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SNR analysis with speckle noise in interferometry using monochromatic expanded source and fringe localization
Optics Communications 455 (2020) 124451
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SNR analysis with speckle noise in interferometry using monochromatic expanded source and fringe localization Wanqi Shang a,b , Wenxi Zhang a,b ,∗, Zhou Wu a,b , Yang Li a , Xinxin Kong a a b
Key Laboratory of Computational Optical Imaging Technology, Academy of Opto-Electronics, Chinese Academy of Sciences, Beijing 100094, China School of Optoelectronics, University of Chinese Academy of Sciences, Beijing 100049, China
ARTICLE Keywords: Speckle noise Signal-to-noise ratio Time scale factor Fringe localization
INFO
ABSTRACT As a strongly coherent light source, laser is widely used in various high-precision interferometry techniques. However, the coherence of laser also produces coherence noise. Laser coherence noise degrades the signal-tonoise (SNR) ratio and the interferogram resolution, leading to difficulties in data classification and information restoration. Here, we aim to derive the expressions of laser speckle contrast and SNR of interferograms through the theory of laser speckle noise. The factors affecting the speckle contrast and SNR of images are analyzed theoretically. The problem of fringe localization caused by dynamic scatter is discussed. We verify the relevant expression by using an experimental system and find that it can serve as a basis of laser interferometry to analyze measurement accuracy and suppress speckle noise.
1. Introduction Speckle is ubiquitous in laser-interference experiments. It degrades the signal-to-noise ratio (SNR) of interferograms and directly affects measurement accuracy. Suppressing the effects of laser speckle noise has become an important branch of interferometry. Such technologies as microlens array [1], deformable mirror, dynamic scatter [2–4], wavelength modulation, and angular diversity [5] are used to suppress these effects. Dynamic scatter and filtering algorithms are common methods of suppressing speckle patterns and increase measurement accuracy. Zhou et al. used power spectral density filtering to attenuate speckle noise [6]. Lee, Frost, and Kuan filtering are widely used to improve the SNR of remote-sensing images, but they are very time consuming and present obvious distortion. Wang proposed a method of using moving scatter to deduce coherent noise and applied it to commercial interferometers [7]. Farrkhi et al. also used electroactive optical diffusers to improve the SNR and suppress speckle noise in quantitative phase imaging [8]. After laser passes through a rotating diffuser, it can be regarded as pseudothermal light. Unlike conventional thermal light, pseudothermal light is a quasi-monochromatic light source generated by simulated radiation. Its phase randomly changes because of the surface undulation on ground glass. Speckle patterns are intensity variations caused by a large number of random phase walking and ultimately form light and dark dots on an image screen. The feasibility of using a rotating diffuser to reduce speckle contrast has been extensively verified; in particular, the speckle intensity-superposition theory proposed by Goodman is widely quoted [9]. The statistical model of ∗
speckle superposition with finite steps has been proposed [10], and the relationship between the pupil size of an imaging optical system and the speckle contrast has been confirmed. Wolf analyzed the related characteristics of Young’s interference of pseudothermal light, but this theory remains to be verified by experiments [11]. Reddy et al. studied the speckle correlation of partially coherent light passing through a dynamic scatter and concluded that this characteristic is related to the surface structure of scatter and the coherence of laser [12]. Li et al. proposed a new coherence theory for laser beams passing through a moving diffuser based on finite observation time; temporal coherence and spatial coherence were considered, but experimental data for a quantitative analysis were lacking [13]. Tiny particles or droplets in a colloidal dispersion are regarded as scattering units, which can time average the coherent noise by Brownian motion [14,15]. The dispersion of multimode optical fiber has also been utilized to produce a time delay and thus broaden the optical spectrum [16]. Using a spatially incoherent light source generated by random laser can obviously reduce speckle noise [17]. However, these methods are expensive and complex and cannot be widely used in interference systems. Numerous studies and experiments have shown that a moving diffuser can effectively improve the system accuracy of interferometry. Nevertheless, limited research has been conducted on the factors (i.e., integral time of matrix detector, rotating period of dynamic scatter, and numerical aperture of optical system) that influence the SNR in interferometry. In this study, the expression of SNR is proposed and an experimental system is built to verify the correctness of this formula. The problem of fringe localization caused by an expanded source is
Correspondence to: Academy of Opto-Electronics, Chinese Academy of Sciences, No.9 Dengzhuang South Road, Haidian District, 100094, Beijing, China. E-mail address: [email protected] (W. Zhang).
https://doi.org/10.1016/j.optcom.2019.124451 Received 28 March 2019; Received in revised form 20 August 2019; Accepted 24 August 2019 Available online 27 August 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
W. Shang, W. Zhang, Z. Wu et al.
Optics Communications 455 (2020) 124451
also confirmed quantitatively. This research can provide theoretical guidance to reduce coherent noise in interferometry and improve the SNR of interferograms. 2. Theoretical model 2.1. Speckle superposition In the stationary case, speckle contrast is expressed as 𝜎𝐼 , 𝐶𝑠𝑡𝑎𝑡𝑖𝑐 = 𝐼𝑚𝑒𝑎𝑛
(1)
Fig. 1. Correspondences between exposure time (a), F number (b) and SNR. In practice, the profile shown in (a) is constant when the exposure time is larger than diffuser rotating velocity.
