Numerical investigation of near-field off-axis scattering in-line recording particle holography

Transcription

1 Numerical investigation of near-field off-axis scattering in-line recording particle holography Xuecheng Wu 1, Gérard Gréhan, Siegfried Meunier-Guttin-Cluzel, Yingchun Wu 1, Linghong Chen 1*, Hao Zhou 1, Kunzan Qiu 1, Kefa Cen 1 1: State Key Laboratory of Clean Energy Utilization, Institute for Thermal Power Engineering, Zhejiang University, 38 #, Zheda road, Hangzhou, 31007, China : LESP, UMR 6614 / CORIA, CNRS, Université et INSA de Rouen, Site du Madrillet, Avenue de l Université, BP Saint Etienne du Rouvray, France Corresponding author: chenlh@zju.edu.cn Abstract: Particle holograms using off-axis scattering light are investigated. A code based on the near field Lorenz-Mie framework capable of calculating particle hologram with arbitrary size, location and refractive index at arbitrary off-axis scattering angle has been developed. Characteristics of off-axis particle holograms are analyzed with special attention to the strong Moiré effect and dual fringe pattern (near field). Reconstruction of off-axis scattering particle hologram is investigated, and the possibility of determining particle location, size and refractive index using near field off-axis digital holography are discussed. It is shown that for transparent particles, the typical off-axis scattering holograms are characterized by strong Moiré effect and dual fringe patterns (in near field) due to the reflected and refracted (1 st or nd order or higher) lights. When near field offaxis scattering holograms with dual fringe patterns are numerically reconstructed with wavelet-transform-based reconstruction approach, two glare spots corresponding to the reflected and refracted (or higher order refracted) light sources coming from the different parts of the particle are clearly visible. It is also shown that in geometric optics, the particle location, size and real part of refractive index can be determined as long as the locations of the two glare spots are known. It has been shown that the measurement values of particle center location, size and real part of refractive index are highly sensitive to the measurement errors of glare spots locations. However, the measurement accuracy of glare spots locations by off-axis scattering digital holographic method for the moment is not higher enough to satisfy this requirements, where further investigation to improve the measurement accuracy is then encouraged. Keywords: digital holography, off axis scattering, near field, refractive index measurement 1. Introduction In recent years, the digital holographic imaging technique has been developed and demonstrated to be a powerful three-dimensional (3D) diagnostic in a variety of applications (Cheong et al. 009, Lee and Grier 007, Malek et al. 008, Pan and Meng 003, Pu et al. 005, Yu et al. 009). Unlike the filmbased holography where off-axis recording mode is used(pu and Meng 000, Pu et al. 000), a forward scattering in-line recording is normally adopted in digital holography, where the particle hologram is produced by the interference between the unscattered reference wave and the forward diffraction wave scattered by the particle. In this way, the bandwidth of the hologram can be suited to the poor spatial sampling rate of digital recording media (e.g., CCD), avoiding the undersampling checked by the Nyquist sampling criteria. Nevertheless, the configuration of forward scattering in-line recording results in a very small angular aperture, which, in turn, causes severe speckle noise and large depth of focus in the hologram reconstruction(meng et al. 1993), affecting the accurate measurements of particle depth position and size(cao et al. 008, Wu et al. 009). Another disadvantage of forward scattering in-line recording configuration is that the measurement volume and location are unable to be controlled, which aggravates the speckle noise

2 In this paper, particle holograms using off-axis scattering light are numerically investigated. Off-axis scattering particle holograms as well as their reconstructed images, and the potential in particle location, size and refractive index measurements are investigated. The paper is organized as follows: the framework of off-axis scattering in-line recording particle holograms based on Lorenz-Mie theory is described in Section. Section 3 discusses typical off-axis scattering particle holograms with special attention to the dual fringe pattern and strong Moiré effect, while in Section 4, reconstructed images of those holograms are studied, as well as the possibility of particle location, size and refractive index measurements. Section 5 will be a conclusion.. Framework of particle hologram calculation Based on generalized Lorenz-Mie theory(gouesbet et al. 1985, Gouesbet et al. 1988) (GLMT), a code capable of accurate calculation of off-axis scattering in-line recording particle hologram has been developed. In the framework of the calculation, a 3D coordinate system XYZO is defined, as shown in.fig. 1. A perfectly spherical, homogeneous and isotropic particle, characterized by its size d and its complex refractive index n, is assumed to be at an arbitrary location (x p, y p, z p ). It is illuminated by a plane wave (optional parallel or vertical polarization) which travels from the negative toward the positive Z. A CCD detector, defined with a resolution of N N and a pixel size of d pix, is assumed to be with its center at (0, 0, R) and perpendicular to the Z-axis (in-line case). The detector may also rotate around Y-axis with an off-axis angle θ, in such a way that the off-axis scattering particle hologram can be recorded. A reference beam is assumed to be the same with the incident plane wave but with adjustable amplitude and has a direction that vertically impinging on the CCD detector. Fig. 1 Optical layout of off-axis particle hologram calculation system. R: the distance between coordinate origin and the detector center; θ: off-axis angle. It is well known that the Lorenz-Mie (LM) theory(mie 1908) provides rigorous solutions of particle scattering problem, which is capable of calculating the scattering intensity of a plane wave by a spherical particle at an arbitrary direction and an arbitrary distance. Here the light scattered by the spherical particle is computed in the framework of the near-field Generalized Lorenz-Mie theory(gouesbet, et al. 1988). The hologram of the particle is the superposition result of the scattered field from the particle with the reference wave field, accordingly all the interference effects are taken into account(chevaillier et al. 1990, Slimani et al. 1984). 3. Discussion of off-axis scattering particle hologram(wu et al. 010) 3.1. Strong Moiré effect - -

