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Vol. 42, No. 5 / March 1 2017 / Optics Letters

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Measuring extinction with digital holography: nonspherical particles and experimental validation MATTHEW J. BERG,1,* NAVA R. SUBEDI,2

AND

PETER A. ANDERSON1

1

Department of Physics, Kansas State University, 1228 North 17th Street, Manhattan, Kansas 66506-2601, USA Department of Physics & Astronomy, Mississippi State University, 355 Lee Boulevard, Starkville, Mississippi 39762, USA *Corresponding author: [email protected]

2

Received 23 December 2016; accepted 6 February 2017; posted 13 February 2017 (Doc. ID 283540); published 1 March 2017

Through simulations and experiment, this Letter shows how a particle’s extinction cross section can be extracted from a digital hologram. Spherical and nonspherical particles are considered covering a range of cross-sectional values of nearly five orders of magnitude. The extracted cross sections are typically less than 10% in error from the true values. It is also shown that holograms encompassing a sufficiently large angular range of scattered light yield an estimate for the absorption cross section. © 2017 Optical Society of America OCIS codes: (090.1995) Digital holography; (290.5850) Scattering, particles; (290.3200) Inverse scattering; (290.2558) Forward scattering; (290.2200) Extinction; (010.1030) Absorption. https://doi.org/10.1364/OL.42.001011

Small, micrometer-sized particles in a beam of light cause attenuation due to absorption and scattering. Extinction describes this effect, i.e., redistribution of the beam’s radiant energy, and is quantified by a particle’s extinction cross section C ext [1]. Accurate measurement of C ext is important in many regards, such as understanding the influence of atmospheric aerosols on the earth’s radiation budget, visibility in urban environments, and remote sensing [2]. Extinction is a consequence of interference between a particle’s scattered light with the incident light and is typically concentrated around the forward-scattering direction [1,3]. This same interference also constitutes a particle’s hologram, which is simply a measurement of the intensity fringe pattern produced by the interference [4]. Thus, there is an inherent connection between C ext and the hologram. In a previous Letter [4], this connection is described for a spherical particle under certain approximations. Here, these approximations are abandoned, and it is shown that C ext and the absorption cross section C abs are connected to a particle’s hologram for a variety of nonspherical particles. Consider a particle in free space illuminated by a linearly polarized plane wave traveling along the nˆ inc direction (z-axis) with wavelength λ, as shown in Fig. 1. This wave approximates an expanded laser beam. The incident and scattered fields Einc , 0146-9592/17/051011-04 Journal © 2017 Optical Society of America

Binc , Esca , and Bsca share the same time-harmonic dependence exp
−iωt, where ω  kc and k  2π∕λ. To see how extinction arises, we begin with Poynting’s theorem [5]: Z W abs  −

V

∇ · hSit dV ;

(1)

where W is the total power absorbed by the particle, hSit 
1∕2μo RefE × B g is the time-averaged Poynting vector, h…it denotes time averaging, and V is an arbitrary volume enclosing the particle. The Poynting vector applies to the total fields, i.e., E  Einc  Esca and B  Binc  Bsca and, thus, factors into three terms: hSit  hSinc it  hSext it  hSsca it , where hSinc it and hSsca it involve the incident and scattered fields only, whereas hSext it involves cross terms. While hSinc it and hSsca it describe the energy flow due to the incident and scattered waves, hSext it describes the energy flow due to the interference of these waves. Thus, via the divergence theorem, Eq. (1) shows that abs

W ext  W abs  W sca ;

(2)

where W and W are the particle’s scattered and extinguished power [3]. This relation expresses the conservation of energy among the incident and scattered waves and that lost due to absorption. Normalizing each term in p Eq. (2) by the ffiffiffiffiffiffiffiffiffiffiffi intensity of the incident wave, I o 
1∕2 ϵo ∕μo jEinc j2 , yields C ext , C abs , and the scattering cross section C sca , respectively. In the following, I o  1 W∕m2 for simplicity. Ordinarily, the cross sections are derived by applying the divergence theorem to Eq. (1) for a large spherical surface centered on the particle in the far-field zone [1]. However, since V is arbitrary and, assuming that the medium surrounding the particle is nonabsorbing, one can choose any closed surface S as long as it contains the particle. Then, consider a measurement of C ext using a planar sensor facing the oncoming incident wave, as described below, and let S consist of the union of the open surfaces S 1 and S 2 in Fig. 1. Surface S 1 is arbitrarily shaped, while S 2 is planar and coincident with the sensor. Using the divergence theorem, W abs [Eq. (1)], W ext , and W sca become Z  Z ˆ  hSit · nda hSit · zˆ da ; (3) W abs  − sca

ext

S1

S2

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Z  Z ˆ  W ext  − hSext it · nda hSext it · zˆ da ; S S Z 1 Z 2 sca sca ˆ  W  hS it · nda hSsca it · zˆ da; S1

