The Poynting vectors, spin and orbital angular momentums of uniformly polarized cosh-Pearcey-Gauss beams in the far zone
doi: 10.37188/CO.EN.2022-0022
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Abstract:
We propose cosh-Pearcey-Gauss beams with uniform polarization, which are mainly modulated by a hyperbolic cosine function (
n ,Ω ) and the angles related to uniform polarization (α ,δ ). Based on angular spectrum representation and the stationary phase method, the Poynting vector, Spin Angular Momentums (SAM) and Orbital Angular Momentums (OAMs) in the far zone are studied. The results show that a largern orΩ in the hyperbolic cosine function can partition the longitudinal Poynting vectors, SAMs and OAMs into more multi-lobed parabolic structures. Different polarizations described by (α ,δ ) can distinguish their Poynting vectors and angular momentums between the TE and TM terms, though this does not affect the patterns of the whole beam. Furthermore, the weight of the left and right sides of longitudinal Poynting vectors, SAMs and OAMs in TE and TM terms can be modulated by left-handed or right-handed elliptical polarization, respectively. The results in this paper may be useful for information storage and polarization imaging.摘要:本文提出了均匀偏振cosh-Pearcey-Gauss 光束，其主要由双曲余弦函数(
n ,Ω )和偏振相关角度(α ,δ )所调制。基于矢量角谱法和稳相法，研究了该光束的远场坡印廷矢量、自旋角动量和轨道角动量。研究结果表明：较大的双曲余弦函数值能将远场坡印廷矢量、自旋角动量和轨道角动量分割成多瓣抛物线结构。虽然左旋和右旋椭圆偏振不能影响整个光束结构，但却可通过调节TE和TM项的左半边和右半边的分布权重，进而分辨出光束的远场坡印廷矢量和角动量分布。本文结果对信息存储与偏振成像技术领域有着潜在的应用价值。-
关键词:
- cosh-Pearcey-Gauss光束 /
- 自旋角动量 /
- 轨道角动量 /
- 坡印廷矢量
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1. Introduction
Three-dimensional surface shape measurement based on grating projection is widely applied in various fields such as computer vision, physical simulation, automatic detection, biology, and medicine. It has many advantages, such as high speed, high accuracy, non-contact, automation, etc[1-3]. Many scholars around the world have researched it and have achieved good results. For example, to reduce phase measurement errors caused by measuring glossy surfaces, Zhou et al. proposed a pixel-by-pixel combination multi-intensity matrix projection method. It has significantly reduced the number of projection operations and time consumption[4]. Due to the influence of too many projected patterns on phase unwrapping, Yang et al. proposed a high-speed measurement method suitable for three-dimensional shapes which uses only three high-frequency internal shift phase modes (70 cycles), which improved measurement accuracy and speed, and obtained wrapping phase and fringe order[5]. Using the gray code method, Lu et al. proposed a method based on staggered gray code light, which avoided step errors without projecting additional gray code modes[6]. Due to overexposure in optical three-dimensional measurement, phase information cannot be obtained reliably. Feng et al. put forward a highly reflective surface measurement method based on pixel-by-pixel modulation, thereby significantly improving the measurement speed and accuracy[7].
During the measurement process, due to the fact that the CCD (Charge Coupled Device) imaging system is not a general translation invariant system but a sampling imaging system with discrete characteristics, it will cause image distortion, resulting in spectra overlapping during the transformation process, and thus bringing errors to the measurement of the three-dimensional surface shape[8-10]. Many scholars have proposed better measurement methods to reduce or eliminate these errors and improve accuracy[11-13]. Due to the nonlinear effect of CCD on high-power laser wavefront detection, Du et al. proposed a method to reduce or eliminate spectra overlapping caused by the CCD nonlinear effect by increasing spatial carrier frequency[11]. Due to the important role played by the nonlinearity of scientific-level CCD in experimental processing, Cheng et al. used two different schemes to experimentally test the nonlinear characteristics of CCD. They achieved good experimental results[12]. Due to CCD’s nonlinear effects, spectra overlapping can occur when measuring complex optical three-dimensional surface shapes. Qiao et al. used the dual-frequency grating projection method to eliminate the nonlinear effects of CCD and improved measurement[13].
The following sections study the influence of sampling on the measurement of three-dimensional surface shapes and provide detailed reasoning and analysis of its basic principles. They also effectively validate the basic principle analysis by conducting simulations and experiments that achieve improved results.
2. Principle analysis
Fig. 1 shows the measurement system schematic diagram.