3 Typical particle holograms using off-axis scattering light are computed with similar recording parameters in experiments(pu, et al. 005). It is found that all of them are characterized with several ghost images known as the Moiré effect(nicolas et al. 007). Fig. shows such holograms (LEFT and MIDDLE) for a 60 µm particle with 90 and 60 scattering lights, while the RIGHT one is with forward scattering light. Fig. Particle holograms at off-axis (LEFT, θ=90 ; MIDDLE, θ=60 ) and in-line (RIGHT, θ=0 ) scattering directions. Particle size d p =60 µm, refractive index n=1.5-0i, recording distance R=5 cm, CCD , size of CCD pixel d pix =10 µm. Particle hologram, typically viewed as a group of concentric circular rings, is originated from the interference of the scattering light in a certain scattering angle range (collection angle) with the reference light. The spatial frequency of those interference fringes depends on the intersection angle between the scattering light and reference light. Due to this fact, the frequency increases gradually from the center to the outside of the circular rings group. As a resolution limited CCD sensor is used, the particle hologram may not always be sufficiently sampled, especially for the outside part of the circular fringes as they have higher frequencies. An undersampling of the hologram may occur according to the Nyquist sampling criteria and gives rise to a Moiré pattern (Lebrun et al. 1991). The Moiré patterns are also exist for forward scattering particle hologram but has a low visibility. As for the forward case, the scattered light is dominated by diffraction which has a strong intensity in the center but drops very quickly (several orders of magnitude), resulting in a rapid decrease of the intensity of the fringe pattern from the center to the outside and suppressing the Moiré effect. On the contrary, for off-axis scattering, the scattered light does not change too much and the amplitude of the interferences with the reference beam will be essentially constant. Then the visibility of the fringes in the hologram will be essentially constant on the detector surface, creating strong Moiré effect. 3.. Near field particle hologram at different off-axis angles In geometric optics, the light scattered by a spherical particle can be described by diffraction (forward), reflection and refraction, which are emerging from different locations of the particle surface. This phenomenon may lead to the off-axis scattering particle hologram to be quite different from the one with forward scattering. If we take a close look at the off-axis scattering particle hologram, i.e., in the near field, the characteristics may be clear. The calculation results have shown that in the near field, most of the particle holograms at various off-axis angles are characterized with dual fringe pattern. Typical near field off-axis particle holograms are shown in Fig. 3, where the recording distance and pixel size are decreased to 1 mm and 0.7µm, and the off-axis angles are 55, 110 and 155 for LEFT, MIDDLE and RIGHT ones, respectively. For LEFT hologram, the off-axis angle is in refracted light domain, leading to the left fringe pattern to be stronger than the right one (Note that the incident beam illuminates the particle from the right to left side, seen by the CCD camera), while it is just the opposite case in MIDDLE hologram where the reflected scattering light is dominant. The dual fringe pattern is - 3 -

