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Abstract
Vortex beam has the potential to significantly improve the performance of lidar (light detection and ranging) and optical communication applications in which low signal-to-noise ratio (SNR) limits the detection/transmission range. The vortex beam method allows for spatially separating the coherent light (laser signal) from the incoherent light (the background radiation and multiple-scattered light) of the received signal. This paper presents results of a simulation model in which the optical vortex acts as an optical filter. We present instrument parameters that describe the filtering effect, e.g., the form of the vortex phase modulation function, the topological charge of the vortex and the focal length of a virtual Fresnel lens that is used for optical filtering. Preliminary experimental results show that the background radiation within the spectral filter bandwidth can be suppressed by as much as 95%. At the same time, we retain 97% of the coherent laser signal. Our simulation model will be used in future design of lidar instruments and optical communication systems in which the optical vortex method is used for optical filtering of the detected signals.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Background radiation and multiple-scattered light decrease the signal-to-noise ratio (SNR) of signals detected by lidar (light detection and ranging) and optical communication systems [1–3]. Spectral filtering is usually used to improve the SNR by removing the background radiation outside the wavelength range that is used by the signal-detection unit. However, it is impossible to remove completely the background radiation which in large part results from the sunlight. It is because the background radiation is distributed across the wavelength range in which lidar/optical communication systems are operated. Besides, the multiple-scattered light within the field-of-view (FOV) of the detector system has the same wavelength as the light-emitting laser. That means, spectral filtering cannot fulfill the requirement either.
Background radiation and multiple-scattered light cause spatial and temporal dispersions which result in image blur and distance-ranging inaccuracy in the case of lidar application. Some approaches are developed to improve the SNR, including the techniques of polarization sensitive detection [4–8], hybrid lidar-radar [9–12], and image processing [13,14]. Polarization sensitive detection is often used to capture the object-reflected light photons without scattering. However, it is impossible to remove background radiation completely, because the background radiation usually contains various polarization states of the detected light. Hybrid lidar-radar combines the coherent techniques of microwave radar and the underwater transmission capability of lidar. The microwave radar signal is superimposed on the lidar optical pulse by a high-speed modulator. The optical carrier then transports the microwave signal through the water. Both the lidar and radar signal are recovered at the detection unit. Hybrid lidar-radar obviously suffers from a complex system setup. The imaging processing methods can improve the image quality of a lidar system, but they cannot remove the unwanted photons out of the detected signals.
Optical filter based on coherence has recently been proposed as a method that allows for reducing the incoherent part of detected light. This filter method uses an optical vortex [15] or an axicon [16,17]. This paper focusses on the optical vortex method. A vortex beam [18,19] has a vortex phase of exp(-ilφ) and shows a spiral wave front. The phase at the center of the beam is uncertain, and it is called an optical singularity. The transverse optical intensity of the vortex beam is annular, with a null-intensity area along the optical axis. A vortex beam carries orbital angular momentum (OAM) lℏ, where l is the topological charge value, and ℏ is the Planck constant. It can be generated by imposing a vortex phase on a plane wave by a spiral phase plate (SPP) [20–22] or a spatial light modulator (SLM) [23,24].
Because of these properties, vortex beam can be used for imaging applications [25–27]. A photon-sieve lidar concept was reported in Ref. [27], with an optical vortex beam as the source and a photon-sieve diffractive filter at the receiver. The plane wave of scattered sunlight and background signals was focused into a “spot” in the center of the focal plane. The light with optical vortex (the emitted laser beam, signal) was focused as a “ring” around the “spot”. As a result, it became possible to separate scattered sunlight from the emitted laser light. Since the optical vortex beam was generated at the lidar source, the maximum power of transmitted laser beam was limit by the phase modulator (SPP or SLM). Thus, the working distance for the lidar was limited. Other researchers used an optical vortex to enhance the ranging accuracy of an underwater lidar, and showed the experimental results of the spatial separation of coherent light and incoherent light [15,28–30]. In their work, the coherent light was focused into a “ring”, while the incoherent light was uniformly distributed on the whole imaging plane without forming a “spot”, because a relatively large receiver FOV of the lidar was used [15]. Therefore, the coherent and incoherent light signals overlapped in space.