$ {P_1}{P_2} $ is the projector’s optical axis,$ {L_0} $ is the distance between the optical center$ {I_2} $ of the CCD imaging system and the reference plane,$A$ and$C$ are the points located on the reference plane, and$D$ is the point on the object surface,$ h $ is the distance from$D$ to the reference surface.
Figure 1. Schematic diagram of the measurement systemBy projecting the grating onto the surface of a three-dimensional object, the signal strength obtained by the CCD imaging system can be expressed as follows
g(x,y)=r(x,y)∞∑n=−∞anexp{j[2πnf0x+nϕ(x,y)]}=∞∑n=−∞qn(x,y)exp(j2πnf0x), (1) where
$r(x,y)$ is the non-uniform reflectivity of the object surface,$n$ is the Fourier series,${a_n}$ is the n-order Fourier coefficient of$g(x,y)$ , and$ {f_0} $ is the fundamental frequency of grating. And$\phi (x,y)$ is the phase of the object, where${q_n}(x,y) = {a_n}r(x,y) \exp [jn\varphi (x,y)]$ .By performing the fast Fourier transform on Eq. (1) and using
${\text{π}} $ phase-shifting technology[14] to eliminate the zero-order spectra components present in the frequency domain, the spectra expression containing the object height information can be obtained as followsG(fx,fy)=∫∞−∞∫∞−∞g(x,y)exp[−j2π(fxx+fyy)]dxdy=∞∑n=−∞Qn(fx−nf0,fy−nf0)+∞∑n=−∞Q∗n(fx−nf0,fy−nf0), (2) where
$G({f_x},{f_{\text{y}}})$ and${Q_n}({f_x},{f_y})$ are the spectra obtained by Fourier transform of$ g(x,y) $ and$ {q_n}(x,y) $ , respectively, and$Q_n^*({f_x},{f_y})$ is the conjugate complex of${Q_n}({f_x},{f_y})$ .Because CCD comprises an array of several small pixels arranged neatly and tightly with a certain geometric size, each CCD has approximately hundreds of thousands or even millions of pixels[15]. Let the pixel shape be a rectangle, represented by the function
$rect({x \mathord{\left/ {\vphantom {x {\Delta x}}} \right. } {\Delta x}},{y \mathord{\left/ {\vphantom {y {\Delta y}}} \right. } {\Delta y}})$ , where$ \Delta x $ and$\Delta y$ present the pixel’s dimensions in the direction of the x-axis and y-axis, respectively. Therefore, the signal strength of each pixel can be represented as the convolution of$ g(x,y) $ and$rect({x \mathord{\left/ {\vphantom {x {\Delta x}}} \right. } {\Delta x}},{y \mathord{\left/ {\vphantom {y {\Delta y}}} \right. } {\Delta y}})$ , its expression is as followsg′(x,y)=g(x,y)∗rect(xΔx,yΔy). (3) Using a comb function to sample Eq. (3), the discrete deformation fringe expression is obtained as follows
g″(x,y)=g′(x,y)comb(xΔx1,yΔy1)=[g(x,y)∗rect(xΔx,yΔy)]comb(xΔx1,yΔy1), (4) where
$ \Delta {x_1} $ and$ \Delta {y_1} $ present the sampling intervals of the fringes in the direction of the x-axis and y-axis, respectively.The spectrum function obtained by performing Fourier transform on Eq. (4) is:
G″(fx,fy)=G(fx,fy)+∞∑nx=−∞+∞∑ny=−∞δ(fx−nxΔx1,fy−nyΔy1)=1sl+∞∑nx=−∞+∞∑ny=−∞sinc(nxπs)sinc(nyπl)Q(fx−f0−nxΔx1,fy−f0−nyΔy1), (5) where
$ {n_x} $ and${n_y}$ present pixel points on the x-axis and y-axis, respectively,$s = {{\Delta {x_1}} \mathord{\left/ {\vphantom {{\Delta {x_1}} {(\Delta x + \Delta {x{'}})}}} \right. } {(\Delta x + \Delta {x{'}})}}$ and$l = {{\Delta {y_1}} \mathord{\left/ {\vphantom {{\Delta {y_1}} {(\Delta y + \Delta y')}}} \right. } {(\Delta y + \Delta y')}}$ present the number of sampling points for each fringe in the corresponding direction, respectively, and$ \Delta {x{'}} $ and$ \Delta {y{'}} $ present the pixel spacing in the corresponding direction, respectively.Thus, the spectra of the sampling function are the infinite repetition for the spectra of the primitive continuous function in the frequency domain. This is commonly known as "spectra island"[15-19]. As a result, in addition to