4 due to the creation of one image by the reflected light (p=0) and one image by the refracted light (p=1). The RIGHT image is a typical particle hologram where the off-axis angle is in rainbow angle domain. It can be found that the intensity distribution of the fringe pattern is strongly affected by the intensity fluctuation in rainbow angle domain. Fig. 3 Typical near field off-axis scattering in-line recording particle holograms. Particle size d p =60 µm, refractive index n=1.5 0i, recording distance R=1 mm, size of CCD pixel d pix =0.7 µm, CCD: 51 51, Vertical polarization. LEFT: off-axis angle θ=55, refraction dominant; MIDDLE: θ=110, reflection dominant; RIGHT: θ=155, 1 st order rainbow angle. 4. Discussion on location, size and refractive index measurement 4.1. Reconstructed particle images from off-axis scattering particle hologram In this section, the reconstructed images of those off-axis scattering in-line recording particle holograms are investigated. A wavelet transform based numerical reconstruction approach (Buraga- Lefebvre et al. 000, Lebrun et al. 1999) is used to reconstruct the scattering field in terms of a series of images along the depth direction. Fig. 4 gives such reconstructed images for an off-axis scattering particle hologram (d p =60 µm, n=1.5-0i, θ=70 ). It is shown in Fig. 4 (LEFT image) that for a transparent particle two glare spots corresponding to the refracted light spot (left) and reflected light spot (right) are clearly reconstructed at a depth close to the particle center (depth position: 963 µm, recording distance of the hologram in this case R=1 mm). Fig. 4 Typical reconstructed images of off-axis scattering particle hologram (d p =60 µm, n=1.5-0i, R=1 mm, θ=70, 51 51, d pix =0.7 µm). LEFT: image at depth position Z=963 µm; RIGHT: image along depth direction (from 750 µm to 150 µm). The RIGHT image in Fig. 4 shows the reconstructed image along depth directions (from 750 µm to 150 µm), which indicates that the scattered light spots have a large depth of focus which seems to be - 4 -

5 similar with the case for in-line scattering particle hologram(pan and Meng 001, Yang et al. 005). But the point is that the 3D locations and their intensity ratio of the two glare spots may be determined in certain accuracy by means of digital imaging processing, which opens a possible way to measure simultaneously the 3D location, size and refractive index (both real and imaginary parts) for sufficiently transparent spherical particles. For totally opaque particles, however, it seems not possible to obtain even the center location and size of the particle since only the location of the reflected light spot can be obtained. 4.. Possibility of particle size, location and refractive index measurements As mentioned in Section 4.1, the reconstructed images of two glare spots provide a possibility to simultaneously measure the 3D location, size and refractive index of transparent spherical particles. Here in geometric optics, we show that the particle center location, size and real part of refractive index can be determined if the locations of two glare spots are known. As schematically shown in Fig. 5, the particle is illuminated by a parallel plane wave, and the CCD receives its scattering light and reference light at an off-axis angle θ. The refracted and reflected light spots and the particle center are marked as A, B and P. A D coordinate system is defined with its origin at the refracted light spot A and Y-axis has a direction perpendicularly towards the CCD camera (parallel with the off-axis angle direction). dp Fig. 5 Geometry optics of the two glare spots. θ i : incident angle, θ t : transmission angle, θ r : reflection angle, n: complex refractive index, (x P,y P ), (x B,y B ): coordinates of particle center and reflected light spot in the coordinate system XOY with an origin at the refracted light spot A. According to the geometric mathematics, the following equations are satisfied to determine the particle size d p and particle center location (x P,y P ) relative to spot A: 1 xp + yp = dp (1) 4 1 xp = xb dpcosθ r () 1 yp = yb dpsinθ r (3) θ r = θ (4) The real part of refractive index Re(n) can be further obtained by the following equations introducing the Snell s law: - 5 -

