarized beam was divided into reference and measurement beams by BS1 and BS2, respectively. After passing through the P-SLM, the measurement beam transformed into fractional VBs with fractional TCs and unique polarization (0.5-order VVB) at point A. The beam was transformed back to a vortex beam with linear polarization by VP2 (0.5-order vortex waveplate) at point B. However, the reference beam maintained linear polarization and was tuned to the identical light intensity as the measurement beam at point C using an NF. Consequently, a fork pattern was obtained, wherein two beams overlapped, as recorded by a CCD. The experiment results of interferometric system are shown in Fig. 4 (F, G, H, I, J). The corresponding theoretical results are shown in Supplementary Fig. 3.
Fig. 4. Experimental results for perfect VBs with TCs $l + 0.5$. Subfigures (a, e, i, m, q) are perfect fractional vortex beam with TCs (a) $l = 0$, (e) $l = 1$, (i) $l = 2$, (m) $l = 3$, (q) $l = 4$, and created by the phase that coded in the P-SLM (A, B, C, D, E). Subfigures (b-d, f-h, j-l, n-p, r-t) indicate the light intensities of perfect fractional VBs (a, e, i, m, q) passing through the polarizer by the purple arrow. The light intensities are normalized to a unit value, and the phase scale is $0\sim 2\pi $ The theoretical results are shown in Supplementary Fig. 2, and (F, G, H, I, J) are the fork patterns forms by overlapping with a reference beam in Supplementary Fig. 2.
5. Discussion and conclusion
5.1 Propagation invariance of fractional perfect VBs
Generally, a light beam with an integer TC is a special solution of the wave function that can propagate stability in free space; for example, the Bessel-Gaussian and Laguerre-Gaussian beams [33]. In contrast, the beam carrying fraction TC is considered to being able to maintain stability in free space [20,22,23], as in case of perfect VBs [21]. According to Figs. 3 and 4, fractional perfect VBs were created based on theory and experiments, and its corresponding TCs were verified. To further prove that this is a new and stable perfect optical vortex that is fundamentally different from the traditional perfect beam with fractional TCs, an experiment with fractional perfect VBs was conducted to prove the propagation invariance.
In Fig. 5, fractional perfect VBs are shown to have the same physical law and stable optical properties with integer optical vortices. Subfigures (a1, a2, a3, a4) are the beam propagation experiment in the yellow dashed box, where fractional perfect VBs were recorded using a CCD along the optical axis from $z = 0cm\textrm{}$ to $z = 20cm$ under $NA = 0.01$. The green dashed box denotes the amplitude distribution at $z = 10cm$ and the experimental and theoretical results are (b, c, d, e) and (b1, c1, d1, e1), respectively. Subfigures (d) and (b, c) are perfect VBs with $m + 0.5$ order and TCs $l + 0.5$ corresponding to Figs. 3 and 4, respectively. According to Equations (2, 3), the perfect VBs are carried by the polarized state of $m + 0.5$ order VVB (Subfigure (d)) and ${\pm} 0.5$ order VVB Subfigure (b, c)), respectively, and the purple arrow is the direction of the polarizer. In principle, l and m satisfy Equations (2, 3), that is, existence of infinite state. In Fig. 5, $m = 2$ and $l = 3$ were set as an example. Subfigures (a1, a2, a3) correspond to Subfigures (b, c, d) of the beam propagation experiment, respectively, with 1000 sets of data collected along the optical axis. They clearly confirm the propagation invariance of fractional perfect VBs. Subfigures (f, g, h, k) are the experimental (red, purple, blue and pink point) and theoretical (black point) results of the intensity distribution along the azimuthal coordinate, corresponding to Subfigures (b, c, d, e) and (b1, c1, d1, e1), respectively. The main reason for the error between theory and experiment is the CCD intensity normalization and pixel damage; however, this does not affect the optical properties of fractional perfect VBs stable propagation because the first is the 0.5 order VVB with arbitrary TCs $l + 0.5$ and the other is a vortex phase (TC is 0.5) with arbitrary $m + 0.5$ order. In general, a perfect VB is the Fourier transform of the Bessel beam, and it is formed in the far field. The traditional fractional perfect VBs exhibit discontinuous light field distribution [21,22]: Subfigure (e) denotes a conventional optical vortex $\textrm{exp}\,[{i({l + 0.5} )} ]$ carrying linear polarization, $l = 3$. The amplitude of this optical vortex was interrupted due to phase discontinuity. In this study, we realized that fractional and integer perfect VBs follow the same physical law. Although the “phase discontinuity” phenomenon still exists, the electric field of the fractional perfect VBs was continuous (as shown in Fig. 5(b)).
