July 1, 2014 / Vol. 39, No. 13 / OPTICS LETTERS
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Using holography to measure extinction Matthew J. Berg,1,* Nava R. Subedi,1 Peter A. Anderson,1 and Nicholas B. Fowler2 1
Department of Physics & Astronomy, Mississippi State University, 355 Lee Boulevard, Mississippi State, Mississippi 39762, USA 2
Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, Kansas 66502, USA *Corresponding author:
[email protected] Received May 8, 2014; accepted May 21, 2014; posted May 29, 2014 (Doc. ID 211556); published June 30, 2014
This work presents a new concept to measure the extinction cross section for a single particle in situ. The concept involves recording the hologram produced by the interference of a particle’s forward-scattered light with the incident light. This interference pattern is fundamentally connected to the energy flow that gives rise to extinction, and, by integrating this measured pattern, one obtains an approximation for the cross section. Mie theory is used to show that this approximation can be as little as 1% in error of the true value for many cases of practical interest. Moreover, since an image of the particle can be computationally reconstructed from a measured hologram using the Fresnel–Kirchhoff diffraction theory, one can obtain the cross section simultaneously with the particle shape and size. © 2014 Optical Society of America OCIS codes: (090.0090) Holography; (290.2200) Extinction; (290.4020) Mie theory; (290.1090) Aerosol and cloud effects; (290.5825) Scattering theory; (290.5850) Scattering, particles. http://dx.doi.org/10.1364/OL.39.003993
Extinction is a ubiquitous phenomenon that describes the attenuation of light traversing a medium due to scattering and absorption [1,2]. Experience shows that a laser beam passing through a colloidal solution emerges with diminished intensity, as does sunlight passing through the atmosphere. In these examples, extinction is attributed to scattering and absorption by the particle or density fluctuations. Thus a particle’s extinction cross section C ext is a fundamental quantity describing the redistribution of incident light via scattering and absorption. Consequently, the ability to accurately measure this cross section is important in many regards, such as understanding the influence of atmospheric aerosols on the Earth’s radiation budget, visibility in urban and rural environments, and remote sensing from terrestrial and space-based platforms [3]. The understanding for how extinction arises is surprisingly nuanced, and careful study reveals some unexpected behavior. For example, if a collimated laser beam were to illuminate a single wavelength-sized particle, one might expect that the intensity of light along the exact forward direction would be reduced due to extinction. However, Ref. [4] shows that the energy flow in this direction can either be decreased or enhanced by the presence of the particle, which is consistent with and requisite for the conservation of energy. The explanation of this behavior relates to the inherent interference origin of extinction: in the far-field zone, a particle’s scattered wave interferes with the incident to produce a pattern of radial-inward and outward energy flow as a function of angle. A detector facing the oncoming beam registers reduced incident power because this oscillating radial flow, which is integrated by the detector, subtracts from the power that would be received in the particle’s absence. This suggests that a detector must cover a solid angle, as measured from the particle, which is suitably larger than that associated with the particle’s geometrical shadow [1,4,5]. A corollary to this fact is that C ext could be measured if one were able to resolve this interference energy-flow across the detector’s surface. This work will carefully show how this interference could be measured using a simple in-line holographic technique for a 0146-9592/14/133993-04$15.00/0
situation consisting of a single particle in a collimated laser beam. This ability could have wide ranging and important implications, as such measurements are traditionally challenging on the single-particle level. Using exact analytical methods, this work will demonstrate the concept for a single, wavelength-sized, spherical particle and discuss practical implementation. Consider a single spherical particle in vacuum illuminated by a linearly polarized plane wave traveling in vacuum along the z axis with wavelength λ. This wave will approximate a collimated laser beam. The particle radius and refractive index are R and m, respectively. The incident fields, Einc and Binc , and scattered fields, Esca and Bsca , all share the same time-harmonic dependence exp−iωt, where ω kc, k 2π∕λ, and this dependence will be suppressed in the following derivation. To see how extinction comes about, begin with Poynting’s theorem [3]: Z W abs − V
∇ · hSit dV ;
(1)