$ {f_0} $ , the higher-order spectral components, such as second and third order, are also generated.Due to the useful information containing changes in object height within
$ {f_0} $ of the spectra, a suitable low-pass filter must be designed to gain the$ {f_0} $ and remove the high-order spectra component.The low-pass filter is a modulation system of a point spread function. Its filtering process is the convolution process of the spectrum function and the point spread function
$ {G_1}({f_x},{f_y}) $ .$ {G_1}({f_x},{f_y}) $ in the frequency domain is presented by a Gaussian filter as followsG1(fx,fy)=12πexp(−fx2+fy22δ2), (6) where
$ \delta $ presents the standard deviation of the filter related to the degree of defocus.The spectra signal obtained through system defocusing is
G″′(fx,fy)=G″(fx,fy)⊗G1(fx,fy). (7) By filtering, the higher-order harmonics can be well separated from
$ {f_0} $ , so that only one of the$({1 / {sl}})\sin c({{\text{π}} /s})\sin c({{\text{π}}/ l})Q({f_x} - {f_0} - {1 / {\Delta {x_1}}},{f_y} - {f_0} - {1 /{\Delta {y_1}}})$ is retained after filtering out the higher-order harmonic components.Then, the inverse Fourier transform is enforced on Eq. (7) to reconstruct the signal strength taken by the CCD imaging system. The measurement system outputs the n-th sine fringe image, which is obtained by the CCD system, as shown below
g∧(x,y)=F−1[G″′(fx,fy)]=∞∑k= - ∞A∧kcos{k[2πf0x+ϕ(x,y)+δn]}, (8) where
$ {F^{ - 1}}[ \cdot ] $ presents inverse Fourier transform,$ A_k^{\land} $ presents the Fourier coefficient of$ g_n^{\land}(x,y) $ , and$ {\delta _n} $ presents the phase-shift amount,$ {\delta _n}{\text{ = }}{{2n{\text{π}} } /{{n_1}}} $ ,$ n = 1,2, \cdots , {n_1} $ .When the n-step phase-shift method is used, we can gain the phase as follows
ϕ∧(x,y)=arctan[N∑n=1g∧n(x,y)sin(δn)N∑n=1g∧n(x,y)cos(δn)], (9) where
$ {\phi ^ \wedge }(x,y) $ is the wrapped phase.Under the conditions of the telecentric projection optical path, considering
${L_o} \gg h(x,y)$ in the real measurement conditions,$h(x,y)$ and$ {\phi ^ \wedge }(x,y) $ will satisfy the relationsh(x,y)=−Lϕ∧(x,y)2πf0d. (10) It has been discussed in the literature [15] that in order to ensure the separation of the
$ {f_0} $ from other periodic spectra components and the separation of spectra components during the same period, the sampling condition$ m > 4 $ (where$m = {{\Delta f} \mathord{\left/ {\vphantom {{\Delta f} {{f_0}}}} \right. } {{f_0}}}$ ,$\Delta f$ presents sampling frequency) must be satisfied. This can avoid overlapping between the$ {f_0} $ and the higher-order spectra components, so as to accurately reconstruct the object shape measured. Otherwise, reconstruction is difficult.Lastly, according to
$s = {{\Delta {x_1}} \mathord{\left/ {\vphantom {{\Delta {x_1}} {{\text{(}}\Delta x + \Delta {x{'}}{\text{)}}}}} \right. } {{\text{(}}\Delta x + \Delta {x{'}}{\text{)}}}}$ and$l = {{\Delta {y_1}} \mathord{\left/ {\vphantom {{\Delta {y_1}} {(\Delta y + \Delta {y{'}})}}} \right. } {(\Delta y + \Delta {y{'}})}}$ , combining the relationship$\Delta {f_x} = {1 \mathord{\left/ {\vphantom {1 {\Delta {x_1}}}} \right. } {\Delta {x_1}}}$ as well as$\Delta {f_y} = {1 \mathord{\left/ {\vphantom {1 {\Delta y}}} \right. } {\Delta y}}$ between sampling frequency and sampling interval, it can be obtained that{mx=1Δx1f0=1s(Δx+Δx′)f0my=1Δy1f0=1l(Δy+Δy′)f0, (11) where
${m_x}$ and${m_y}$ present the sampling frequency ratio in the direction of x-axis and y-axis to${f_0}$ , respectively.It can be seen that the method of reducing the sampling interval, i.e., the number of sampling points per fringe, to increase
$ m $ , can be used to increase the accuracy of object surface shape measurements.3. Simulation and experiment
Simulation and experiment were executed to validate the basic principle analysis.