In this paper, we present a simulation model of an optical filter based on the theory of optical vortex. We mainly use Fourier optics in this model to get an analytical solution of light intensity distribution on the image plane. Different from the work presented in Ref. [27], a regular Gaussian beam is used as laser source, and the phase modulator is positioned at the receiver. The spatial separation of coherent and incoherent light is improved by optimizing the parameters of the phase modulation function, including the form of the vortex phase modulation function, the topological charge of the vortex and the focal length of a virtual Fresnel lens. We thus obtain an improved SNR. The simulation model of the optical filter can be used in future design of lidar instruments and optical communication systems in which the optical vortex method is used for optical filtering of the detected signals.
The simulation model is presented in section 2. The computational results are given in section 3. The experimental set-up is illustrated in section 4. The experimental results and the comparison of both computational and experimental results are given in section 5. Conclusions are finally given in section 6.
2. Simulation model
The signals detected by a lidar or an optical communication system can be written as
where the parameter I denotes the total signal, Icoherent denotes the coherent part, referring to the single-scattered light, and Iincoherent denotes the incoherent part, which includes the background radiation and the multiple-scattered light.Figure 1 illustrates the model of an optical filter. A Gaussian-beam laser is used as coherent light source. The beam waist is described by ω0. The field distribution of the coherent light wave just before the aperture is E0(x1, y1). A circular aperture is placed in the optical light path. The distance between the light source and the aperture is z. We used a halogen lamp as incoherent light source. A spatial light modulator (SLM) is placed immediately after the aperture to implement the phase modulation. The field distribution of the coherent light wave just after the SLM is E1(x1, y1). The distance between the SLM and the lens is d. The focal length of the lens is f. The light beam is eventually focused onto the image plane. The field distribution of the coherent light wave on the image plane is E(x, y).
The field distribution E1(x1, y1) can be written as
where we use the aperture function P(x1, y1) and the phase modulation function S(x1, y1).The information of the phase modulation function S(x1, y1) is uploaded to the micro-processor of the SLM. The phase profile of the incident coherent light beam is modulated by the function S(x1, y1), while that of the incoherent part remains random after the SLM. The function S(x1, y1) is the product of the vortex phase modulation function V(x1, y1) which is used to generate optical vortex, and an additional function A(x1, y1) which is used to further increase the spatial separation of coherent and incoherent light. The additional function A(x1, y1) is described as a virtual Fresnel lens which only affects the coherent light. As a result, the coherent light and incoherent light are focused at different locations along the direction of light propagation.
In polar coordinates, the phase modulation function can be written as
Fourier optics [31,32] can be used for deriving an analytical model that allows us to describe the coherent light on the image plane. We find the following expression
In Eq.(4), the focal length f, which is a constant coefficient, affects both the coherent and incoherent light in the same way. The distance ratio d/f in the second-order phase factor does not affect the coherent light intensity distribution on the image plane. However, if the light beam travels through optical elements after the image plane, the value d/f either needs to be considered or we can simply let d = f to eliminate this term.
The coherent intensity distribution on the image plane is given by
where E*(x, y) is the complex conjugate of E(x, y).Different from the coherent light (see Eq.(4) and Eq.(5)), the analytical model of the incoherent light is given by
Based on Fourier optics, we have
where I0(x, y) represents the intensity distribution of the ideal optical image of the object on the (x, y) plane.Since we consider only the normal incident plane wave which indicates object at infinity, we have I0(x, y)=δ and $\mathcal{F}\left\{E_1\left(x_1, y_1\right)\right\}=1$. Then we obtain
where hC(x, y) represents the coherent point-spread function. In polar coordinates, it can be described as where P(r) is the aperture function. C is a constant coefficient. D is the diameter of the circular aperture. J1(r) denotes the first-order Bessel function.Substituting Eq.(9) into Eq.(8), we obtain
where I0 is the intensity in the center of the image on the (x, y) plane.The radius of the incoherent spot, i.e., the Airy disk, is described as
The analytical model of the incoherent light was also discussed in Ref. [32]. In our case, the SLM only modulates the phase of the coherent light, leaving the incoherent light unmodulated. Hence, the intensity distribution of the incoherent part, a “spot” with the Airy disk radius, remains unchanged when the phase function S changes.
3. Simulation results
3.1 Coherent light
We used MATLAB software for a simulation of the light field, intensity, and phase distribution. In our computations, we consider an interested area of 30 mm × 30 mm of the incident light beam on the (x1, y1) plane. The number of sample points is 8192 × 8192. That means the grid spacing, or the simulation resolution, on the (x1, y1) plane is 30 mm/8192 = 3.7 µm. The size and grid spacing of the (x, y) plane, namely the imaging plane, are determined by the coordinate transformation of the Fourier transform. The parameters used in the numerical simulation example are given in Table 1.
