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Abstract
Vortex beams (VBs), a type of light beam with a spiral wavefront, have unique properties, such as the orbit angular momentum (OAM), and diverse applications in optical communications and optical trapping and tweezers. Therefore, accurate measurements and estimations of the phase distribution and topological charge are essential for their applications to ensure VB quality. In this paper, we employed a sinusoidal phase modulation (SPM) interferometry to measure the phase distributions of VBs and the topological charge of VBs were estimated by mean of a method of the process of unwrapped phase. The phase measurement of optical vortices generated by a spatial light modulator (SLM) demonstrated that the SPM interferometry-based technique had a high measurement accuracy with a simplified configuration. The estimation errors of the topological charges for various orders of VBs were within approximately 4%. The fluctuation in the surface of the SLM leading to the flatness of the wavefront was estimated to be 0.06 rad by 10 consecutive measurements
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1. Introduction
In recent years, there has been considerable interest in the generation of a VB (i.e., optical vortices or Kummer beams [1–4]) and their applications [5]. In 1992, Allen et al. demonstrated that each photon of the VB possesses the OAM, derived from a characteristic phase structure with a helical wavefront characterized by exp(ilσ), where σ is the azimuthal angle and l denotes the topological charge. This property yields a phase singularity point in the optical axis with an undefined phase and the amplitude of zero [6]. Various applications have been developed using these features. For instance, the stimulated emission depletion (STED) microscopy utilizes the constructed light beam with a spiral wavefront as the saturated depletion light. It realizes super-resolution imaging of fluorescence with high contrast [7]. Additionally, in the field of optical communication, OAM has been used as an alternative degree of freedom for multiplexing modulation, thereby enlarging the capacity of optical communication [8], which is also referred to as mode/spatial-division multiplexing [9]. Moreover, in optical tweezers [10] and spanners [11], OAM can be transferred to the trapped particle, causing it to rotate around the center of the VBs. In addition, the characteristics of VB can be used for the fabricating the chiral micro-scale structures [12]. Considering these applications, the phase distribution of the wavefront of a VB, which is characterized by a topological charge related to OAM, is one of the most crucial parameters [13–15]. Therefore, the accurate measurements of the phase profiles of the wavefronts of VBs are required to guarantee the beam quality in such applications.
At first, many researchers have been attracted more attention to study in the estimation of the topological charge of a VB and many methods have been proposed. These methods can be classified mainly into two types. One type is based on the interferometric and diffractive techniques by detecting the interference or diffraction pattern [16–26]. Other type is based on a computing process of Fourier transform performed for intensity (or diffractive intensity) of VBs [27,28]. Nevertheless, these measurement methods roughly estimated the value of the topological charge from the diffraction or interference pattern, because phase information was uncertain. With the development of optical communication and micromanipulation, the quantitative measurements of the topological charge and phase distribution of VBs become to be essential. To ascertain the detailed vortex wavefront, the Shack-Hartmann wavefront sensors (SHWFSs) based on geometrical optics have been employed for the phase measurement of VBs [29]. However, this method of SHWFSs involves a complicated configuration and a high cost of the camera. In recent years, a phase sensitive interferometry with algorithms of time-domain phase shift and spatial carried frequency have been adopted in the measuring the phase of VBs [30,31]. For instance, the time-domain phase shift technique with four-step phase shift method was proposed [30]. However, this method requires the least three predefined accurate phase shift and thus corresponding interferograms. A slight deviation in the phase shift will lead to enormous noise in the measurement results. The spatial carrier frequency algorithm enables the extraction of phase of a VB from a single interferogram [31]. However, accurate results require very dense fringes (i.e., at least 200 interference fringes per millimeter) and a minimally sized camera [32], which are susceptible to disturbances under a condition of low sampling frequency.
In this study, we propose a method to directly obtain an accurate VB phase distribution by employing the SPM method [33,34] in a more convenient configuration. Interferometry involves a SPM with a reference light to obtain the interference phase of the wavefront. Compared with the phase-shifting method or heterodyne interferometer [32], the calibration is more straightforward, because the sinusoidal modulation of the reference beam generates a repetitive signal that is separated from the noise with a simple and convenient implementation. The spatial phase distribution detected by the charge coupled device (CCD) camera can be reconstructed by mean of a suitable phase unwrapping method, enabling the quantitative measurement of the phase difference to estimate the topological charge with high accuracy.