where 𝐼𝑚𝑒𝑎𝑛 is the averaged intensity of speckle image, and 𝜎𝐼 is the standard deviation of speckle intensity; they are written as 𝐼𝑚𝑒𝑎𝑛 =
𝑁 1 ∑ 𝐼, 𝑁 𝑖=1 𝑖
√ √ 𝑁 √1 ∑ )2 ( 𝐼 − 𝐼𝑚𝑒𝑎𝑛 , 𝜎𝐼 = √ 𝑁 𝑖=1 𝑖
(2)
where ∅𝑑 is the random phase of diffuser, 𝑣 is the rotating velocity, 𝜔 is the laser frequency, and 𝜃 is the phase of laser beam. ∅𝑑 and 𝜃 satisfy the Gaussian distribution. Eq. (7) is substituted into Eq. (6), and the temporal autocorrelation function is
(3)
( ) 𝛤 𝛼1 , 𝛽1 , 𝑡; 𝛼2 , 𝛽2 , 𝑡 − 𝜏 = 𝑃 (0, 0, 𝑡) 𝑃 ∗ (0, 0, 𝑡 − 𝜏) ( ) ( ) ( ) ( ) = 𝑘 𝛼1 , 𝛽1 𝑘 𝛼2 , 𝛽2 𝐸 𝛼1 , 𝛽1 , 𝑡 𝐸 ∗ 𝛼2 , 𝛽2 , 𝑡 − 𝜏 ∬∞ ∬∞
where 𝑖 is the 𝑖th pixel on the interferogram, and 𝑁 is the total number of image pixels. When the laser beam actively illuminates the scatter, the phenomenon that a moving scatter causes the speckle patterns to change with time is called dynamic speckle [18]. The contrast of dynamic speckle is ( { ) ]})1∕2 [ ( 𝜏2 𝜏𝑐 2𝑇 𝐶𝑑𝑦𝑛𝑎𝑚𝑖𝑐 = 𝛽 + 𝑐 exp − −1 , (4) 𝑇 𝜏𝑐 2𝑇 2
∬∞
[ 2
]} 1 − 𝜇𝑑 (𝛥𝛼 − 𝑣𝜏, 𝛥𝛽)
𝐾 (𝛥𝛼, 𝛥𝛽) exp −𝜎𝑑 ∬ [ ]} exp −𝜎𝜃2 1 − 𝜇𝜃 (𝛥𝛼, 𝛥𝛽, 𝜏) 𝑑𝛥𝛼𝑑𝛥𝛽 {
where 𝜎𝑑 , 𝜇𝑑 denote the variance and mean of the scattering surface, respectively; 𝜎𝜃 , 𝜇𝜃 are the variance and mean of the phase fluctuation in laser. The fluctuation frequency of the laser is greater than the diffuser and is independent of position; hence, the second exponential term can be moved out of Eq. (8) as a constant. The correlation functions of the scattering surface and phase fluctuation are assumed to totally satisfy the negative exponential distribution, and the correlation function of laser can be expressed as [ ( ) ] 2 𝜏 , (9) 𝜇𝜃 (𝛥𝛼, 𝛥𝛽, 𝜏) = exp − 𝜏𝑙
where 𝜎∅ is the phase standard deviation of the scattering surface, 𝑣 is the linear velocity of the rotating diffuser, and 𝑟∅ is the radius of phase correlation area. The correlation between two points in this region decreases with increased distance between them, and the correlation between points outside the region can be neglected. 𝜏 is the observation time, and 𝜏𝑙 is the coherence time of laser. When setting 𝜏 = 0, the mutual coherence function only reveals the spatial coherence of laser beam that relates to the space coordinate (𝛥𝛼, 𝛥𝛽). The coherence time is generally shorter than the observation time; thus, the temporal coherence is determined by the laser linewidth factor. When the laser beam is monochromatic with an infinite coherence time, the temporal coherence is determined by the characteristic of scattering surface. Mutual coherence function shows the coherence characteristic when the laser passes through a rotating diffuser and is a significant parameter to define the correlation time of speckles. The coherence characteristic of the light is researched, and the speckle contrast and SNR of interferograms are analyzed. The laser beam is assumed to be a quasi-monochromatic partially coherent light, and the influence of the point spread function is considered; the image field can be written as [19] 𝑘 (𝑥 + 𝛼, 𝑦 + 𝛽) 𝐸 (𝛼, 𝛽, 𝑡) 𝑑𝛼𝑑𝛽,
{
2 = ||𝐸0 ||
where 𝛽 is a constant related to the system parameters, 𝜏𝑐 is the coherence time of speckle patterns, and 𝑇 is the integral time of a matrix detector. The contrast of dynamic speckle patterns is merely related to 𝑇 and 𝜏𝑐 when the entire system is fixed. Considering the laser coherence time and the fluctuation in scattering surface, the normalized mutual correlation function [13] is { [ { }]} ( ) (𝛥𝛼 − 𝑣𝜏)2 + 𝛥𝛽 2 𝜏2 2 ̃ 𝛾 = exp −𝜎𝜙 1 − exp − exp − , (5) 𝑟2𝜙 𝜏𝑙2
𝑃 (𝑥, 𝑦, 𝑡) =
(8)
,
𝑑𝛼1 𝑑𝛼2 𝑑𝛽1 𝑑𝛽2
where 𝜏 represents the integral time, and 𝜏𝑙 represents the laser coherence time. For general non-single-frequency narrow linewidth laser, 𝜏 ∼ 10−3 , 𝜏𝑙 ∼ 10−6 ; consequently, Eq. (9) tends to zero. Assuming 𝜎𝜃 ≪ 1, the third term of integration in Eq. (8) tends to one. Thus, Eq. (8) is simplified as 2 𝛤 (𝛥𝛼, 𝛥𝛽, 𝜏) = ||𝐸0 || 𝐾 (𝛥𝛼, 𝛥𝛽) ∬ { ∞ [ (
exp
−𝜎𝑑2 1 − exp −
(𝛥𝛼 − 𝑣𝜏)2 𝑟2𝑑
)]}
,
(10)
𝑑𝛥𝛼𝑑𝛥𝛽
where 𝑟𝑑 is the radius of phase correlation area of diffuser. 𝐾 (𝛥𝛼, 𝛥𝛽) is the deterministic autocorrection function of the amplitude point-spread function of the optical system; its Fourier transform is the squared magnitude of the pupil function of the lens, and its form is the same as 𝑘 and can be written as 𝐽 (𝜉) , (11) 𝐾 (𝛥𝛼, 𝛥𝛽) = 2𝐶 1 𝜉 where C is a constant, 𝐽1 is the first-order Bessel function, and 𝜉 is √ 𝜋𝐷 𝛥𝛼 2 + 𝛥𝛽 2 𝜉= . (12) 𝜆𝑧 The normalized autocovariance coherence function must be calculated to obtain the degree of coherence. The size of the first and second terms in Eq. (10) is different, and a comparative analysis is needed. Eq. (12) shows that the width of 𝐾 (𝛥𝛼, 𝛥𝛽) is approximately 𝜆𝑧∕𝐷, where 𝜆 is the laser wavelength, 𝑧 is the distance from pupil to
(6)
where 𝑘 represents the amplitude point-spread function of the optical system, and electric field 𝐸 is changing with time because of the phase fluctuation of laser and the moving diffuser. For simplicity, the reflectivity of the object is assumed to be constant, and 𝐸 is given by [ ] 𝐸 (𝛼, 𝛽, 𝑡) = 𝐸0 exp 𝑗𝜙𝑑 (𝛼 − 𝑣𝑡, 𝛽) exp {𝑗 [𝜔𝑡 − 𝜃 (𝛼, 𝛽, 𝑡)]} , (7) 2
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Optics Communications 455 (2020) 124451
time, and partial speckle patterns are related to one another. Thus, Eq. (18) should be revised as √ ⎧ 4𝜆𝑧𝐹 𝑇𝑟 ⎪ ,𝑇 > 𝑇 ⎪ 3𝜋 3 𝑓 ′ 𝑇 𝑅 𝑟 , (19) 𝐶𝑛 = ⎨ √ 4𝜆𝑧𝐹 ⎪ ⎪ 3𝜋 3 𝑓 ′ 𝑅 , 𝑇𝑟 ≤ 𝑇 ⎩ where 𝑅 is the distance from laser point to the rotation center, 𝑇𝑟 is the rotating period, and 𝑇 is the integral time. 2.2. SNR
Fig. 2. Fringe location of Mach–Zehnder optical path.