Table 1. Parameters used for the simulation example
Figure 2 shows the plane wave of the incident laser beam E0(x1, y1). We see a Gaussian-type intensity distribution (in Fig. 2(a)) and a uniform phase distribution (in Fig. 2(b)).
Figure 3 shows the pattern of the phase modulation function S(x1, y1). It can be described by exp(-ilφ) in which the topological charge l = 10. The effective area of the SLM is 7.5 mm×7.5 mm, with 1024 × 1024 pixels and 8-bit. The size of a single unit is 7.5 mm/1024 = 7.3 µm, twice the grid spacing on the (x1, y1) plane.
The intensity and phase distribution on the (x1, y1) plane just after the SLM can be derived by the use of Eq.(2) and Eq.(3). The results are shown in Fig. 4. A comparison to Fig. 2 shows that both the intensity and phase distribution are limited by the aperture. A spiral wave front is generated by the phase modulation.
Fig. 4. The incident coherent light after aperture and phase modulation. Shown are (a) intensity distribution and (b) phase distribution
The intensity and phase distributions on the (x, y) plane (image plane) are obtained by Eq.(4) and Eq.(5), shown in Fig. 5. We can see that a ring-shape-like intensity distribution appears on the image, see Fig. 5(a). The concentric “rings” outside the main “ring” result from the diffraction effect due to the circle aperture. At the same time, the phase distribution (see Fig. 5(b)) changes significantly after light propagation, compared to the image shown in Fig. 4(b). Since the intensity distribution pattern is quite small, Fig. 5(a) does not show the complete interested area on the image plane.
Fig. 5. Image of coherent light. Shown are (a) intensity distribution (partial enlarged) and (b) phase distribution
Figure 6 shows the cross-section A-B of the intensity distribution shown in Fig. 5(a). The radius of the central null-intensity area is R = 33 µm. The boundary of the null-intensity area is defined by the location (radius) where the relative intensity drops to 1/e2.
3.2 Incoherent light
Computational results of the incoherent light of the detected signal (described by Eq.(10)) are shown in Fig. 7. The radius of the Airy disk, which describes the minimum image spot of an object at infinity (e.g., sunlight background radiation) due to light diffraction, is found to be 13 µm, see Fig. 7(a). It is consistent with the theoretical result by Eq.(11). Since the incoherent “spot” is smaller than the coherent “ring” (see Fig. 7(b)), the coherent and incoherent signal are separated spatially. Thus, it is possible to remove the incoherent part of light, i.e., sunlight, completely from the total signal.
Fig. 7. Image of the incoherent light. Shown are (a) incoherent light only and (b) both coherent and incoherent light
It is important to emphasize that only the zero-FOV-case is considered in this paper. If a detection system has a small but non-zero FOV, which is the typical case, the incident light with a small incident angle is focused on the image plane with a transverse displacement. The radius of incoherent light spot will increase with increasing FOV. Besides, the diffraction limit is usually hard to achieve. That means the radius of incoherent light spot for a real lidar or an optical communication system will be much larger than the theoretical values calculated in this section. At the same time, the width of the coherent light ring-pattern will also increase with increasing FOV, resulting in a smaller central null-intensity area R.
3.3 Optimization
We investigated different options for optimizing the spatial separation of the coherent and incoherent part of the incident light, including the form of the vortex phase modulation function, the topological charge of the vortex and the focal length of a virtual Fresnel lens.
(1) Form of the vortex phase modulation function
First, we investigated the vortex phase modulation function V(r, φ) in Eq.(3). Figure 3 shows the simplest form of a vortex phase distribution, which is described as exp(-ilφ). More complicated shapes of vortex phase modulation functions can be used, e.g., a laser beam with a complex-amplitude distribution in terms of Laguerre-Gauss-function VLG(r, φ) or Bessel-Gauss-function VBG(r, φ). The phase terms of VLG(r, φ) and VBG(r, φ) are shown in Eq.(12) and Eq.(13), respectively.
Here, the parameter p denotes the number of nodes in radial direction, z’ describes the distance of light propagation, and z0’ is the Rayleigh range.