3.1 Simulation
We performed computer simulation verification on the analysis of basic principles. Assuming that the geometric parameter of the measurement system is
$ {{{L_0}} \mathord{\left/ {\vphantom {{{L_0}} d}} \right. } d} = 4 $ . The simulated object surface shape is shown in Fig. 2 (color online), with a size of 512×512 pixels.
Figure 2. The simulated object surface shapeWe projected a digital projector onto a simulated object and used a CCD camera system to obtain deformation fringes. If 40 fringes are taken, then
${f_0} = {{40} \mathord{\left/ {\vphantom {{40} {512}}} \right. } {512}}$ fringe/pixel. Using MATLAB to process the fringes, the sampling intervals of both the x-axis and y-axis directions were 8 pixels, thus,$m = 1.600\;0$ can be obtained from Eq. (11). Reducing the sampling interval of the fringes to 0.5 times the original sampling interval, i.e., the sampling interval was 4 pixels. From$s = {{\Delta {x_1}} \mathord{\left/ {\vphantom {{\Delta {x_1}} {{\text{(}}\Delta x + \Delta {x{'}}{\text{)}}}}} \right. } {{\text{(}}\Delta x + \Delta {x{'}}{\text{)}}}}$ and$l = {{\Delta {y_1}} \mathord{\left/ {\vphantom {{\Delta {y_1}} {(\Delta y + \Delta {y{'}})}}} \right. } {(\Delta y + \Delta {y{'}})}}$ , it can be seen that the number of sampling points for each fringe was 0.5 times that of the original, and from Eq. (11),$m = 3.200\;0$ can be obtained. It can be seen that neither of the two situations satisfied the sampling condition$ m > 4 $ . The obtained spectra diagrams are shown in Fig. 3 (a) and 3(b) (color online), respectively.
Figure 3. The spectra diagramsThen, we made the sampling points of each fringe
${1 \mathord{\left/ {\vphantom {1 4}} \right. } 4}$ and${1 \mathord{\left/ {\vphantom {1 8}} \right. } 8}$ times the original, i.e., the sampling interval was 2 pixels and 1 pixel, respectively.$m = 6.400\;0$ and$m = 12.800\;0$ can be obtained, respectively, indicating that the sampling condition$ m > 4 $ was satisfied. The obtained spectra diagrams are shown in Fig. 3 (c) and 3(d) (color online), respectively.In Figs. 3 (a) and 3(b), the components of
${f_0}$ in the spectra diagrams overlap with the higher-step frequency components because the sampling condition$ m > 4 $ was not satisfied. But in Figs. 3 (c) and 3(d), the corresponding frequency components are separated because the sampling condition was satisfied. In the four sub-figures, the smaller the number of sampling points, the larger the$m$ , and the better the separation effect.Fig. 4 shows the surface shape errors between the reconstructed and simulated objects in the above four situations.
Figure 4. The errors in the surface shape between the reconstructed and simulated objectsThe maximum absolute error value (MAEV) and average absolute error value (AAEV) of each sub-image in Fig. 4 are shown in Tab. 1. The MAEV obtained from the last three sampling intervals are 88.80%, 38.38%, and 31.50%, and the AAEV are 71.84%, 43.27%, and 32.26%, respectively, of the first sampling interval.
Table 1. Error between reconstructed object and simulated objectSampling interval 8 pixels 4 pixels 2 pixels 1 pixels MAEV 1.3246 1.1762 0.5084 0.4173 AAEV 0.4758 0.3418 0.2059 0.1535 | Show Table
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As shown by Fig. 4 and Tab. 1, when sampling condition
$ m > 4 $ is not satisfied, it is difficult to reconstruct the object’s surface shape and has relatively large errors. On the contrary, the error is relatively small. The larger$m$ , the smaller the error, but improving the resolution of the CCD imaging system is necessary.3.2 Experiment
To further validate the impact of CCD imaging system sampling on measuring three-dimensional surface shapes, an actual measurement experiment of a hemispherical object was carried out. As shown in Fig. 5, the simple experimental device system uses a digital projector and a low-distortion CCD camera.
Figure 5. The simple experimental device systemThe actual experiment used the same method as the computer simulation to obtain the deformation fringes of the experimental object. 600×480 pixels and 40 fringes were taken, the fundamental frequencies were
${f_x} = {{40} \mathord{\left/ {\vphantom {{40} {600}}} \right. } {600}}$ fringe/pixel and${f_y} = {{40} \mathord{\left/ {\vphantom {{40} {480}}} \right. } {480}}$ fringe/pixel, respectively. Using MATLAB to process the fringes, the sampling intervals of two directions were 8 pixels. From Eq. (11),${m_x} = 1.875\;0$ and${m_y} = 1.500\;0$ can be obtained.Using the same method as computer simulation, the fringes’ sampling interval was reduced to 0.5 times that of the original fringes, changing it to 4 pixels. Similarly,
${m_x} = 3.750\;0$ and${m_y} = 3.000\;0$ can be obtained from Eq. (11).Neither of the above situations satisfies the sampling condition
$ m > 4 $ . The reconstructed object surface shapes are shown in Fig. 6 (a) and 6(b) (color online).