Interferograms are generally superposed coherent noise on the signal fringes, and thus, they can be simplified into a two-beam interference model. One beam is the signal light, and the other is the noise light. According to interference theory, the intensity is √ (20) 𝐼 = 𝐼𝑛 + 𝐼𝑠 + 2 𝐼𝑛 𝐼𝑠 cos (𝛥𝜑) ,
screen, and 𝐷 is the diameter of pupil. In consideration of the integral convergence, the normalized correlation radius of diffuser is [9] ] [ ( 2) ( ) 𝐸𝑖 𝜎𝑑 − ln 𝜎𝑑2 − 𝑒 , (13) 𝑟𝑐 = 𝑟𝑑 ( ) exp 𝜎𝑑2 − 1
where 𝐼𝑛 represents the intensity of coherent noise, and 𝐼𝑠 is the intensity of desired signal. We define 𝐼𝑠 ∕𝐼𝑛 = 𝑟, and the contrast is written as √ √ 2 𝐼𝑛 𝐼𝑠 2 𝑟 = . (21) 𝐶𝑠 = 𝐼𝑛 + 𝐼𝑠 1+𝑟
where e is the Euler–Mascheroni constant, and 𝐸𝑖 is the exponential integral that can be written as ( −𝜀 𝑡 𝑥 𝑡 ) ( ) 𝑒 𝑒 𝐸𝑖 𝜎𝑑2 = lim 𝑑𝑡 + 𝑑𝑡 . (14) ∫𝜀 𝑡 𝜀→+0 ∫−∞ 𝑡
The ratio of signal contrast to speckle contrast is expressed as SNR = 𝐶𝑆 ∕𝐶𝑛 , which is called SNR of the interferogram. Eqs. (19) and (21) are substituted into the above expression, and the final equation of SNR is
In general case, 𝜆𝑧∕𝐷 ∼ μm ≫ 𝑟𝑐 ∼ nm; the final normalized autocovariance function [20] can be expressed as 𝜇𝑐 (𝜏) =
𝐽 (𝜌) 𝐾 (𝑣𝜏) =2 1 , 𝐾 (0) 𝜌
(15) √ ⎧ 3𝜋 3 𝑟𝑓 ′ 𝑡𝑣 𝑅 ⎪ 2 , 𝑡𝑣 < 1 ⎪1 + 𝑟 SNR = ⎨ , √ 4𝜆𝑧𝐹 ⎪ 2 3𝜋 3 𝑟𝑓 ′ 𝑅 , 𝑡𝑣 ≥ 1 ⎪ 4𝜆𝑧𝐹 ⎩1 + 𝑟
where 𝜋𝐷𝑣𝜏 . 𝜆𝑧
𝜌=
(16)
The correlation time of the speckle intensity is ∞
𝜏𝑐 =
|𝜇 (𝜏)|2 𝑑𝜏 = 8𝜆𝑧 . ∫−∞ | 𝑐 | 3𝜋 2 𝑣𝐷
(22)
where 𝑡𝑣 = 𝑇 ∕𝑇𝑟 denotes the time scale factor, which is used to characterize the relationship between observation time and the period of rotating diffuser. When 𝑡𝑣 ≥ 1 (partial speckle patterns are related to one another), SNR is independent of 𝑡𝑣 and changes with other system parameters. When 𝑡𝑣 < 1, Eq. (22) mainly embodies that SNR is influenced by the time scale factor and the F number of lens. As shown in Fig. 1(a), the SNR gradually increases with increased 𝑡𝑣 , the same as with increased exposure time and rotating velocity. Thus, the reasonable configuration between the two characters is necessary. In the case of a smaller F number (larger pupil), the limited optical depth of field causes the speckle patterns to be blurred, which improves the SNR of interferogram. Fig. 1(b) reveals the relationship between the SNR and the F number; a large F number indicates a low SNR.
(17)
The intensity superposition of multiple speckle patterns in the time domain can significantly reduce the coherent noise, and the speckle contrast is presented as √ √ 𝜏𝑐 1 8𝜆𝑧 𝐶= √ = = . (18) 2 𝑣𝐷𝑇 𝑇 3𝜋 𝑁 The relationship between the motion period of the diffuser and the integral time of a matrix detector is considered, and the F number that can be written as 𝐹 = 𝑓 ′ ∕𝐷 is substituted, where 𝑓 ′ is the focus length of lens. When 𝑇𝑟 ≤ 𝑇 , the diffuser rotates over one round in this integral
Fig. 3. Schematic of an experimental setup, F is a neutral density filter; M1, M2, M3 are mirrors; RD is a rotating ground glass(diffuser); L is an aspheric condenser lens; BS1 and BS2 are beam splitters.
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Optics Communications 455 (2020) 124451
Fig. 4. Fringes on target. (a) laser, (b) with rotating diffuser.
Fig. 5. Variation in the speckle contrast of images with exposure time and F number; the color of the histogram from blue to yellow indicates that F number is from 0.95 to 16, and dotted lines are shown in legend. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Comparison of experimental data-fitting result with theoretical simulation. (a), (b), (c), (d) are the curves in different 𝑡v values.
𝐷 along the optical axis from P0 , the two beams are separated by a distance along the vertical direction, which can be written as
2.3. Fringe location
𝛥𝑑 = 𝐷𝜃 = 𝐷
The rotating diffuser smoothens the speckle noise in finite exposure time. After the laser passes through a diffuser, the original highly spatial coherence point source becomes a partially coherent plane extension light source. With this source of finite size, localized fringes can observed with a Mach–Zehnder interferometer (see Fig. 2). In consideration of the case in which the interferometer is illuminated by this circular and expanded source, the partially coherent light is approximately collimated by lens L, the two mirrors are adjusted to let the center of localization be at P0 , and the angle between the interfering beams is 𝜃. At position P, which is located at a distance
𝜆 , 2𝑒
(23)
where 𝑒 is the fringe width at position P, and 𝜆 is the wavelength of this quasi-monochromatic expanded source. The degree of coherence can be given by [21] ) ) ( ( 2𝐽1 𝑘𝑅 𝛥𝑑 2𝐽1 𝜋𝐷𝑟 𝑓 𝑒𝑓 𝛾= = , (24) 𝜋𝐷𝑟 𝑘𝑅 𝛥𝑑 𝑓 𝑒𝑓 where 𝑘 is the wavenumber, 𝑅 is the radius of expanded source, 𝑓 is the focal length of collimated lens, and 𝐽1 is the first-order Bessel function.
Fig. 6. Trend of speckle contrast (a) and SNR (b) with different exposure times and F numbers.