Since we are interested only in the intensity distribution on the image plane, the terms without r and φ can be ignored. The Bessel-Gauss-type phase term VBG(r, φ) can be written as
while the Laguerre-Gauss-type phase term is given byIf z’=0 or ∞, VLG(r, φ) can also be written as exp(-ilφ). When $z\mathrm{^{\prime}\ =\ }z{^{\prime}_0}^{1/2}$, the term $z\mathrm{^{\prime}/}({z{\mathrm{^{\prime}}^2} + z{^{\prime}_0}} )$ reaches its maximum value. As an example, we set z0’=49 m and z’=7 m. Figure 8 shows the computational results. Compared with Fig. 3 and Fig. 5(a), we see that the latter phase term in Eq.(15) creates a slight pattern distortion of the phase modulation function, while the intensity distribution on the image plane remains unchanged. Hence, for future simulations of a laser beam with complex-amplitude distribution of Laguerre-Gauss-type or Bessel-Gauss-type, we can use the simplest phase modulation function exp(-ilφ).
Fig. 8. Computational results with Laguerre-Gauss-type phase term. Shown are (a) phase modulation function and (b) intensity distribution on the image plane
(2) Topological charge
We used the topological charge l = 10 for the example discussed up to this point. Figure 9 shows the results for different topological charges l. Figure 9(a)-(f) show the patterns of the phase modulation function, for l = 0, 1, 10, 40, 80, 200, respectively. Figure 9(g)-(l) show the corresponding intensity distributions on the image plane. We find that the central null-intensity area exists when l≥1. The radius of the null-intensity area R linearly increases if the topological charge l increases. This result indicates that a better separation of coherent and incoherent light is possible if we increase the topological charge l.
Fig. 9. Simulation results for different topological charge l. Shown are (a)-(f) phase modulation function and (g)-(l) intensity distribution on the image plane
The maximum value of the topological charge l is determined by the resolution of the phase modulator SLM. Figure 9(f) shows that the phase modulation function cannot be resolved correctly anymore for l = 200, due to the small widths of the fine stripes of the phase pattern. It is hard to decide on the specific threshold value of l from which the pattern of the phase modulation function begins to distort. In the numerical simulation, we use the maximum l = 80 for a phase modulator with 1024 × 1024 pixels.
(3) Focal length of a virtual Fresnel lens
In another step we investigate the function A(r, φ) which describes a virtual Fresnel lens. Figure 10(a) and (i) show the case without the use of a virtual Fresnel lens. Figure 10(b)-(h) show the phase modulation function, for the virtual Fresnel lens fF = 10000 mm, 8000 mm, 6000 mm, 4000 mm, 2000 mm, 1000 mm and 500 mm, respectively. Figure 10(i)-(p) show the corresponding light intensity distributions on the image plane. All the results shown in Fig. 10 represent l = 10. The results show that R increases if fF decreases. Figure 10 furthermore shows a bright spot that appears within the central null-intensity area of the intensity distributions. This result indicates that the function A(r, φ), i.e., the virtual Fresnel lens, breaks the phase uncertainty of the optical singularity.
Fig. 10. Simulation results for different focal length of the virtual Fresnel lens fF (topological charge l = 10). Shown are (a)-(h) phase modulation function and (i)-(p) intensity distribution on the image plane
Figure 11 shows the cross-section C-D of the intensity distribution of Fig. 10(o). The central peak of the intensity indicates the bright spot in the null-intensity area. This part of coherent light will be removed together with the incoherent light spot, which eventually causes the loss of signal intensity. The tooth-shape-like envelops on both sides indicate the diffraction rings.
Figure 12 shows that the relative intensity of the central bright spot increases significantly when fF < 2000 mm, which indicates a large signal loss of the coherent part of the detected light. Thus, we select fF = 2000 mm. The loss of the coherent signal is less than 5%.
According to the simulation results shown before, we conclude that for a detection system with parameters shown in Table 1, the optimized parameters for an optical filter are topological charge l = 80 and virtual Fresnel lens with fF = 2000 mm, see in Table 2.
Table 2. Optimized parameters
Figure 13 shows the simulation results. The radius of the central null-intensity area is R = 594 µm, which is about 18 times the value of the original value R = 33 µm. This result implies that we obtain a better spatial separation of the coherent and incoherent light.
Fig. 13. Optimized results with (a) phase modulation function and (b) intensity distribution on the image plane (coherent only)
Figure 14 shows the simulation results of both coherent and incoherent light. A comparison to Fig. 7 shows a better spatial separation.