In our experiment, a VB was generated by a SLM for measurement purposes and vortex wavefronts of various orders of VBs were successfully measured by the proposed SPM method as the interference phase. This study discusses the accuracy of the topological charge derivation based on realistic VB phase distributions. Furthermore, the wavefront error due to the interference of light from SLM and plane mirrors with Gaussian beams was discussed.
2. Principle and setup for SPM interferometer
To precisely measure the phase distribution of the VBs, Michelson interferometer with a reference mirror attached to a piezoelectric transducer (PZT) was used for the SPM method as shown in Fig. 1. A VB was generated by an SLM (HOLOEYE PLUTO, Phase Only). A laser diode with a center wavelength of 780 nm was employed as the incident light, which was divided into two beams by a nonpolarizing beam splitter (BS). Spatial phase modulation was applied to the beam injected into the SLM to convert the initial plane wave of the Gaussian beam profile into a helical phase structure to generate the VB. The optical phase of the reference wave was temporally modulated via a reference mirror attached to the PZT, controlled by a sinusoidal signal from the signal generator (SG). The produced a VB and modulated reference beams were recombined by BS and interfered with. The interference pattern was acquired by using a CCD image sensor (IMPERX ICL-B0620, 640 × 480 pixels). The modulation frequency fm of the sinusoidal signal and frame rate of the CCD, fFPS = 1000 Hz, are synchronized and in a multiplicative relationship. In this experiment, we set the modulation frequency to fm = 125 Hz, which is equivalent to one eighth the frame rate of fFPS
Fig. 1. Experimental setup of the interferometer with SPM configuration for measuring phase distributions of the optical vortices. LD, a laser diode with a wavelength of 780 nm; CL, collimator lens; PC, polarization controller; R, reference mirror with a PZT for conducting phase modulation; L1 and L2, lens with the focal the length of f1 = 10 cm and f2 = 15 cm respectively. Gaussian beams from LD were converted to VBs via SLM.
The spatial phase difference between the surface of the SLM and the reference mirror can be obtained as the phase distribution representing the helical profile of the generated VB by adopting the SPM method. To obtain the SPM signal, the phase of the reference wave was modulated by a sinusoidal signal acos(ωct + θ), where a, ωc, and θ are the amplitude, angle frequency, and initial phase, respectively.
Hence, the optical interference signal containing sinusoidal modulation can be represented by
where z = (4π/λ)a is the modulation amplitude in terms of the radian phase value. S0 denotes the amplitude of the interference signal. Furthermore, x and y represent the spatial coordinate of lateral and vertical axes, respectively. By Fourier transform of Eq. (1), we obtained the signal of the m-order component in the frequency domain as follows;Equation (3) indicates that the value of z is required and must be determined in advance. First, the value of z can be determined from the relation between the measured values of F(3ωc)/F(ωc) and R31 = J3(z)/J1(z) [33]. The practical value of z should have a limited region of approximately 2.5–2.7 rad. The theoretical analysis of s(t) containing the additive noise reveals that z = 2.63 rad as the most suitable amplitude for practical measurements [33]. Owing to the monotonically decreasing curve of R31 as a function of z, the amplitude value of z can be uniquely and accurately determined in this range.
3. Results
3.1 Intensity and phase distributions of VBs
We carried out the experiment as the described method in section 2 to generate various orders of VBs and measured the corresponding phase distributions. The predesigned computer-generated hologram of the phase mapping was uploaded onto the SLM, which allows flexible control of the spatial transverse mode [35]. Thus, high-quality VBs with five different integer orders of topological charge were produced as shown in Figs. 2(a)-(e), recorded by the CCD with the corresponding with topological charge of l = 1, 2, 3, 4, and 5, respectively. Figs. 2(f)-(e) show the interference pattern with the vortex and reference Gaussian beams.
Fig. 2. (a)-(j): The intensity profiles of vortex beams and the corresponding interference patterns for the topological charge 1, 2, 3, 4 and 5 respectively. (a), (f): l = 1. (b), (g): l = 2. (c), (h): l = 3. (d), (i): l = 4. (e),(j): l = 5. (k): The temporal sinusoidal phase modulated signal from Fig. 2(f) at the position of x = 23 and y = 22 pixel.
A temporal interference signal with sinusoidal vibration at each pixel of the CCD image sensor was first detected, as shown in Fig. 2 (k). Owing to the sinusoidal vibration on the reference mirror with a frequency of f = 125 Hz which is equal to 1/8 of the frame rate, each pixel on the interference fringe can be represented as a series of time-varying periods constructed by 8 frames of 2-D interference pattern. To apply SPM method, we collected 8 periods of discrete points over the acquisition time of 64 ms.