where W abs 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, the asterisk denotes complex conjugation, and V is an arbitrary volume enclosing the particle. The Poynting vector depends on the total fields, E Einc Esca and B Binc Bsca , and factors into three terms: hSit hSinc it hSsca it hSext it ;
(2)
where hSinc it and hSsca it involve the incident and scattered fields only, whereas hSext it involves cross terms, hSext it
1 RefEinc × Bsca Esca × Binc g; 2μo
(3)
and represents the interference of the incident and scattered waves. Combining Eqs. (1)–(3) shows that © 2014 Optical Society of America
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W ext W abs W sca ;
(4)
where W sca and W ext are the scattered and extinguished power [3]. This well-known relation expresses the conservation of energy among the incident and scattered waves and the energy that is lost due to absorption within the particle. Not present here is the contribution due to hSinc it alone, which, by using the divergence theorem in Eq. (1), can be seen to be zero as the incident wave carries as much energy into V as it carries out. Ordinarily, the next step would be to define V as the volume enclosed by a large spherical surface S sph centered on the particle. Then, as just mentioned, the divergence theorem can be used to transfer the volume integral in Eq. (1) to a surface integral over S sph . In that case, one has the familiar expressions for the extinction, absorption, and scattering cross sections, respectively, as C ext
W ext ; I inc
C abs
W abs ; I inc
C sca
W sca ; I inc
Z
Z S1
hSit · rˆ da
S2
hSit · zˆ da ;
Z I det o
(6)
where S 1 and S 2 are the portions of the spherical and detector surfaces, respectively, which together form the closed surface shown in Fig. 1. Now let the distance between the particle and detector be d and consider the combination of Eqs. (2) and (6). The response of the detector is given by the integral over S 2 of the component of hSit directed into the detector, i.e., hSit · zˆ , which is the last integral in Eq. (6). Also, notice that hSsca it decays as d−2 on S 2 , whereas hSinc it is constant, and hSext it decays as d−1 . This decay behavior is due to the far-field decay of the scattered fields. Then
Z S2
hSinc it · zˆ da and
I det
S2
hSit · zˆ da; (7)
where hSit is given by Eq. (2). Neglecting the contribution due to hSsca it across S 2 , as argued above, and using the fact that the integral of hSinc it over the closed surface S 1 ∪S 2 is zero, shows that the difference of these two measurements is Z
(5)
where I inc is the intensity of the incident wave. However, since V is arbitrary in Poynting’s theorem, one can choose any closed surface so long as it contains the particle. Then, consider the measurement of C ext , using a planar detector facing the oncoming incident light, and let S sph be large enough to intersect a portion of this detector. Applying the divergence theorem to the volume enclosed by the union of this spherical surface, the detector gives for Eq. (1): W abs −
the detector can be assumed to be far enough from the particle that the contribution of hSsca it to the last integral in Eq. (6) can be neglected. Last, as implied earlier, extinction results in the reduction of light in a beam due to absorption and scattering by the particle and, thus, tacitly compares two measurements: the detector’s response when no particle is present and the response det denote when the particle is present. Then let I det o and I the former and latter measurements, respectively, which are given by
I det o
−I
det
W
ext
S1
hSext it · rˆ da:
(8)
In arriving at Eq. (8), the analog of Eq. (6) involving only hSinc it or hSsca it is used to re-express the integrals over S 2 in Eq. (7) in terms of integrals over S 1 . Equation (8) is the central result of this work: it shows that the difference between the two measurements described is equivalent to the extinguished power plus an additional term, which involves the interference energy flow hS ext it passing through the partial spherical surface S 1 . That is, this additional term does not involve the detector surface S 2 . As will be shown below, in many cases of practical interest this last term in Eq. (8) is negligible; thus the difference measurement combined with Eq. (5) gives good approximation for C ext . Note that if the detector is position sensitive, such as a 2D CCD or CMOS array, the integrand in Eq. (7) is known. Thus one can computationally evaluate the integral in Eq. (7) for surfaces S 2 of increasing angular size θdet beginning at zero and increasing up to the actual angular size of the detector. To demonstrate this explicitly, define f θdet
1 det I det θdet ; o θdet − I I inc
(9)
and δθdet −
1 I inc
Z S1
hSext it · rˆ da;
(10)
where θdet is the angular size of the detector S 2 as shown in Fig. 1. Then Eqs. (5) and (8) show that C ext f θdet δθdet :
Fig. 1. Sketch of the spherical surface S sph and composite surface S 1 ∪S 2 in relation to the particle and incident wave. Also shown is particle–detector separation d and detector angle θdet .