4. Experimental setup
The experimental setup is shown in Fig. 15. A 532 nm pulsed laser is used as a coherent light source. A laser beam with Gaussian-type intensity distribution is emitted. A halogen lamp and a 532 nm filter are used for generating the incoherent light source. The information of the phase modulation function S is uploaded to the micro-processor chip of the SLM (Hamamatsu, LCOS-SLM X13138). A beam analyzer (WinCamD-LCM, CMOS beam profiler) is placed in the focus plane to record the beam intensity distributions of both the coherent and incoherent light. The geometrical parameters are the same as in Table 1.
5. Experimental results and discussion
5.1 Spatial separation of coherent and incoherent light
In the first step we investigated the influence of l on the radius of the central null-intensity area R. The laser was used as the only source, i.e., we did not use the halogen lamp.
Figure 16 shows the experimental results of the intensity distribution on the image plane for different l, detected by the beam analyzer. In order to show the details in the image patterns for l = 0 and l = 1, we magnified 2-fold the image of the intensity distribution patterns. We find the same linear relationship that we already observed from our simulation results. However, there is deviation between the computational and experimental results of the R value. The difference increases with increasing l, see Fig. 17. One possible reason is that the widths of the fine stripes in the center of the pattern of phase modulation function are comparably small for large l, i.e., for l = 80. If the widths of the stripes are equivalent to, or even smaller than, the size of the SLM unit, the phase modulation function will not be resolved correctly. On the contrary, when the widths of stripes are larger, i.e., for l = 10, the experimental result fits well with the simulation results.
Fig. 16. Experimental results of the intensity distribution of coherent light on the image plane for different topological charge l
Fig. 17. Comparison of computational and experimental results of R for different topological charge l
In the second step we investigated the influence of fF. Again, we did not use the halogen lamp. Figure 18 shows the experimental results, all of which for l = 10. Both experimental and computational results show an exponential relationship of R versus fF, see Fig. 19. Again, the deviation may be caused by the fine stripes of the phase modulation function pattern that cannot be resolved correctly on the SLM. The deviation increases from 45 µm to 77 µm, when fF decreases from 10000 mm to 1000 mm. Central peaks in the intensity distributions can be seen in all cases shown in Fig. 18. The relative intensity of the peaks is higher than what we find from our simulations. Still, the trend is consistent, i.e., we find an increase of the peak values for decreasing fF. The reason of the existence of the peak for high values of fF, i.e., for fF = 10000 mm, is that the laser we used for our experiments was not an ideal coherent source. It means both coherent and incoherent parts were present in the incident light. The incoherent part was not phase modulated, and was focused as a “spot” in the center of each of the images.
Fig. 18. Experimental results of intensity distribution of coherent light for different values of fF
We did not use the halogen lamp up to this point. In this next step, we used both laser source and the halogen lamp for the experiment. The optimized parameters we used for the experiment are shown in Table 2 and the results are shown in Fig. 20.
Fig. 20. Experimental results of spatial separation of coherent light and background radiation for l = 80 and fF = 2000mm
Figure 20(a) is the case with the coherent light only (laser), (b) with the incoherent light source only (halogen lamp), and (c) with both light sources. When we switched off the SLM in example (c), we obtained the results shown in (d). This latter case is the typical case of an ordinary detection system that does not separate coherent and incoherent light. In Fig. 20(a), we find a mean value of R = 407 µm. This value has been determined using a decrease of the signal strength by 1/e2. This value, again, is smaller than what we find from our simulations, see Fig. 13. The radius of the intensity distribution pattern of the incoherent light in Fig. 20(b) is 180µm. This value is significantly larger than the radius 13µm of an ideal Airy disk. It is mainly caused by the fact that the halogen lamp was an extended light source rather than an ideal point source, and the receiver FOV of the detector unit was not strictly limited. Besides, the aberrations of the optics were not considered in our simulation.
5.2 SNR
In the next step, we discuss the envelopes of the intensity distribution of both coherent and incoherent light. These envelopes are essential to the SNR improvement analysis. The intensity distributions for the cases of coherent light only (shown in Fig. 20(a)) and incoherent light only (shown in Fig. 20(b)) are related to the radial parameter r as
where ${\rho _{coherent}}({r,\varphi } )$ and ${\rho _{incoherent}}({r,\varphi } )$ are the light intensity density of coherent light and incoherent light, respectively.Figure 20(c) shows the results if we use the optical filter. The image plane is divided in two parts, the coherent channel and the incoherent channel. The radius of the incoherent channel is r0. The light intensity detected in each channel is given by
We can use a baffle of radius r0 to block the incoherent channel. In that case only the light in the coherent channel is detected. We define the SNR as the ratio of the coherent intensity to the incoherent intensity in the coherent channel.