To obtain the phase and intensity distributions of the VB, the interference signals, containing with 64 frames of 2-D interference in the 8 periods as shown in Fig. 2(k), were analyzed by a fast discrete Fourier transform (FFT) at all points with irradiance above a certain threshold observed at the CCD receiving surface. Consequently, the intensity and wrapped phase distributions of VBs in the order of l = 1, 2, 3, 4, and 5 were directly obtained using Eq. (3) with F(ωc) and F(2ωc), respectively.
The results of SPM signal processing are shown in Fig. 3 and Fig. 4. Figures 4(a)–(e) show that the phase changed gradationally to 2π, 4π, 6π, 8π, and 10π along the azimuthal angle of 2π, corresponding to the topological charges of l = 1, 2, 3, 4, and 5, respectively.
Fig. 3. The measured intensity distributions of the VB with topological charge of (a) l = 1, (b) l = 2, (c) l = 3, (d) l = 4, and (e) l = 5, respectively.
Fig. 4. The wrapped phase distribution of the VB with topological charge of (a) l = 1, (b) l = 2, (c) l = 3, (d) l = 4, and (e) l = 5, respectively.
3.2 Phase unwrapping process for VBs
The topological charge is characterized by l=φmax/2π ,where the φmax means that the maximum continuous variation value of phase along with the azimuth angle on the clockwise or anticlockwise direction according the sign of topological charge. Owing to the property of the phase distribution of a VB, the phase step can be found in the phase distribution of the VB between the phase value of the azimuth angle of 0 rad and 2π rad at the same radius and the corresponding with the value of the phase step obtained by phase difference can be regarded as the φmax. Hence, the phase unwrapping is required to quantify and evaluate the phase step. However, the conventional unwrapped phase algorithm is based on Cartesian coordinates along with the x-axis and y-axis, leading to discontinuity and ambiguity in the resultant phase distributions [32]. Therefore, we proposed this method for the unwrapped phase algorithm by transforming Cartesian coordinates into polar coordinates and the process of unwrapped phase algorithm is shown in Fig. 5 and Fig. 6.
Fig. 5. Process of phase unwrapping method specialized for optical vortices with the order of l = 2, (a) Obtained wrapped phase distribution by SPM, (b) Coordinate transformed phase distribution. The horizontal axis represents the sequence of points was obtained along to the azimuth angle on the clockwise direction at the same radius. (c) Unwrapped phase distribution with new coordinate.
Fig. 6. Re-transformed unwrapped phase distribution of a VB with the order of l = 2, reveals the approximately 4π phase step, marked by red arrows.
Based on the proposed process of unwrapped phase algorithm of the VB, the phase distributions of the VB with the order of l = 1,2,3,4 and 5 were unwrapped as shown in the Figs. 7(a)-(e). A threshold value of 50% of the maximum irradiance of the measured intensity distributions in Fig. 3 was set in the unwrapped phase of VBs. Only the phase structure was highlighted without the background light and the part of phase singularity by removing the area below this threshold from the two-dimensional phase distribution. This phase unwrapping enabled the detection of phase differences at the phase step, allowing the automatic quantitative measurement of the topological charge.
Fig. 7. The unwrapped phase distribution of the VB showing only the light intensity components above 50% of maximum value of topological charge of (a) l = 1, (b) l = 2, (c) l = 3, (d) l = 4, and (e) l = 5.
3.3 Estimation of topological charge
We successfully determined the topological charges of the integer and fractional orders of VBs. Figure 8(a) shows the result of the unwrapping phase distribution of the first order of a VB, where the phase step was marked by a red arrow and the phase distribution can be represented under the polar coordinates with the clockwise direction, as shown in Fig. 8(b) which shows only the phase distribution on the part of phase step within the radius of 25 to 35 pixels. Hence, the phase value in Fig. 8(b) of the azimuth angle of 0 rad and the azimuth of 2π were obtained and the corresponding with the phase difference [PD] were determined, as shown in the Table 1 and Table 2. Give that these values of the PD are equivalent to φmax, by employing this data analysis automatically with the equation of l=φmax/2π, the topological charge of the VB with l = 1 was quantitatively determined, as shown in Fig. 9. Similarly, the results of estimating the topological charge for the second to fifth order based on the phase difference obtained by applying the above process with the same manner were shown in Fig. 10.
Fig. 8. Example of topological charge estimation for first order VB. (a) Extracted phase distribution of the VB with a first-order topological charge. (b) Phase distribution of Fig. 8 (a) was represented by the polar coordinates along to the clockwise direction within the radius of 25 to 35 pixels.