S2

Letter

(4) (5)

where the negative sign in Eq. (4) is explained in [1,3]. While there are multiple ways to measure C ext , the operational definition above, i.e., attenuation of a beam of light, suggests measuring C ext as the difference of the net response of a sensor looking into the oncoming incident wave when a particle is not present and when it is present [3,4]. Thus, we define Z 1 I sen
θ  hSinc it · zˆ da; o I o S2 Z 1 hSit · zˆ da; (6) I sen
θ  I o S2 sen where I sen represent the normalized net response of o and I the sensor without and with a particle present, respectively. Here, “response” is the total power received by the sensor due to the energy flow hSinc it or hSit integrated across S 2 , which subtends an angle θsen , as shown in Fig. 1. Note, that there are subtleties to this definition in regard to the angular size of the sensor; see [6]. Using Eqs. (2)–(5), the difference between these measurements can be expressed as sen f
θ  I sen o
θ − I
θ 2 3 Z Z 7 16 ˆ − W sca  ˆ  W ext 7 hSsca it · nda hSext it · nda  6 5: I o 4 S1 S1 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} I1

I2

(7) To understand the significance of Eq. (7), notice that if θ ≪ 1, S 2 is small, and S 1 is nearly a closed surface surrounding the particle. In that case, I1 is nearly equal to W sca , which is canceled by the second term in Eq. (7). Meanwhile, I2 is nearly equal to −W ext via Eq. (4), which is canceled by the last term in Eq. (7). Thus, when θ ≪ 1, f ≃ 0. Now consider the opposite limit: as θ becomes large. It is shown in [3] that hSext it rapidly oscillates along the rˆ direction from positive to negative with θ. Then, as θ is made larger, I2 ≈ 0 in Eq. (7) due to this oscillation provided that nˆ ≃ rˆ on S 1 .

Fig. 1. Sketch of the surfaces S 1 and S 2 , along with θ and θsen , used to derive Eqs. (7) and (8). Also shown are the particle shapes considered.

Meanwhile, I1 decreases from its θ  0 value of W sca . Using Eqs. (2)–(5), Eq. (7) can also be expressed with the same I2 as 2 3 Z Z 7 1 6 abs ˆ 7 W  hSsca it · zˆ da − W sca  hSext it · nda f
θ  6 4 5: Io S2 S1 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} I3

I2

(8) Given that particles larger than λ typically scatter most strongly around θ  0, as θ grows, I3 in Eq. (8) becomes approximately equal to W sca , which is canceled by the third term in Eq. (8). With the recognition that I2 ≃ 0 due to its oscillatory integrand, Eq. (8) shows that f ≃ W abs as θ becomes large. What all of this means is that as θ increases from zero up to θsen , f will trace a curve that quickly rises from zero to a peak, oscillates, and then decays to an asymptote. If the particle is nonabsorbing, W abs  0, and the curve will asymptotically approach zero. If the particle is absorbing, the curve will asymptotic approach W abs for large θ. In [4], a similar analysis is given that neglects hSsca it , assuming that the particle sensor separation is large enough that hSsca it has little impact in Eq. (6). However, for large sensors, this is not justified since the area of S 2 with increasing θ compensates the inverse square dependence of hSsca it on the particle sensor separation. This approximation is abandoned here, and Eqs. (7) and (8) are exact. Nevertheless, much of the discussion of f above is qualitative, and its real significance is not yet obvious. Inspired by the optical theorem [3], the hypothesis here is that the average of f with θ over the domain where f oscillates approximates W ext when extrapolated to θ  0. Based on Eq. (8), this average also approximates W abs for large θ. Equations (7) or (8) then provide an operational means to measure C abs and C ext . To test this hypothesis, two standard codes are used to simulate light scattering from a variety of particles. The particle shapes include a sphere, prolate and oblate spheroids, and a cube; see Fig. 1. For particles with rotational symmetry, the transition matrix (TM) code of [7] is used to calculate the scattering matrix. Then, the scattered wave
Esca ; Bsca  is generated from this matrix and superimposed with the incident wave in the particle’s far-field zone. From this, hSit is calculated across S 2 allowing evaluation of f via Eq. (6). To simulate scattering from the cube, the discrete dipole approximation (DDA) is used as the TM method has difficulties for particles with points [8]. Both simulations are validated by considering a spherical particle of radius R and comparing the scattered intensity to the Mie theory via [1]. For each particle, a variety of sizes are considered as quantified by kR for the sphere and kR ve for nonspherical particles, where R ve is the sphere volume-equivalent radius. Figure 2 plots f normalized by C ext for an oblate spheroid oriented with respect to nˆ inc by the Euler rotation angles α  β  π∕4 described in [7] and shown in the Fig. 1 inset. The particle’s aspect ratio is a∕b  2.0, where a and b are the horizontal and rotational radii of the spheroid, respectively, and kR ve  15.7. Both nonabsorbing (a) and absorbing (b) refractive indices of m  1.33  0i and 1.33  0.1i are considered, respectively. For a comparison with [4], f is also presented with hSsca it  0 in Eq. (7). Envelopes are spline-fit to the f -curve’s maxima, f top , and minima, f bot , shown in dash. These envelopes define the average trend for f as
f top − f bot ∕2  f bot , shown by the red curve. Note that because f top is defined by the

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Fig. 2. Transition matrix simulations of Eq. (6), i.e., the “f -curve,” for an oblate spheroidal particle with size parameter kR ve  15.7 and the orientation shown in Fig. 1. In (a), a nonabsorbing refractive index of m  1.33  0i is considered, whereas (b) presents the same simulation for an absorbing refractive index of m  1.33  0.1i. The contrast hologram yielding f in (a) is shown in the inset where portions are outlined as I–IV. Following integration, these portions give the corresponding f -values for the θ values indicated on the horizontal axis. Note that S 2 is square, resulting in the square outlines I-IV denoting the joining contour of S 1 and S 2 ; recall Fig. 1.

maxima of f , the trend curve terminates at the first maximum. By extrapolating this endpoint of the trend curve to θ  0, an estimate for C ext is obtained. Note that f is not itself evaluated at θ  0. This extrapolation is denoted by the horizontal arrow in the plots. Both particles in Fig. 2 show an estimate that is less than 10% in error. For the large θ limit, f should approach C abs , which is seen Fig. 2(b). The connection to digital holography can be understood by recalling how a Gabor-type hologram of a particle is formed [9]. Consider Fig. 3, where a frequency-doubled Nd:YLF laser emitting at 526.5 nm is expanded to approximately one cm in diameter by a spatial filter. The beam illuminates a particle deposited on an anti-reflection coated glass window. Most of the beam passes the particle, while a small portion is scattered. The scattered and unscattered waves interfere across a CCD sensor. The hologram is the resulting (intensity) interference fringe pattern, which is proportional to the integrand of I sen in Eq. (6) evaluated at each sensor pixel. If a particle-free measurement is also performed, then the sensor’s response is proportional to the integrand of I sen o in Eq. (6). Often in holography, the difference between the particle-free and particle-present measurements is called the contrast hologram, which is shown in the Fig. 2(a) inset for the nonabsorbing spheroid [10]. Thus, an integration of the contrast hologram is f , Eq. (7). To further test the hypothesis that the extrapolated average of the f envelope yields the cross section, contrast holograms are measured following Fig. 3 for a single 50 μm diameter soda-lime glass sphere (Duke Scientific, 9050) and a cluster of ragweed pollen grains. Equation (7) is then evaluated, and the average trend is found. Plots of f for these particles are shown in Fig. 4. Since the true cross section for these particles is not known, an estimate C ext est is used to normalize the f -curve as in Fig. 2. Each particle, or cluster, in this case, is much larger than λ and, thus, the cross section is approximately given by geo , where C geo is the particle or cluster’s geometric C ext est ≃ 2C cross section in the nˆ inc direction. This relationship is known as the extinction paradox; see [1,11]. To determine C geo , silhouette-like images of the particles are reconstructed from the contrast hologram following [12]. These images are shown in the inset in Fig. 4. Since the spherical particle’s size is well known from the manufacturer (49.0  1.4 μm diameter), the

Fig. 3. Measurement of a particle’s contrast hologram. An expanded beam from a pulsed laser illuminates a particle deposited on a glass window or “stage.” The resulting hologram is resolved by a CCD sensor, while a photodiode monitors the pulse intensity, as explained below.

holographic image is used to map the image pixels to real dimensions in μm. Binarization of the images then yields C geo in μm2 for each particle. Inspection of Fig. 4 reveals that the same conclusion is found as for the simulated particles in Fig. 2. That is, extrapolation of the f trend curve to its θ  0 value provides an estimate for C ext. For the sphere, the agreement between ext C ext holo , i.e., that extrapolated from f , is within 2% of C est . For the ragweed cluster, the agreement is within 5%. The analysis behind Fig. 4 involves a subtle detail regarding the measurements in Fig. 3. Recall that the contrast hologram requires two measurements: without and with a particle present. Thus, the sensor in Fig. 3 must be illuminated twice. This is readily done by simply queuing two pulses before and after installing the particle. However, the pulse-to-pulse intensity of the laser is not exactly constant. This means that the sensor may be illuminated with more intensity during one measurement than another. Thus, for the contrast measurement to be consistent with the definition in Eq. (6), the intensity of each pulse must be known such that the sensor response may be scaled. The scaling adjusts the response so that each measurement corresponds to the same illumination intensity. If this is not done, then the f curve in Fig. 4 may be shifted vertically. Scaling is achieved in Fig. 3 by diverting a small portion of the illumination beam to a photodiode.

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Fig. 4. Evaluation of f , Eq. (7), for experimentally measured contrast holograms of (a) a 50 μm diameter glass sphere and (b) a cluster of ragweed pollen grains. The experimental arrangement is shown in Fig. 3. Hologram-derived images of these particles are shown in the inset. The plot (b) inset also shows the binarized particle image used to establish C geo in that case.

While Figs. 2 and 4 show that extrapolation of the f curve approximates C ext well, one may wonder how general the result is. To test this, Fig. 5 presents C ext holo for 71 particles using TM

Letter or DDA simulations. The experimental data of Fig. 4 are included as well. For the simulated data, the particle morphologies are the same as those in Fig. 1, and the range of kR, kR ve , and m is indicated in the key. Here, one can see that C ext holo agrees with C ext within 10% in almost all cases, over a range of nearly five orders of magnitude. In the context of applying this technique to particles in the atmosphere, it should be noted that such particles scatter light as an ensemble of random particle orientations. This Letter, however, considers particles in fixed orientations. Yet, extraction of C ext from a hologram could be extended to randomly oriented particles by performing a series of hologram measurements as a particle takes on different orientations. Lastly, work by [13] rediscovers the connection between C ext and a hologram presented in [4], and considers extension of the concept to non-plane wave illumination of a particle. In summary, to estimate C ext , a particle-free measurement of the illumination-beam profile is performed in an arrangement such as Fig. 3. The measurement is then repeated with the particle present, where the difference between the measurements constitutes a contrast hologram. Integrating this hologram as a function of θ yields the f -curve presented in Figs. 2 and 4 via Eqs. (6) and (7). The average of this curve is then found by fitting envelopes to the extrema, which together define a trend curve. Finally, C ext is estimated by extrapolating the trend curve to θ  0. As demonstrated by Fig. 5, this analysis is valid for a large variety of particle size, shape, and refractive index. Funding. U.S. Army Research Office (ARO) (W911NF15-1-0549); Directorate for Geosciences (GEO) (1453987, 1665456). Acknowledgment. The authors are thankful for discussions with Michael Mishchenko, Vinod Kumarappan, Gorden Videen, and Chris Sorensen, and for access to the High Performance Computing Collaboratory at Mississippi State University. REFERENCES

Fig. 5. Survey of the hologram-extracted cross section C ext holo versus the true cross section C ext for 71 particles simulated with TM or DDA. The size parameter values for each particle shape are shown in the key. Also shown are C ext holo for the experimental data in Fig. 4 versus their geo . The filled (open) symbols estimated true cross section C ext est ≃ 2C correspond to the absorbing (nonabsorbing) refractive indices indicated, and the dashed lines represent 10% error bounds.

1. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983). 2. P. Kulkarni, P. A. Baron, and K. Willeke, eds., Aerosol Measurement: Principles, Techniques, and Applications, 3rd ed. (Wiley, 2011). 3. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, J. Opt. Soc. Am. A 25, 1504 (2008). 4. M. J. Berg, N. R. Subedi, P. A. Anderson, and N. B. Fowler, Opt. Lett. 39, 3993 (2014). 5. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002). 6. M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. M. van der Mee, J. Quant. Spectrosc. Radiat. Transfer 110, 323 (2009). 7. M. I. Mishchenko, Appl. Opt. 39, 1026 (2000). 8. M. A. Yurkin and A. G. Hoekstra, J. Quant. Spectrosc. Radiat. Transfer 106, 558 (2007). 9. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999). 10. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley, 2005). 11. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, J. Quant. Spectrosc. Radiat. Transfer 112, 1170 (2011). 12. M. J. Berg and N. R. Subedi, J. Quant. Spectrosc. Radiat. Transfer 150, 36 (2015). 13. E. W. Marengo, M. V. Bunn, and J. Tu, Forum Electromag. Res. Methods Appl. Tech. 16, 1 (2016).