(11)
In the so-called resonance or Mie particle-size range, the angular distribution of scattered light becomes more concentrated around the forward direction as R increases relative to λ. Then, one would expect that δ is small compared with f for a detector of sufficiently large size, since δ captures the energy flow over all directions
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except this forward region. To show that this is true, Mie theory is used to calculate f and δ exactly for a uniform spherical particle. This is done here following the formulation in [2] and implemented using the scientific software Mathematica. To validate this simulation, scattered intensity patterns for spheres with a wide range of R and m are compared to the well-verified BHMIE simulation also given in [2]. Figure 2 shows the evolution of f and δ with increasing detector size θdet as provided by Mie theory simulation, in which a variety of particle sizes are considered. Here the refractive index ranges from m 1.33 0i in plots (a)–(c), and m 1.55 0.1i in plot (d); each plot shows the cross-section normalized curves, i.e., f ∕C ext and δ∕C ext , for the given size ratio R∕λ. The particle–detector separation is d 10R in all plots. One can see that both f and δ oscillate strongly with θdet , which is due to the interference of the incident and scattered waves in Eq. (3). At θdet 0, f 0 since there is no detector area S 2 and, consequently, δ C ext . As θdet increases, corresponding to an enlarging detector area, f begins to grow at the expense of δ and oscillations set in. Depending on the particle size, the oscillations attenuate with θdet at different rates, and f converges to C ext , whereas δ converges to zero. The explanation for this behavior is that, as the particle size grows, the scattering becomes increasingly concentrated around the forward direction. This is why the curves in plot (a), where R λ, converge slowly compared with the larger particles in plots (b)–(d). The interference energy flow also varies with the particle-detector separation d. To examine this dependence, Fig. 3 shows plots of f ∕C ext and δ∕C ext as a function of θdet for increasing values of d ranging from d 2R to 100R. Thus this range covers the near- to far-field zone transition. One can see that, generally, the farther away
Fig. 2. Behavior of the f and δ curves of Eqs. (9) and (10) as a function of θdet . Plots (a)–(c) show the curves for nonabsorbing spheres with m 1.33 0i and R λ, 2λ, and 5λ, respectively, whereas the sphere in plot (d) is absorbing with m 1.55 0.1i and R 5λ. Also shown are the points along the f curve, after which its error in approximating C ext drops below 10% (all plots) and 1% in plots (c) and (d).
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Fig. 3. Same as Fig. 2 except here the plots show the behavior of f and δ for increasing particle-detector separation d, ranging from the near field to the far field as shown. The particle is the same in each plot, with m 1.33 0i and R 2λ.