Table 1. Phase difference (PD) between the phase value in the azimuth angle (0) [rad] and the azimuth angle (2π) [rad] on the clockwise direction
Table 2. Phase difference (PD) between the phase value in the azimuth angle (0) [rad] and the azimuth angle (2π) [rad] on the clockwise direction
Table 3 shows the error between the mean value of the measured topological charge and the set value at each order. The detection error of the topological charge is the largest for the second and fourth order, approximately 4%. The standard deviation shows the scatter of the phase difference of the detected phase discontinuity lines, which is less than 0.02 rad in all cases.
Table 3. The measured errors of the topological charges
3.4 Measurement for VB with fractional orders
Give that the SPM method is an interferometric measurement of arbitrary wavefronts, it also can be employed to precisely measure the phase distribution of the fractional order of a VB if a crisp interferogram is obtained. To verify the feasibility of the fractional order of a VB, we generated the fractional order of VBs with l = 2.5, 3.5, and 4.5 from the SLM and measured the interference pattern with a Gaussian beam. The corresponding wrapped phase distributions were measured as shown in Fig. 11.
Fig. 11. Obtained phase distributions of vortex beam with fractional order of (a) l = 2.5, (b) l = 3.5, and (c) l = 4.5.
Next, we conducted unwrapped phase algorithm for the phase distributions of Fig. 11. In the unwrapping process, the background or diffraction light from the SLM was eliminated and reduced in the same manner described in previous section. Figure 12 shows the results of phase unwrapping and indicates that crisp phase distribution containing defect in the phase distribution along the azimuth angle.
Fig. 12. The unwrapped phase distribution of the VBs with the topological charge of (a) l = 2.5, (b) l = 3.5 and (c) l = 4.5, respectively.
In theory, the phase variation for the VB is along the azimuth angle. Hence, we can determine the value of φmax by determining the value of the maximum variation of phase along to the azimuth angle on the clockwise direction. Consequently, we found that the average values of maximum variation of phase φmax with the fractional topological charges were estimated to be approximately 16.005, 25.425 and 27.961 rad, and the corresponding topological charges were 2.5487, 3.6322 and 4.5432, respectively, as shown in Table 4. Compared to the theoretical value, the maximum measurement error was approximately less than 4%.
Table 4. The measured errors of the topological charges
4. Discussion
The measurement accuracy of VB phase measurement is affected by fluctuations in the Gaussian beam of the reference light (i.e., the wavefront of the light collimated from the laser). The tilt with respect to the reference plane was detected directly as the overall tilt of the phase distribution.
To investigate the accuracy due to the wavefront error, we measured the interference between the Gaussian beam used as the reference and a similar Gaussian beam reflected from the SLM (the surface of the SLM was set flat without hologram). The phase profile fluctuation and flatness obtained from this system were used as indicators of the wavefront error in the VB measurement.
Ten consecutive measurements were conducted in the same manner, and the correspondence with the mean value of the phase distribution was obtained, as shown in Fig. 13(a). An ideal plane of the mean distribution of the measured wavefront (shown in Fig. 13(a)) was fitted using the least-squares method, as shown in Fig. 13(b), and the difference between the measured and fitting distributions was calculated, as shown in Fig. 13(c). The flatness of the wavefront was 0.06 rad according to the root-mean-square (RMS) of Fig. 13(c), revealing the potential wavefront error in the spatial propagation of the VB.
Fig. 13. Average value of phase distribution of Gaussian beam with 10 measurements. (a) The measured phase distribution. (b). The fitting result of phase distribution. (c). The phase different between the measured phase distribution and fitting result.
5. Conclusion
In this paper, we demonstrated the SPM method using a Michelson interferometer to measure the phase distributions of the high integer and fractional orders of VBs. Optical vortices were generated from the SLM, and their phase distributions were successfully measured. A phase unwrapping method was proposed to detect and automatically extract the phase steps of discontinuous phase distributions. Consequently, topological charges were precisely determined from the phase steps. The standard deviation of the measurement of the integer order of VBs was estimated to be approximately in the range of 0.01-0.02 rad. By comparing the theoretical results, the maximum errors of the integer and fractional orders of the VBs were approximately 4% and 3.8%, respectively. Moreover, the flatness of wavefront was evaluated by 10 consecutive measurements of the fluctuation between the surface of SLM and the reference mirror, which revealed that the estimated flatness error was approximately 0.06 rad in terms of RMS. These results indicate that SPM interferometry is a useful technique for the precise spiral phase distribution and topological charge estimation of optical vortices.
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.
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