Figure 20(d) shows the result for the case of a detection system that does not phase modulation, corresponding to r0 = 0. The SNR is given by
Figure 21 shows the angular integration results of Eq.(16) and Eq.(17). The solid line curves are the angular integration results, while the dotted ones (arbitrarily scaled) show the intensity profile of E-F and G-H in Fig. 20. We find two peaks on the solid line curve in the coherent channel. They correspond to the “ring” image of coherent light and the first-order diffraction ring, respectively. The curves clearly show the spatial separation of coherent and incoherent light. We can set the radius r0 to separate the two channels. A larger r0 results in a lower crosstalk caused by incoherent light, which in turn helps with increasing the SNR. But at the same time, the loss of coherent signal also increases, which leads to a low SNR. So, in order to find the best SNR, we need to optimize the parameter r0.
It is important to understand that the SNR curve is related to the ratio of the coherent and incoherent light intensity. For an example, if the detection system is used in daytime conditions, we recommend using a larger r0. More sunlight can be filtered out, though it will be at the cost of losing more signal from the coherent light. If it is used at night, or if the incoherent light noise is low, a smaller r0 is suitable as it allows us to detect as much coherent laser-light signals as possible.
Figure 22 shows the SNR curve calculated by Eq.(20). When r0 = 0, all the signals are detected in the coherent channel, and we find the SNR = 0.9. When r0 increases, the SNR value increases significantly, because the incoherent light is filtered out. When r0 is in the range of 260 µm-400 µm, we find a flat region of the SNR curve, where we find the mean value of SNR = 19 and the maximum value of SNR = 19.4. Thus, the SNR value increases by a factor of 21. The flat region on the curve is caused by the spatial gap of the intensity distributions, see Fig. 20(c) and Fig. 21. The larger the spatial separation of coherent and incoherent light, the larger the range of the flat region of the SNR curve we have. This results in a large tolerance range for dimensional and positioning errors of the baffle.
5.3 Transmission percentage and cross-talk
Finally, we calculate the transmission percentage and cross-talk in the coherent channel. We set r0 = 300µm, and find the transmission percentage of coherent laser signal
So, the coherent light loss is about 3.0%, which is mainly caused by the central peak of the intensity distribution we find in our simulations, and the fact that the light emitted by the laser source is not perfectly coherent.
The crosstalk by the incoherent light is calculated as
In our experiment, only 4.8% of total incoherent light are detected in the coherent channel. That means the incoherent light noise, i.e., background radiation, is suppressed by an order of magnitude. If we do not use the optical filter, i.e., no phase modulation is imposed by the SLM, the incoherent part of detected signal will be superimposed on the coherent part, as shown in Fig. 20(d). In that case, the crosstalk is 100%.
6. Conclusion
We presented a simulation model of an optical filter that is based on the use of an optical vortex. We applied Fourier optics in our simulation model. In order to have a larger spatial separation of coherent and incoherent light, we need to use a larger topological charge l and a virtual Fresnel lens with short focal length fF. The topological charge l shows a linear relationship regard to the radius of the central null-intensity area R. In contrast, fF shows an exponential relationship with R. The optimized parameters are l = 80 and fF = 2000 mm in our simulation example, and they can change according to the specific instrument parameters for different applications. Preliminary experiment confirms the simulation results. We find that the background radiation within the spectral filter bandwidth can be filtered out by an order of magnitude if we choose a proper incoherent channel radius r0. In our experiment, the SNR value increased from 0.9 to 19, i.e., by a factor of 21. The crosstalk of incoherent light can be reduced to 4.8% compared to the situation where we do not use this filter technique. In contrast, the loss of coherent light is approximately 3.0%. The values show the importance of using such a filter technology for, e.g., measurements under daytime conditions. The improved SNR helps to increase the working distance for lidar and optical communication systems.
Funding
National Natural Science Foundation of China (62105248).
Acknowledgments
The authors would like to express our appreciation to Prof. Detlef Müller for valuable suggestions for improving our manuscript.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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