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Optics Communications 455 (2020) 124451
Fig. 8. Origin 2D data and FFT analysis of 1D data. (a) Interference fringe in different positions caused by localization; (b) the intensity of central line in different distance; (c) FFT of central line of different image; (d)–(f) 2560th row, 1280th row, 3840th row of three images in different localized positions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Parameter configuration. Exposure time
10 ms, 30 ms, 50 ms, 80 ms, 100 ms
F number
0.95, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16
SNR of the entire system. When the rotating velocity is constant, the correspondence between the time proportional coefficient and the SNR is studied by changing the integral time. The large aperture, which has a small value of point-spread function, can obviously smoothen speckle patterns. A large radius leads to a small localized depth due to the destruction of spatial coherence caused by an expanded light source. A diode-pumped solid-state laser (DJ532-10, 10 mw, 532 nm, Thorlabs, Inc.) is used for this experiment. The experimental setup is shown in Fig. 3. The laser beam passing through a round variable ND filter (NDC-50C-2M-A, diameter 50 mm, optical density 0–2.0) can be adjusted continually. A ground glass diffuser (DFB1-50C02-600, roughness #600, Sigma, Inc.) mounted on a step motor can scatter this beam, and the velocity is changeable with input voltage. An aspheric condenser lens (ACL7560U-A, diameter 75 mm, focal length 60 mm, NA 0.61, Thorlabs, Inc.) is used to collect and collimate the scattering light. An iris diaphragm (D25S, max aperture 25 mm, Thorlabs, Inc.) is used to control the diameter of the laser beam. When beams go through the Mach–Zehnder interferometer, interference fringes can be captured on the target printed with logo and text. A CCD camera (LXC-250, resolution 5120 × 5120, pixel size 4.5 μm, max rate 32 fps, Baumer, GmbH.) connected with graphics workstation can record images in different parameters. The two mirrors on the two arms of Mach–Zehnder interferometer are adjusted, and the image is monitored in real time through a computer screen. The plane of localized fringes should be located on target, then the zoom ring of the lens is turned to make the edges of
The experimental results in the next section quantitatively verify the varying trend of 𝛾. 3. Experiment configuration The Mach–Zehnder interferometer is built to verify the effects of speckle reduction and SNR improvement, as well as the localized fringe caused by the destruction of spatial coherence with rotating diffuser. The speckle contrast and SNR are analyzed in different conditions of F number and exposure time in case of fixed working distance. Before the experiment, whether the depth of field can meet the demand at a maximum aperture (minimum F number) and clearly distinguish the target details is determined. The parameters in Table 1 are set with existing hardware and conditions to study the effect of exposure time and F number of active imaging lens on speckle contrast and SNR of interferogram. Improving the rotating velocity of diffuser or exposure time is equivalent to the climbing of time scale factor, which can effectively suppress the coherent noise in the interferometry and improve the 5
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Optics Communications 455 (2020) 124451
of speckle contrast and exposure time when the F number is constant. Speckle noise can be significantly reduced by increasing the exposure time in a small F number. Curves approach flatness in the situation of large F number. A 2D difference method is used to construct part of the data to reflect the macroscopic trend because of the limited amount of data. Fig. 6 illustrates the combined influence of F number and integral time on speckle and SNR. The speckle contrast is lowest under the conditions of long exposure time and small F number (Fig. 6(a)). Fig. 6(b) illustrates that SNR can remain at a high level when F number is small, and the trend is smooth on the central region of this map. The trend becomes worse in the situation of large F number and short exposure time. The data of 30, 50, 80, and 100 ms are fitted by the power function model, and the theoretical curve is simulated by setting different time proportional coefficients and 𝑟 values (the ratio of the signal intensity to the noise intensity). Most of system parameters in Eq. (22) are proportional coefficients; hence, all is normalized to compare fitting data with simulation data. The integral time is adjusted in four times and the rotating period of a dynamic diffuser is fixed. We can get 𝑡𝑣 = 0.015, 0.025, 0.04, 0.05 and the corresponding r = 10, 50, 100, 200 which illustrates the ratio between intensity of signal and intensity of noise. Base on these theoretical parameters, we can numerically simulate and plot curves by dashed lines (Fig. 7). The range of data along the vertical axis gradually increases from Fig. 7(a) to (d), and the conclusion is that increasing 𝑡𝑣 benefits the improvement in the SNR of the entire system. The experimental data is consistent with the theoretical simulation curve, which verifies the correctness of Eq. (22). When the laser beam is illuminated onto the scattering surface and has a certain diameter, a circular extended source is formed. Each illuminating element is equivalent to a wave source, and the spatial dispersion of each other causes the spatial coherence to be broken into partially coherent light source, which eventually leads to localized fringes. Fig. 8 shows the interferogram in different distances and the Fourier analysis of these 1D data. The contrast of interference fringes is optimal at 0 mm (Fig. 9(c) and (d)), and fringes are distributed throughout the full field. With increased distance, the fringe contrast (intensity of FFT) of the 2560th and 3840th rows obviously decreases, indicating that the degree of coherence gradually degrades from 0 mm to 60 mm. The interferograms at different positions are shown in Fig. 9. The contrast is best at 590 mm, and the localized depth is approximately 120 mm. The fringe contrast of red and green dots on the line is approximately 0.65, and the position is 590 mm. The distance between the two positions is approximately 60 mm, and the fringe width is approximately 120 pixels (e = 120 × 5.5 μm = 0.66 mm). The radius of an equivalent surface source is 350 μm because of the aperture. The initial parameters are integrated into Eq. (24) for simulation, 𝛥𝐷𝑠 = 657.3 mm − 590 mm = 67.3 mm (see Fig. 10); system errors in this experiment should be considered. The above experimental results are consistent with the theoretical values. Localized depth of the interference fringes is quantitatively verified using the spatially extended light source.
Fig. 9. Interferogram in different positions and data of central line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 10. Theoretical simulation result.
image and text clear and sharp. Changing the aperture of lens (F/# 0.95 the first time), rotating the variable filter, and setting proper velocity of the diffuser were performed to ensure that images are not overexposed. After the multiple data of exposure time in current F number are recorded, the above steps are repeated by changing the F number. 4. Result and analysis A comparison of the two images in Fig. 4 shows that the speckle noise is obvious without rotating ground (Fig. 4(a)), and the texts below that logo are difficult to recognize. When the laser beam illuminates onto scattering surface, it is regarded as an extended and circular source phase modulated randomly and has an averaging effect in a period of time. Coherent noise is effectively restrained, and the fringe contrast is slightly reduced (Fig. 4(b)). The experimental data are processed, which contains 45 (9 of F number × 5 of exposure time) images. The speckle contrast and SNR can be revealed from them. Each of histogram clusters in Fig. 5 shows the relationship between the speckle contrast and the F number of lens at a constant integral time. The speckle contrast is sensitive to F number in a long exposure time. Broken lines indicate the correlation curve
5. Conclusion In this study, many factors that affect the SNR of interference system are researched; the quantitative formula is given; and the correctness of the correlation parameters, such as integral time, period of rotating diffuser, and F number of lens, is also confirmed. Research shows that increasing the time proportional coefficient and reducing the F number of lens can effectively improve the SNR of interferograms. This approach can be applied in interferometry, coherent tomography measurement, fringe projection for 3D measurement, and quantitative phase imaging to suppress speckle noise and improve measurement accuracy. 6
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Optics Communications 455 (2020) 124451
Acknowledgments
[9] J.W. Goodman, Statistical Optics, John Wiley & Sons, 2015. [10] J.W. Goodman, Speckle with a finite number of steps, Appl. Opt. 47 (2008) A111–A118. [11] E. Wolf, Coherence of two interfering beams modulated by a uniformly moving diffuser, J. Modern Opt. 47 (2000) 1569–1573. [12] G. Reddy, V. Rao, Correlation of speckle patterns generated by a diffuser illuminated by partially coherent light, Opt. Acta: Int. J. Opt. 30 (1983) 1213–1216. [13] G. Li, Y. Qiu, H. Li, Coherence theory of a laser beam passing through a moving diffuser, Opt. Express 21 (2013) 13032–13039. [14] F. Riechert, G. Bastian, U. Lemmer, Laser speckle reduction via colloidaldispersion-filled projection screens, Appl. Opt. 48 (2009) 3742–3749. [15] B. Redding, G. Allen, E.R. Dufresne, H.J.A.o. Cao, Low-loss high-speed speckle reduction using a colloidal dispersion, Appl. Opt. 52 (2013) 1168–1172. [16] J.G. Manni, J.W. Goodman, Versatile method for achieving 1% speckle contrast in large-venue laser projection displays using a stationary multimode optical fiber, Opt. Express 20 (2012) 11288–11315. [17] B. Redding, M.A. Choma, H. Cao, Spatial coherence of random laser emission, Opt. Lett. 36 (2011) 3404–3406. [18] D. Briers, D.D. Duncan, E.R. Hirst, S.J. Kirkpatrick, M. Larsson, W. Steenbergen, T. Stromberg, O.B. Thompson, Laser speckle contrast imaging: theoretical and practical limitations, J. Biomed. Opt. 18 (2013) 066018. [19] J.W. Goodman, Speckle Phenomena in Optics: Theory and Applications, Roberts and Company Publishers, 2007. [20] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge university press, 1995. [21] M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Elsevier, 2013.