Figure 6. Results of object surface shape reconstructed under different sampling conditionsWhen the sampling intervals for each fringe in the directions of the x-axis and y-axis are set to 2 pixels and 1 pixel, respectively, then
${m_x} = 5.625\;0$ ,${m_y} = 4.500\;0$ , and${m_x} = 7.500\;0$ ,${m_y} = 6.000\;0$ can be obtained, respectively. Both situations satisfy the sampling condition$ m > 4 $ . The reconstructed object surface shapes are shown in Fig. 6 (c) and 6(d) (color online).Comparing the experimental reconstruction results of the four sub-images in Fig. 6 yields the same conclusions as the computer simulation results mentioned above.
4. Conclusions
Due to the impact of sampling on the accuracy of three-dimensional surface shape measurements, the basic principles of spectra overlapping, spectra separation, and measurement accuracy caused by sampling were analyzed and discussed.
The comb function was used to sample the signal intensity of each pixel in the CCD, and discrete deformation fringes were obtained. After the Fourier transform of the deformation fringes, the "spectra island" is generated in the frequency domain. When the method of reducing the CCD sampling interval is used, i.e., the sampling frequency is increased to satisfy the sampling condition
$ m > 4 $ , then the adjacent " spectra islands" will not overlap with the${f_0}$ ; thus, reconstruction of the three-dimensional shape can be improved.Reducing the sampling interval of pixels causes the ratio
$m$ to increase. When the sampling conditions are satisfied, the larger the$m$ , the better the recovery of the object’s three-dimensional shape. However, further improvement in the resolution of the CCD imaging system is necessary.In the computer simulation and practical experiment, the smaller the sampling interval for each fringe, i.e., the fewer sampling points, the better the reconstruction of the object’s surface shape. When sampling condition
$ m > 4 $ is satisfied, it effectively reconstructs the object’s three-dimensional shape. -
Figure 1. The initial intensities and angular spectra of the cPeG beams with different initial polarization states for n=1, 2 and 4. (a), (d): α=π/4, δ=π/8; (b), (e): α=0, δ=0; (c), (f): α=π/4, δ=−3π/4. The phase distribution and polarization states are described in the top form left to right in Figs.1 (a)−(c). Blue: left-handed elliptical polarization; Red: right-handed elliptical polarization; Black: linear polarization. The parameters are x0=y0=100 μm and Ω=1.4
Figure 2. The theoretical design scheme of the cPeG beams with uniform polarization. BS: Beam Splitter; SLM: Spatial Light Modulator
Figure 3. Normalized longitudinal Poynting vector (backgrounds) and transversal Poynting vector (arrows) of cPeG beams for different uniform polarizations (α, δ) at z=50z0. (a), (d), (g): TE term; (b), (e), (h): TM term; (c), (f), (i): whole beam. The parameters are n=2 and Ω=1.4. The red point symbolizes topological charge l=+1, and the white point denotes l=−1
Figure 4. Normalized longitudinal Poynting vector (backgrounds) and transverse Poynting vector (arrows) of cPeG beams for different n and Ω at z=50z0, where the uniform polarization is (α, δ)=(π/4,π/8)
Figure 5. Normalized longitudinal SAM (3D and 2D) and transverse SAM (arrows) of cPeG beams at z=50z0. (a)−(c): α=π/4, δ=π/8; (d)−(f): α=π/4, δ=−3π/4. The other parameters are the same as those in Fig. 3
Figure 6. Normalized longitudinal (3D and 2D) SAM and transverse SAM (arrows) of cPeG beams (n=1, 4) at z=50z0. (a)−(c): n=1, Ω=3; (d)−(f): n=4, Ω=1. The other parameters are the same as those in Figs. 5 (α=π/4, δ=π/8)
Figure 7. Normalized longitudinal OAM (3D and 2D) and transverse OAM (arrows) of cPeG beams (Ω=1.4) at z=50z0. (a)−(c): α=π/4, δ=π/8; (d)-(f): α= 0, δ= 0; (g)−(i): α=π/4, δ=−3π/4; The other parameters are the same as those in Fig. 3. (n=2)
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