Fig. 5. Experimental results for the propagation invariance of fractional perfect VBs. Subfigures (a1, a2, a3, a4) are the beam propagation experiment in the yellow dashed box. The green dashed box denotes the result along the optical axis at $z = 10cm$ and the experimental and theoretical results are (b, c, d, e) and (b1, c1, d1, e1), respectively. Subfigures (f, g, h, k) are the intensity distribution along the azimuthal coordinate, $\varphi $: experimental (red, purple, blue and pink point); simulated (black point). The light intensities are normalized to a unit value, and the purple arrow is the direction of the polarizer (Subfigure a1, a2 and a3 see Visualization 1, 2 and 3, respectively.). Note: Subfigure (e) denotes a conventional optical vortex $\textrm{exp}\,[{i({l + 0.5} )\varphi } ]$ carrying linear polarization, $l = 3$.
5.2 Significance of fractional perfect VBs
(R1-Q2) In recent years, many breakthroughs have been made on fractional vortex beams. The vortex beam with half-integer TC can be generated by the amplitude of a constituent beam in the superposition of two Bessel-Gaussian beams [34], this work demonstrates that when the amplitudes of the constituent beams tend to be equal, superposition of two BG beams has a half-integer TC. Kotlyar et al. realized tightly focusing vector beams containing V-point polarization singularities, this work shows an nth-order source vector field has a central V-point [35]. Stafeev et al. demonstrated tight focusing cylindrical vector beams with fractional order and analyzed the Poynting vector of light beams [36].
Fractional perfect VBs, with its unique spin and orbital angular momentum, can provide a new degree of freedom for many applications. However, in such applications, the stable propagation characteristics of optical beam is the cornerstone of solving scientific and engineering problems, and the instability characteristics of traditional fractional perfect VBs in free space is a natural barrier that restricts its application. In optical communication, improving the information capacity of optical communication systems has always been a significant challenge. Perfect VBs with continuous integer and non-integer TCs can overcome the aperture size limitation and thus expand the communication capacity [37]. In ultrafine optics, fractional perfect VBs may provide new ideas for the control of micro-laser emissions [38]. Further, in the field of quantum sensing, fractional perfect VBs offers more possibilities for the interaction between light and atoms because of its unique spin angular momentum [39]. In addition, fractional perfect VBs that can be stably propagated can also influence optical imaging [40], optical trapping [41], spatiotemporal optical vortices [42], and other scientific research related to spin and orbital angular momentum.
Thus, this study demonstrated that it is possible to create a fractional perfect vortex beam in free space based on theory and experiments. The amplitude and vortex phases with fractional TCs can be kept stable in the process of propagation proposed in this study. This new fractional perfect vortex beams is expected to break the cognition of traditional perfect beams and promote the application of beam spin and orbital angular momentum as well as the application in basic sciences such as electricity and magnetism.
Funding
National Key Research and Development Program of China ((2018YFC1313803).); National Natural Science Foundation of China ((62022059/11804232)).
Disclosures
The authors declare no conflicts of interest.
Data availability
All data supporting the findings of this study are available from the corresponding author on request.
Supplemental document
See Supplement 1 for supporting content.
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