6 1 d ( 1 sinθ ) ( cosθ ) x y 4 sinθ = Re n sinθ p + t + t = B + B i ( ) ( ) = i t θ θ θ t (5) (6) (7) Error of depth position, σz [µm] 10 3 σz vs σx A σz vs σy A σz vs σx B 10 σz vs σy B d p =60 µm n=1.5-0i θ=55 ο Error of particle size, σd p [µm] 10 3 σd p vs σx A σd p vs σy A σd p vs σx B 10 σd p vs σy B d p =60 µm n=1.5-0i θ=55 ο Position errors of glare spots, σx A, σy A, σx B, σy B [µm] Position errors of glare spots, σx A, σy A, σx B, σy B [µm] Fig. 6 Sensitivity analysis of measurement errors of glare spots to errors of particle depth position (LEFT) and size (RIGHT) (d p =60 µm, n=1.5-0i, θ=55 ). Error of refractive index (real), σre(n) d p =60 µm n=1.5-0i θ=55 ο σre(n) vs σx A σre(n) vs σy A σre(n) vs σx B σre(n) vs σy B Position errors of glare spots, σx A, σy A, σx B, σy B [µm] Error of refractive index (real), σre(n) 1.0 σre(n) vs σx d A p =60 µm σre(n) vs σy n=1.5-0i A 0.8 σre(n) vs σx θ=55 ο B σre(n) vs σy B Position errors of glare spots, σx A, σy A, σx B, σy B [µm] Fig. 7 Sensitivity analysis of measurement errors of glare spots to error of real part of refractive index (d p =60 µm, n=1.5-0i, θ=55 ). The above equations (1)-(7) indicate that the center location, size and real part of refractive index of the particle are known as long as the locations of glare spots can be determined correctly. Considering that in digital holography, the location of particle is commonly determined with a certain measurement error, especially in depth direction (here in Y direction), the sensitivity of the location variation of glare spots to the errors of determining particle center, size and real part of refractive index need to be tested. The particle is assumed with a size and a refractive index of 60 µm and 1.5-0i, respectively, while the off-axis angle is to be 55. The absolute errors of depth position (Y direction, σz), particle size (σd p ) and real part of refractive index (σre(n)) due to the absolute errors of glare spots locations, σx A, σx B, σy A, σy B, are calculated, as shown in Fig. 6 and Fig. 7. The variation of these measurement errors is from -100 µm to 100 µm. As for the depth position error, shown in the LEFT image of Fig. 6, it s found to be much more sensitive to the Y direction errors of glare spots than the X direction. The position error in Y direction results in at least the same order of magnitude of error in particle depth position. The similar situation can be found for particle size, as shown in the RIGHT image of Fig. 6. Here the X direction errors of glare spots have stronger influence on the error of particle size

7 As shown in Fig. 7 (the RIGHT image is a zoom view of the LEFT one around 0 of horizontal axis), the measurement result of real part of refractive index is also sensitive to the position errors of glare spots. If an admissible error of ±0.1 is required for Re(n), the error of Y direction location of glare spots can only be no larger than about µm, while for X direction location, an error of about 40 µm would be allowable. To sum up, the successful measurements of particle location, size and refractive index do heavily rely on the measurement accuracy of the glare spots locations, especially at Y direction. It seems to be a big challenge to determine their locations with the required accuracy by means of digital holography. To check the possible precision level that the off-axis scattering digital holography could reach, various cases has been numerically tested, where the particle size, recording distance, off-axis angle and real part of refractive index are within 40 µm -10 µm, 300 µm-1 cm, 55-80, and , respectively. The cases are organized by the Fraunhofer criterion ( CF = π dp 4λz). The glare spots positions are determined from the wavelet-based numerical reconstruction (mentioned in section 4.1) and digital imaging processing. The position errors of glare spots are expressed as spot B relative to spot A. As shown in Fig. 8, most of the measurement errors in X direction are smaller than 1 µm, while unfortunately in depth direction (Y direction), the errors are several orders of magnitude larger than those in X direction. Only a few cases that the position errors in depth direction are smaller than 10 µm can be found, but the minimal error is around 4 µm which is still not small enough to make the real part of refractive index to be accurately determined. Position errors of glare spots σx B A, σy B A, [µm] Condition: 10 3 σx B A σy B A λ: 53 nm d p : µm z: 300 µm-1 cm n: θ: 55 o -80 o Fig. 8 Numerical testing of measurement errors of glare spots location (λ=53 nm, d p =40-10 µm, Re(n)=1.-1.8, θ=55-80 ). To realize accurate measurements of particle size, location, and refractive index with off-axis scattering digital holography, a further investigation on precise location measurement of glare spots is desirable. The imaginary part of the refractive index Im(n), however, cannot be derived from geometric optics. Note that the scattering intensity from the refracted light spot may be sensitive to the absorption by the particle(arnold et al. 1995), the imaginary part of the refractive index may probably be obtained by the intensity ratio of the two glare spots. Thus a further investigation on this issue is also desirable. 5. Conclusion With a code based on the near field Lorenz-Mie framework, off-axis scattering particle holograms were analyzed. It is shown that for transparent particles, the typical off-axis scattering holograms are characterized by strong Moiré effect and dual fringe patterns (in near field) due to the reflected and refracted (1 st or nd order or higher) lights. When near field off-axis scattering holograms with dual fringe patterns are numerically reconstructed with wavelet-transform-based reconstruction approach, two glare spots corresponding to the reflected and refracted (or higher order refracted) light sources C F - 7 -