the detector is, the more rapidly the curves oscillate. However, in all cases it is clear that f ∕C ext is able to approximate the cross section well, even in the near field where d 2R, i.e., plot (a), provided that θdet is beyond 50°, for example. What remains to be seen is exactly how this extinction behavior is related to holography. To explain this, consider a Gabor-type, or in-line, holographic arrangement intended to image a single particle. Here a laser beam directly illuminates a position-sensitive detector facing the oncoming light, i.e., what is shown in Fig. 1. If a particle is introduced into the beam, the light scattered will interfere with the incident light and produce a fringe pattern across the detector. This interference pattern constitutes a digital hologram from which an image of the particle can be computationally reconstructed by applying the Fresnel–Kirchhoff diffraction theory, see [6,7]. Yet this is precisely the energy flow given by Eq. (2). Following the same argument as earlier, the contribution of hSsca it to this flow can be neglected if d is sufficiently large. In addition, the hologram can be subtracted from the corresponding particle-free measurement to yield a contrast hologram, the result of which is identical to the difference measurement of Eq. (8). Thus, by simply integrating the contrast hologram from the forwarddirection out, one can generate the f curves in Fig. (2) and obtain an estimate for C ext . The accuracy of the estimate will depend on the particle size to wavelength ratio R∕λ and the angular size of the detector θdet , but, as shown in Figs. 2 and 3, this estimate can be within a several-percent error from the true cross-section value. Thus, from a single contrast-hologram measurement, it is possible to extract an unambiguous image of the particle simultaneously with a measurement of its extinction cross section. The reader is referred to [7] for an example of such imaging on an aerosol particle in situ.
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Returning to Figs. 2 and 3, the oscillations in f might suggest that an accurate determination of C ext is problematic, as these oscillations can give large errors even for large detector sizes, e.g., θdet ∼ 65° . However, provided that the detector is position sensitive, such that f can be computed from a difference measurement, one can compute the average of the oscillations in the f curve, which, as Figs. 2 and 3 show, would yield a far better estimate for C ext . For example, consider the case shown in Fig. 2, plot (d). Here one could use a small detector with angular size θdet ∼ 15° and have an error of 9%. Yet, provided that the first few oscillations in f are generated from the measurement by evaluating the integral in Eq. (9), one could obtain C ext to much better accuracy. Moreover, if the detector size is increased to θdet ∼ 45° , and the oscillations were averaged, an error less than 1% could be expected. What this work does not consider is the effect of particle shape. Overall, one would expect the same general trends depicted in Figs. 2 and 3 to be seen for nonspherical particles since the concentration of scattering around the forward direction with increasing particle size is a general effect. Moreover, the work in [8], which investigates the interference energy flow for cubical particles, suggests that this concentrated forward scattering is only enhanced by the departure from a spherical particle shape. This suggests that the difference measurement described above for C ext would be at least as effective, if not potentially more so, for nonspherical particles as compared with the spherical particles studied here. In conclusion, this work shows that the same interference pattern associated with a particle’s hologram is inherently related to extinction. By integrating the contrast hologram, which is simply a difference between
the hologram measurement without and with the particle present in a beam, one obtains an approximation for the particle’s extinction cross section. While the accuracy of this value depends on the detector’s angular size, Mie theory simulations are used to show that the error in the cross section value can be less than 1% for a particle with a size-to-wavelength ratio greater than five, roughly, and a detector angular size of around 45°. Such results establish the theoretical feasibility of simple single-particle extinction cross-section measurements in a contact-free manner using digital holography. The authors are grateful for useful discussions with Alex Yuffa. This work was supported by the U.S. Army Research Office under the Young Investigator Program, grant number W911NF-12-1-0032. References 1. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1957). 2. C. F. Borhen and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983). 3. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002). 4. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, J. Opt. Soc. Am. A 25, 1504 (2008). 5. M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. M. van der Mee, J. Quant. Spectrosc. Radiat. Transfer 110, 323 (2009). 6. T. Kreis, Handbook of Holographic Interferometry; Optical and Digital Methods (Wiley, 2005). 7. M. J. Berg and G. Videen, J. Quant. Spectrosc. Radiat. Transfer 112, 1776 (2011). 8. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, J. Opt. Soc. Am. A 25, 1514 (2008).