This work was supported by Youth Innovation Promotion Association, CAS (No.2015127) and National Natural Science Foundation of China (Grant No. 61605217). References [1] J. Pauwels, G. Verschaffelt, Speckle reduction in laser projection using microlens-array screens, Opt. Express 25 (2017) 3180–3195. [2] Y. Kuratomi, K. Sekiya, H. Satoh, T. Tomiyama, T. Kawakami, B. Katagiri, Y. Suzuki, T. Uchida, Speckle reduction mechanism in laser rear projection displays using a small moving diffuser, J. JOSA A 27 (2010) 1812–1817. [3] H.-A. Chen, J.-W. Pan, Z.-P. Yang, Speckle reduction using deformable mirrors with diffusers in a laser pico-projector, Opt. Express 25 (2017) 18140–18151. [4] Z. Cui, A.-t. Wang, Z. Wang, S.-l. Wang, C. Gu, H. Ming, C.-q. Xu, Speckle suppression by controlling the coherence in laser based projection systems, J. Disp. Technol. 11 (2015) 330–335. [5] H. Yamada, K. Moriyasu, H. Sato, H. Hatanaka, Effect of incidence/observation angles and angular diversity on speckle reduction by wavelength diversity in laser projection systems, Opt. Express 25 (2017) 32132–32141. [6] L. Zhou, J. Gan, X. Liu, L. Xu, W. Lu, Speckle-noise-reduction method of projecting interferometry fringes based on power spectrum density, Appl. Opt. 51 (2012) 6974–6978. [7] S. Wang, A contactless method for characterization of diffusers for coherent noise reduction, in: Optical Fabrication and Testing, Optical Society of America, 2010), p. JMB4. [8] H. Farrokhi, J. Boonruangkan, B.J. Chun, T.M. Rohith, A. Mishra, H.T. Toh, H.S. Yoon, Y.-J. Kim, Speckle reduction in quantitative phase imaging by generating spatially incoherent laser field at electroactive optical diffusers, Opt. Express 25 (2017) 10791–10800.
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SNR analysis with speckle noise in interferometry using monochromatic expanded source and fringe localization Wanqi Shang a,b , Wenxi Zhang a,b ,∗, Zhou Wu a,b , Yang Li a , Xinxin Kong a a b
Key Laboratory of Computational Optical Imaging Technology, Academy of Opto-Electronics, Chinese Academy of Sciences, Beijing 100094, China School of Optoelectronics, University of Chinese Academy of Sciences, Beijing 100049, China
ARTICLE Keywords: Speckle noise Signal-to-noise ratio Time scale factor Fringe localization
INFO
ABSTRACT As a strongly coherent light source, laser is widely used in various high-precision interferometry techniques. However, the coherence of laser also produces coherence noise. Laser coherence noise degrades the signal-tonoise (SNR) ratio and the interferogram resolution, leading to difficulties in data classification and information restoration. Here, we aim to derive the expressions of laser speckle contrast and SNR of interferograms through the theory of laser speckle noise. The factors affecting the speckle contrast and SNR of images are analyzed theoretically. The problem of fringe localization caused by dynamic scatter is discussed. We verify the relevant expression by using an experimental system and find that it can serve as a basis of laser interferometry to analyze measurement accuracy and suppress speckle noise.
1. Introduction Speckle is ubiquitous in laser-interference experiments. It degrades the signal-to-noise ratio (SNR) of interferograms and directly affects measurement accuracy. Suppressing the effects of laser speckle noise has become an important branch of interferometry. Such technologies as microlens array [1], deformable mirror, dynamic scatter [2–4], wavelength modulation, and angular diversity [5] are used to suppress these effects. Dynamic scatter and filtering algorithms are common methods of suppressing speckle patterns and increase measurement accuracy. Zhou et al. used power spectral density filtering to attenuate speckle noise [6]. Lee, Frost, and Kuan filtering are widely used to improve the SNR of remote-sensing images, but they are very time consuming and present obvious distortion. Wang proposed a method of using moving scatter to deduce coherent noise and applied it to commercial interferometers [7]. Farrkhi et al. also used electroactive optical diffusers to improve the SNR and suppress speckle noise in quantitative phase imaging [8]. After laser passes through a rotating diffuser, it can be regarded as pseudothermal light. Unlike conventional thermal light, pseudothermal light is a quasi-monochromatic light source generated by simulated radiation. Its phase randomly changes because of the surface undulation on ground glass. Speckle patterns are intensity variations caused by a large number of random phase walking and ultimately form light and dark dots on an image screen. The feasibility of using a rotating diffuser to reduce speckle contrast has been extensively verified; in particular, the speckle intensity-superposition theory proposed by Goodman is widely quoted [9]. The statistical model of ∗
speckle superposition with finite steps has been proposed [10], and the relationship between the pupil size of an imaging optical system and the speckle contrast has been confirmed. Wolf analyzed the related characteristics of Young’s interference of pseudothermal light, but this theory remains to be verified by experiments [11]. Reddy et al. studied the speckle correlation of partially coherent light passing through a dynamic scatter and concluded that this characteristic is related to the surface structure of scatter and the coherence of laser [12]. Li et al. proposed a new coherence theory for laser beams passing through a moving diffuser based on finite observation time; temporal coherence and spatial coherence were considered, but experimental data for a quantitative analysis were lacking [13]. Tiny particles or droplets in a colloidal dispersion are regarded as scattering units, which can time average the coherent noise by Brownian motion [14,15]. The dispersion of multimode optical fiber has also been utilized to produce a time delay and thus broaden the optical spectrum [16]. Using a spatially incoherent light source generated by random laser can obviously reduce speckle noise [17]. However, these methods are expensive and complex and cannot be widely used in interference systems. Numerous studies and experiments have shown that a moving diffuser can effectively improve the system accuracy of interferometry. Nevertheless, limited research has been conducted on the factors (i.e., integral time of matrix detector, rotating period of dynamic scatter, and numerical aperture of optical system) that influence the SNR in interferometry. In this study, the expression of SNR is proposed and an experimental system is built to verify the correctness of this formula. The problem of fringe localization caused by an expanded source is
Correspondence to: Academy of Opto-Electronics, Chinese Academy of Sciences, No.9 Dengzhuang South Road, Haidian District, 100094, Beijing, China. E-mail address: [email protected] (W. Zhang).
https://doi.org/10.1016/j.optcom.2019.124451 Received 28 March 2019; Received in revised form 20 August 2019; Accepted 24 August 2019 Available online 27 August 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.
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Optics Communications 455 (2020) 124451
also confirmed quantitatively. This research can provide theoretical guidance to reduce coherent noise in interferometry and improve the SNR of interferograms. 2. Theoretical model 2.1. Speckle superposition In the stationary case, speckle contrast is expressed as 𝜎𝐼 , 𝐶𝑠𝑡𝑎𝑡𝑖𝑐 = 𝐼𝑚𝑒𝑎𝑛
(1)
Fig. 1. Correspondences between exposure time (a), F number (b) and SNR. In practice, the profile shown in (a) is constant when the exposure time is larger than diffuser rotating velocity.
where 𝐼𝑚𝑒𝑎𝑛 is the averaged intensity of speckle image, and 𝜎𝐼 is the standard deviation of speckle intensity; they are written as 𝐼𝑚𝑒𝑎𝑛 =
𝑁 1 ∑ 𝐼, 𝑁 𝑖=1 𝑖
√ √ 𝑁 √1 ∑ )2 ( 𝐼 − 𝐼𝑚𝑒𝑎𝑛 , 𝜎𝐼 = √ 𝑁 𝑖=1 𝑖
(2)
where ∅𝑑 is the random phase of diffuser, 𝑣 is the rotating velocity, 𝜔 is the laser frequency, and 𝜃 is the phase of laser beam. ∅𝑑 and 𝜃 satisfy the Gaussian distribution. Eq. (7) is substituted into Eq. (6), and the temporal autocorrelation function is
(3)
( ) 𝛤 𝛼1 , 𝛽1 , 𝑡; 𝛼2 , 𝛽2 , 𝑡 − 𝜏 = 𝑃 (0, 0, 𝑡) 𝑃 ∗ (0, 0, 𝑡 − 𝜏) ( ) ( ) ( ) ( ) = 𝑘 𝛼1 , 𝛽1 𝑘 𝛼2 , 𝛽2 𝐸 𝛼1 , 𝛽1 , 𝑡 𝐸 ∗ 𝛼2 , 𝛽2 , 𝑡 − 𝜏 ∬∞ ∬∞
where 𝑖 is the 𝑖th pixel on the interferogram, and 𝑁 is the total number of image pixels. When the laser beam actively illuminates the scatter, the phenomenon that a moving scatter causes the speckle patterns to change with time is called dynamic speckle [18]. The contrast of dynamic speckle is ( { ) ]})1∕2 [ ( 𝜏2 𝜏𝑐 2𝑇 𝐶𝑑𝑦𝑛𝑎𝑚𝑖𝑐 = 𝛽 + 𝑐 exp − −1 , (4) 𝑇 𝜏𝑐 2𝑇 2
∬∞
[ 2
]} 1 − 𝜇𝑑 (𝛥𝛼 − 𝑣𝜏, 𝛥𝛽)
𝐾 (𝛥𝛼, 𝛥𝛽) exp −𝜎𝑑 ∬ [ ]} exp −𝜎𝜃2 1 − 𝜇𝜃 (𝛥𝛼, 𝛥𝛽, 𝜏) 𝑑𝛥𝛼𝑑𝛥𝛽 {
where 𝜎𝑑 , 𝜇𝑑 denote the variance and mean of the scattering surface, respectively; 𝜎𝜃 , 𝜇𝜃 are the variance and mean of the phase fluctuation in laser. The fluctuation frequency of the laser is greater than the diffuser and is independent of position; hence, the second exponential term can be moved out of Eq. (8) as a constant. The correlation functions of the scattering surface and phase fluctuation are assumed to totally satisfy the negative exponential distribution, and the correlation function of laser can be expressed as [ ( ) ] 2 𝜏 , (9) 𝜇𝜃 (𝛥𝛼, 𝛥𝛽, 𝜏) = exp − 𝜏𝑙
where 𝜎∅ is the phase standard deviation of the scattering surface, 𝑣 is the linear velocity of the rotating diffuser, and 𝑟∅ is the radius of phase correlation area. The correlation between two points in this region decreases with increased distance between them, and the correlation between points outside the region can be neglected. 𝜏 is the observation time, and 𝜏𝑙 is the coherence time of laser. When setting 𝜏 = 0, the mutual coherence function only reveals the spatial coherence of laser beam that relates to the space coordinate (𝛥𝛼, 𝛥𝛽). The coherence time is generally shorter than the observation time; thus, the temporal coherence is determined by the laser linewidth factor. When the laser beam is monochromatic with an infinite coherence time, the temporal coherence is determined by the characteristic of scattering surface. Mutual coherence function shows the coherence characteristic when the laser passes through a rotating diffuser and is a significant parameter to define the correlation time of speckles. The coherence characteristic of the light is researched, and the speckle contrast and SNR of interferograms are analyzed. The laser beam is assumed to be a quasi-monochromatic partially coherent light, and the influence of the point spread function is considered; the image field can be written as [19] 𝑘 (𝑥 + 𝛼, 𝑦 + 𝛽) 𝐸 (𝛼, 𝛽, 𝑡) 𝑑𝛼𝑑𝛽,
{
2 = ||𝐸0 ||
where 𝛽 is a constant related to the system parameters, 𝜏𝑐 is the coherence time of speckle patterns, and 𝑇 is the integral time of a matrix detector. The contrast of dynamic speckle patterns is merely related to 𝑇 and 𝜏𝑐 when the entire system is fixed. Considering the laser coherence time and the fluctuation in scattering surface, the normalized mutual correlation function [13] is { [ { }]} ( ) (𝛥𝛼 − 𝑣𝜏)2 + 𝛥𝛽 2 𝜏2 2 ̃ 𝛾 = exp −𝜎𝜙 1 − exp − exp − , (5) 𝑟2𝜙 𝜏𝑙2
𝑃 (𝑥, 𝑦, 𝑡) =
(8)
,
𝑑𝛼1 𝑑𝛼2 𝑑𝛽1 𝑑𝛽2
where 𝜏 represents the integral time, and 𝜏𝑙 represents the laser coherence time. For general non-single-frequency narrow linewidth laser, 𝜏 ∼ 10−3 , 𝜏𝑙 ∼ 10−6 ; consequently, Eq. (9) tends to zero. Assuming 𝜎𝜃 ≪ 1, the third term of integration in Eq. (8) tends to one. Thus, Eq. (8) is simplified as 2 𝛤 (𝛥𝛼, 𝛥𝛽, 𝜏) = ||𝐸0 || 𝐾 (𝛥𝛼, 𝛥𝛽) ∬ { ∞ [ (
exp
−𝜎𝑑2 1 − exp −
(𝛥𝛼 − 𝑣𝜏)2 𝑟2𝑑
)]}
,
(10)
𝑑𝛥𝛼𝑑𝛥𝛽
where 𝑟𝑑 is the radius of phase correlation area of diffuser. 𝐾 (𝛥𝛼, 𝛥𝛽) is the deterministic autocorrection function of the amplitude point-spread function of the optical system; its Fourier transform is the squared magnitude of the pupil function of the lens, and its form is the same as 𝑘 and can be written as 𝐽 (𝜉) , (11) 𝐾 (𝛥𝛼, 𝛥𝛽) = 2𝐶 1 𝜉 where C is a constant, 𝐽1 is the first-order Bessel function, and 𝜉 is √ 𝜋𝐷 𝛥𝛼 2 + 𝛥𝛽 2 𝜉= . (12) 𝜆𝑧 The normalized autocovariance coherence function must be calculated to obtain the degree of coherence. The size of the first and second terms in Eq. (10) is different, and a comparative analysis is needed. Eq. (12) shows that the width of 𝐾 (𝛥𝛼, 𝛥𝛽) is approximately 𝜆𝑧∕𝐷, where 𝜆 is the laser wavelength, 𝑧 is the distance from pupil to
(6)
where 𝑘 represents the amplitude point-spread function of the optical system, and electric field 𝐸 is changing with time because of the phase fluctuation of laser and the moving diffuser. For simplicity, the reflectivity of the object is assumed to be constant, and 𝐸 is given by [ ] 𝐸 (𝛼, 𝛽, 𝑡) = 𝐸0 exp 𝑗𝜙𝑑 (𝛼 − 𝑣𝑡, 𝛽) exp {𝑗 [𝜔𝑡 − 𝜃 (𝛼, 𝛽, 𝑡)]} , (7) 2
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Optics Communications 455 (2020) 124451
time, and partial speckle patterns are related to one another. Thus, Eq. (18) should be revised as √ ⎧ 4𝜆𝑧𝐹 𝑇𝑟 ⎪ ,𝑇 > 𝑇 ⎪ 3𝜋 3 𝑓 ′ 𝑇 𝑅 𝑟 , (19) 𝐶𝑛 = ⎨ √ 4𝜆𝑧𝐹 ⎪ ⎪ 3𝜋 3 𝑓 ′ 𝑅 , 𝑇𝑟 ≤ 𝑇 ⎩ where 𝑅 is the distance from laser point to the rotation center, 𝑇𝑟 is the rotating period, and 𝑇 is the integral time. 2.2. SNR
Fig. 2. Fringe location of Mach–Zehnder optical path.
Interferograms are generally superposed coherent noise on the signal fringes, and thus, they can be simplified into a two-beam interference model. One beam is the signal light, and the other is the noise light. According to interference theory, the intensity is √ (20) 𝐼 = 𝐼𝑛 + 𝐼𝑠 + 2 𝐼𝑛 𝐼𝑠 cos (𝛥𝜑) ,
screen, and 𝐷 is the diameter of pupil. In consideration of the integral convergence, the normalized correlation radius of diffuser is [9] ] [ ( 2) ( ) 𝐸𝑖 𝜎𝑑 − ln 𝜎𝑑2 − 𝑒 , (13) 𝑟𝑐 = 𝑟𝑑 ( ) exp 𝜎𝑑2 − 1
where 𝐼𝑛 represents the intensity of coherent noise, and 𝐼𝑠 is the intensity of desired signal. We define 𝐼𝑠 ∕𝐼𝑛 = 𝑟, and the contrast is written as √ √ 2 𝐼𝑛 𝐼𝑠 2 𝑟 = . (21) 𝐶𝑠 = 𝐼𝑛 + 𝐼𝑠 1+𝑟
where e is the Euler–Mascheroni constant, and 𝐸𝑖 is the exponential integral that can be written as ( −𝜀 𝑡 𝑥 𝑡 ) ( ) 𝑒 𝑒 𝐸𝑖 𝜎𝑑2 = lim 𝑑𝑡 + 𝑑𝑡 . (14) ∫𝜀 𝑡 𝜀→+0 ∫−∞ 𝑡
The ratio of signal contrast to speckle contrast is expressed as SNR = 𝐶𝑆 ∕𝐶𝑛 , which is called SNR of the interferogram. Eqs. (19) and (21) are substituted into the above expression, and the final equation of SNR is
In general case, 𝜆𝑧∕𝐷 ∼ μm ≫ 𝑟𝑐 ∼ nm; the final normalized autocovariance function [20] can be expressed as 𝜇𝑐 (𝜏) =
𝐽 (𝜌) 𝐾 (𝑣𝜏) =2 1 , 𝐾 (0) 𝜌
(15) √ ⎧ 3𝜋 3 𝑟𝑓 ′ 𝑡𝑣 𝑅 ⎪ 2 , 𝑡𝑣 < 1 ⎪1 + 𝑟 SNR = ⎨ , √ 4𝜆𝑧𝐹 ⎪ 2 3𝜋 3 𝑟𝑓 ′ 𝑅 , 𝑡𝑣 ≥ 1 ⎪ 4𝜆𝑧𝐹 ⎩1 + 𝑟
where 𝜋𝐷𝑣𝜏 . 𝜆𝑧
𝜌=
(16)
The correlation time of the speckle intensity is ∞
𝜏𝑐 =
|𝜇 (𝜏)|2 𝑑𝜏 = 8𝜆𝑧 . ∫−∞ | 𝑐 | 3𝜋 2 𝑣𝐷
(22)
where 𝑡𝑣 = 𝑇 ∕𝑇𝑟 denotes the time scale factor, which is used to characterize the relationship between observation time and the period of rotating diffuser. When 𝑡𝑣 ≥ 1 (partial speckle patterns are related to one another), SNR is independent of 𝑡𝑣 and changes with other system parameters. When 𝑡𝑣 < 1, Eq. (22) mainly embodies that SNR is influenced by the time scale factor and the F number of lens. As shown in Fig. 1(a), the SNR gradually increases with increased 𝑡𝑣 , the same as with increased exposure time and rotating velocity. Thus, the reasonable configuration between the two characters is necessary. In the case of a smaller F number (larger pupil), the limited optical depth of field causes the speckle patterns to be blurred, which improves the SNR of interferogram. Fig. 1(b) reveals the relationship between the SNR and the F number; a large F number indicates a low SNR.
(17)
The intensity superposition of multiple speckle patterns in the time domain can significantly reduce the coherent noise, and the speckle contrast is presented as √ √ 𝜏𝑐 1 8𝜆𝑧 𝐶= √ = = . (18) 2 𝑣𝐷𝑇 𝑇 3𝜋 𝑁 The relationship between the motion period of the diffuser and the integral time of a matrix detector is considered, and the F number that can be written as 𝐹 = 𝑓 ′ ∕𝐷 is substituted, where 𝑓 ′ is the focus length of lens. When 𝑇𝑟 ≤ 𝑇 , the diffuser rotates over one round in this integral
Fig. 3. Schematic of an experimental setup, F is a neutral density filter; M1, M2, M3 are mirrors; RD is a rotating ground glass(diffuser); L is an aspheric condenser lens; BS1 and BS2 are beam splitters.
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Optics Communications 455 (2020) 124451
Fig. 4. Fringes on target. (a) laser, (b) with rotating diffuser.
Fig. 5. Variation in the speckle contrast of images with exposure time and F number; the color of the histogram from blue to yellow indicates that F number is from 0.95 to 16, and dotted lines are shown in legend. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 7. Comparison of experimental data-fitting result with theoretical simulation. (a), (b), (c), (d) are the curves in different 𝑡v values.
𝐷 along the optical axis from P0 , the two beams are separated by a distance along the vertical direction, which can be written as
2.3. Fringe location
𝛥𝑑 = 𝐷𝜃 = 𝐷
The rotating diffuser smoothens the speckle noise in finite exposure time. After the laser passes through a diffuser, the original highly spatial coherence point source becomes a partially coherent plane extension light source. With this source of finite size, localized fringes can observed with a Mach–Zehnder interferometer (see Fig. 2). In consideration of the case in which the interferometer is illuminated by this circular and expanded source, the partially coherent light is approximately collimated by lens L, the two mirrors are adjusted to let the center of localization be at P0 , and the angle between the interfering beams is 𝜃. At position P, which is located at a distance
𝜆 , 2𝑒
(23)
where 𝑒 is the fringe width at position P, and 𝜆 is the wavelength of this quasi-monochromatic expanded source. The degree of coherence can be given by [21] ) ) ( ( 2𝐽1 𝑘𝑅 𝛥𝑑 2𝐽1 𝜋𝐷𝑟 𝑓 𝑒𝑓 𝛾= = , (24) 𝜋𝐷𝑟 𝑘𝑅 𝛥𝑑 𝑓 𝑒𝑓 where 𝑘 is the wavenumber, 𝑅 is the radius of expanded source, 𝑓 is the focal length of collimated lens, and 𝐽1 is the first-order Bessel function.
Fig. 6. Trend of speckle contrast (a) and SNR (b) with different exposure times and F numbers.
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Optics Communications 455 (2020) 124451
Fig. 8. Origin 2D data and FFT analysis of 1D data. (a) Interference fringe in different positions caused by localization; (b) the intensity of central line in different distance; (c) FFT of central line of different image; (d)–(f) 2560th row, 1280th row, 3840th row of three images in different localized positions. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 1 Parameter configuration. Exposure time
10 ms, 30 ms, 50 ms, 80 ms, 100 ms
F number
0.95, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16
SNR of the entire system. When the rotating velocity is constant, the correspondence between the time proportional coefficient and the SNR is studied by changing the integral time. The large aperture, which has a small value of point-spread function, can obviously smoothen speckle patterns. A large radius leads to a small localized depth due to the destruction of spatial coherence caused by an expanded light source. A diode-pumped solid-state laser (DJ532-10, 10 mw, 532 nm, Thorlabs, Inc.) is used for this experiment. The experimental setup is shown in Fig. 3. The laser beam passing through a round variable ND filter (NDC-50C-2M-A, diameter 50 mm, optical density 0–2.0) can be adjusted continually. A ground glass diffuser (DFB1-50C02-600, roughness #600, Sigma, Inc.) mounted on a step motor can scatter this beam, and the velocity is changeable with input voltage. An aspheric condenser lens (ACL7560U-A, diameter 75 mm, focal length 60 mm, NA 0.61, Thorlabs, Inc.) is used to collect and collimate the scattering light. An iris diaphragm (D25S, max aperture 25 mm, Thorlabs, Inc.) is used to control the diameter of the laser beam. When beams go through the Mach–Zehnder interferometer, interference fringes can be captured on the target printed with logo and text. A CCD camera (LXC-250, resolution 5120 × 5120, pixel size 4.5 μm, max rate 32 fps, Baumer, GmbH.) connected with graphics workstation can record images in different parameters. The two mirrors on the two arms of Mach–Zehnder interferometer are adjusted, and the image is monitored in real time through a computer screen. The plane of localized fringes should be located on target, then the zoom ring of the lens is turned to make the edges of
The experimental results in the next section quantitatively verify the varying trend of 𝛾. 3. Experiment configuration The Mach–Zehnder interferometer is built to verify the effects of speckle reduction and SNR improvement, as well as the localized fringe caused by the destruction of spatial coherence with rotating diffuser. The speckle contrast and SNR are analyzed in different conditions of F number and exposure time in case of fixed working distance. Before the experiment, whether the depth of field can meet the demand at a maximum aperture (minimum F number) and clearly distinguish the target details is determined. The parameters in Table 1 are set with existing hardware and conditions to study the effect of exposure time and F number of active imaging lens on speckle contrast and SNR of interferogram. Improving the rotating velocity of diffuser or exposure time is equivalent to the climbing of time scale factor, which can effectively suppress the coherent noise in the interferometry and improve the 5
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Optics Communications 455 (2020) 124451
of speckle contrast and exposure time when the F number is constant. Speckle noise can be significantly reduced by increasing the exposure time in a small F number. Curves approach flatness in the situation of large F number. A 2D difference method is used to construct part of the data to reflect the macroscopic trend because of the limited amount of data. Fig. 6 illustrates the combined influence of F number and integral time on speckle and SNR. The speckle contrast is lowest under the conditions of long exposure time and small F number (Fig. 6(a)). Fig. 6(b) illustrates that SNR can remain at a high level when F number is small, and the trend is smooth on the central region of this map. The trend becomes worse in the situation of large F number and short exposure time. The data of 30, 50, 80, and 100 ms are fitted by the power function model, and the theoretical curve is simulated by setting different time proportional coefficients and 𝑟 values (the ratio of the signal intensity to the noise intensity). Most of system parameters in Eq. (22) are proportional coefficients; hence, all is normalized to compare fitting data with simulation data. The integral time is adjusted in four times and the rotating period of a dynamic diffuser is fixed. We can get 𝑡𝑣 = 0.015, 0.025, 0.04, 0.05 and the corresponding r = 10, 50, 100, 200 which illustrates the ratio between intensity of signal and intensity of noise. Base on these theoretical parameters, we can numerically simulate and plot curves by dashed lines (Fig. 7). The range of data along the vertical axis gradually increases from Fig. 7(a) to (d), and the conclusion is that increasing 𝑡𝑣 benefits the improvement in the SNR of the entire system. The experimental data is consistent with the theoretical simulation curve, which verifies the correctness of Eq. (22). When the laser beam is illuminated onto the scattering surface and has a certain diameter, a circular extended source is formed. Each illuminating element is equivalent to a wave source, and the spatial dispersion of each other causes the spatial coherence to be broken into partially coherent light source, which eventually leads to localized fringes. Fig. 8 shows the interferogram in different distances and the Fourier analysis of these 1D data. The contrast of interference fringes is optimal at 0 mm (Fig. 9(c) and (d)), and fringes are distributed throughout the full field. With increased distance, the fringe contrast (intensity of FFT) of the 2560th and 3840th rows obviously decreases, indicating that the degree of coherence gradually degrades from 0 mm to 60 mm. The interferograms at different positions are shown in Fig. 9. The contrast is best at 590 mm, and the localized depth is approximately 120 mm. The fringe contrast of red and green dots on the line is approximately 0.65, and the position is 590 mm. The distance between the two positions is approximately 60 mm, and the fringe width is approximately 120 pixels (e = 120 × 5.5 μm = 0.66 mm). The radius of an equivalent surface source is 350 μm because of the aperture. The initial parameters are integrated into Eq. (24) for simulation, 𝛥𝐷𝑠 = 657.3 mm − 590 mm = 67.3 mm (see Fig. 10); system errors in this experiment should be considered. The above experimental results are consistent with the theoretical values. Localized depth of the interference fringes is quantitatively verified using the spatially extended light source.
Fig. 9. Interferogram in different positions and data of central line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 10. Theoretical simulation result.
image and text clear and sharp. Changing the aperture of lens (F/# 0.95 the first time), rotating the variable filter, and setting proper velocity of the diffuser were performed to ensure that images are not overexposed. After the multiple data of exposure time in current F number are recorded, the above steps are repeated by changing the F number. 4. Result and analysis A comparison of the two images in Fig. 4 shows that the speckle noise is obvious without rotating ground (Fig. 4(a)), and the texts below that logo are difficult to recognize. When the laser beam illuminates onto scattering surface, it is regarded as an extended and circular source phase modulated randomly and has an averaging effect in a period of time. Coherent noise is effectively restrained, and the fringe contrast is slightly reduced (Fig. 4(b)). The experimental data are processed, which contains 45 (9 of F number × 5 of exposure time) images. The speckle contrast and SNR can be revealed from them. Each of histogram clusters in Fig. 5 shows the relationship between the speckle contrast and the F number of lens at a constant integral time. The speckle contrast is sensitive to F number in a long exposure time. Broken lines indicate the correlation curve
5. Conclusion In this study, many factors that affect the SNR of interference system are researched; the quantitative formula is given; and the correctness of the correlation parameters, such as integral time, period of rotating diffuser, and F number of lens, is also confirmed. Research shows that increasing the time proportional coefficient and reducing the F number of lens can effectively improve the SNR of interferograms. This approach can be applied in interferometry, coherent tomography measurement, fringe projection for 3D measurement, and quantitative phase imaging to suppress speckle noise and improve measurement accuracy. 6
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Acknowledgments
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