8 coming from the different parts of the particle are clearly visible. It is also shown that in geometric optics, the particle location, size and real part of refractive index can be determined as long as the locations of the two glare spots are known. It has been shown that the measurement values of particle center location, size and real part of refractive index are highly sensitive to the measurement errors of glare spots locations. However, the measurement accuracy of glare spots locations by off-axis scattering digital holographic method for the moment is not higher enough to satisfy this requirements. Acknowledgements The authors gratefully acknowledge the financial supports from the National Natural Science Foundation of China (NSFC) projects (grants ), the National Basic Research Program of China (grants 009CB1980) and the Program of Introducing Talents of Discipline to University (B0806). References Arnold S, Holler S, Li JH, Serpengüzel A, Auffermann WF, Hill SC (1995) Aerosol particle microphotography and glare-spot absorption spectroscopy. Optics Letters 0: Buraga-Lefebvre C, Coetmellec S, Lebrun D, Ozkul C (000) Application of wavelet transform to hologram analysis: three-dimensional location of particles. Optics and Lasers in Engineering 33: Cao L, Pan G, de Jong J, Woodward S, Meng H (008) Hybrid digital holographic imaging system for three-dimensional dense particle field measurement. Applied Optics 47: Cheong FC, Sun B, Dreyfus R, Amato-Grill J, Xiao K, Dixon L, Grier DG (009) Flow visualization and flow cytometry with holographic video microscopy. Opt Express 17: Chevaillier J-P, Fabre J, Gréhan G, Gouesbet G (1990) Comparison of diffraction theory and generalized Lorenz-Mie theory for a sphere located on the axis of a laser beam. Applied Optics 9: Gouesbet G, Grehan G, Maheu B (1985) Scattering of a Gaussian beam by a Mie scatter center using a Bromwich formalism. J Optics (Paris) 16:83-93 Gouesbet G, Maheu B, Grehan G (1988) Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation. J Opt Soc Am A 5: Lebrun D, Belad S, zkul C (1999) Hologram Reconstruction by use of Optical Wavelet Transform. Applied Optics 38: Lebrun D, Ozkul C, Allano D, Leduc A (1991) Use of the moire effect to improve diameter measurements with charge coupled imagers. Journal of Optics : Lee S-H, Grier DG (007) Holographic microscopy of holographically trapped three-dimensional structures. Optics Express 15: Malek M, Lebrun D, Allano D (008) Digital In-Line Holography System for 3D- 3C Particle Tracking Velocimetry. In: Particle Image Velocimetry. (ed A. Schroeder, CEW). Vol. 11, pp : Springer-Verlag Berlin Heidelberg Meng H, Anderson WL, Hussain F, Liu DD (1993) Intrinsic speckle noise in in-line particle holography. J Opt Soc Am A 10: Mie G (1908) Beiträge fur optik truber medien, speziell kolloidaler metallösungen. Ann Phys 5:377 Nicolas F, Coëtmellec S, Brunel M, Lebrun D (007) Suppression of the Moiré effect in subpicosecond digital in-line holography. Optics Express 15: Pan G, Meng H (001) Digital in-line holographic PIV for 3D particulate flow diagnostics. In: 4th International Symposium on Particle Image Velocimetry, Göttingen, Germany - 8 -

9 Pan G, Meng H (003) Digital holography of particle fields: reconstruction by use of complex amplitude. Applied Optics 4: Pu SL, Allano D, Patte-Rouland B, Malek M, Lebrun D, Cen KF (005) Particle field characterization by digital in-line holography: 3D location and sizing. Experiments in Fluids 39:1-9 Pu Y, Meng H (000) An advanced off-axis holographic particle image velocimetry (HPIV) system. Experiments in Fluids 9: Pu Y, Song X, Meng H (000) Off-axis holographic particle image velocimetry for diagnosing particulate flows. Experiments in Fluids 9:S117-S18 Slimani F, Grehan G, Gouesbet G, Allano D (1984) Near-field Lorenz-Mie theory and its application to microholography. Applied Optics 3: Wu XC, Gréhan G, Meunier-Guttin-Cluzel S, Chen LH, Cen KF (009) Sizing of particles smaller than 5 µm in digital holographic microscopy. Optics Letters 34: Wu XC, Gréhan G, Meunier-Guttin-Cluzel S, Qu RY, Gu ML, Xu JP, Chen LH, Qiu KZ, Cen KF (010) Off-axis scattering particle holography: a numerical study. In: The 7 th Progress In Electromagnetics Research Symposium (PIERS) 010 Xi'an, China Yang W, Kostinski AB, Shaw RA (005) Depth-of-Focus reduction for digital in-line holography of particle fields. Optics Letters 30:1-3 Yu L, Mohanty S, Zhang J, Genc S, Kim MK, Berns MW, Chen Z (009) Digital holographic microscopy for quantitative cell dynamic evaluation during laser microsurgery. Opt Express 17: