## Microwave measurements of the full amplitude scattering matrix of a complex aggregate: a database for the assessment of light scattering codes

Optics Express, Vol. 18, Issue 3, pp. 2056-2075 (2010)

http://dx.doi.org/10.1364/OE.18.002056

Acrobat PDF (1198 KB)

### Abstract

We present an extensive experimental study of microwave scattering by a fully characterized complex aggregate. We measured the full amplitude scattering matrix (amplitude and phase of the four elements) for a wide range of configurations. The presented results are of special interest to the light scattering community. Our experiments offer the possibility to validate numerical methods against experiments, since the geometrical and dielectric properties of the complex target are known to a high degree of precision, a situation difficult to attain in the optical regime. We analyze in detail the behaviour of amplitude and phase as a function of the scattering angle and target orientation. Furthermore, we compare different computational methods for a specific experimental configuration.

© 2010 Optical Society of America

## 1. Introduction

1. M. I. Mishchenko, “Scale invariance rule in electromagnetic
scattering,” J. Quant. Spectrosc. Radiat. Transf. **101**, 411–415
(2006). [CrossRef]

*r*) and incident wavelength

*λ*by the same factor, keeping fixed their size parameter

*x*= 2

*πr*/

*λ*and complex refractive index, the scattering behaviour of the analog object is identical to the original one. This so-called Microwave Analogy principle has been applied since the fifties in order to analyze optical scattering properties of non-spherical particles and systems [2

2. J. M. Greenberg, N. E. Pedersen, and J. C. Pedersen, “Microwave analog to the scattering of light by
nonspherical particles,” J. Appl. Phys. **32**, 233–242
(1961). [CrossRef]

3. B. Gustafson, *Light Scattering by Nonspherical Particles: Theory,
Measurements, and Applications* (Academic
Press, 2000), chap. Microwave Analog to
Light-Scattering Measurements, pp. 367–390. [CrossRef]

2. J. M. Greenberg, N. E. Pedersen, and J. C. Pedersen, “Microwave analog to the scattering of light by
nonspherical particles,” J. Appl. Phys. **32**, 233–242
(1961). [CrossRef]

3. B. Gustafson, *Light Scattering by Nonspherical Particles: Theory,
Measurements, and Applications* (Academic
Press, 2000), chap. Microwave Analog to
Light-Scattering Measurements, pp. 367–390. [CrossRef]

4. B. A. S. Gustafson, “Microwave analog to light scattering measurements: a
modern implementation of a proven method to achieve precise
control.” J. Quant. Spectrosc. Radiat. Transf. **55**, 663–672
(1996). [CrossRef]

4. B. A. S. Gustafson, “Microwave analog to light scattering measurements: a
modern implementation of a proven method to achieve precise
control.” J. Quant. Spectrosc. Radiat. Transf. **55**, 663–672
(1996). [CrossRef]

5. R. H. Zerull, B. Gustafson, K. Schulz, and E. Thiele-Corbach, “Scattering by aggregates with and without an
absorbing mantle: Microwave analog experiments,” Appl.
Opt. **32**, 4088–4100
(1993). [PubMed]

6. Y.-L. Xu and B. A. S. Gustafson, “A generalized multiparticle mie-solution: further
experimental verification,” J. Quant. Spectrosc. Radiat.
Transf. **70**, 395–419
(2001). [CrossRef]

7. L. Kolokolova and B. A. S. Gustafson, “Scattering by inhomogeneous particles: microwave
analog experiments and comparison to effective medium theories,”
J. Quant. Spectrosc. Radiat. Transf. **70**, 611–625
(2001). [CrossRef]

8. J. E. Thomas-Osip, B. A. S. Gustafson, L. Kolokolova, and Y.-L. Xu, “An investigation of titan’s aerosols using microwave
analog measurements and radiative transfer modeling,”
Icarus **179**, 511–522
(2005). [CrossRef]

4. B. A. S. Gustafson, “Microwave analog to light scattering measurements: a
modern implementation of a proven method to achieve precise
control.” J. Quant. Spectrosc. Radiat. Transf. **55**, 663–672
(1996). [CrossRef]

5. R. H. Zerull, B. Gustafson, K. Schulz, and E. Thiele-Corbach, “Scattering by aggregates with and without an
absorbing mantle: Microwave analog experiments,” Appl.
Opt. **32**, 4088–4100
(1993). [PubMed]

7. L. Kolokolova and B. A. S. Gustafson, “Scattering by inhomogeneous particles: microwave
analog experiments and comparison to effective medium theories,”
J. Quant. Spectrosc. Radiat. Transf. **70**, 611–625
(2001). [CrossRef]

8. J. E. Thomas-Osip, B. A. S. Gustafson, L. Kolokolova, and Y.-L. Xu, “An investigation of titan’s aerosols using microwave
analog measurements and radiative transfer modeling,”
Icarus **179**, 511–522
(2005). [CrossRef]

6. Y.-L. Xu and B. A. S. Gustafson, “A generalized multiparticle mie-solution: further
experimental verification,” J. Quant. Spectrosc. Radiat.
Transf. **70**, 395–419
(2001). [CrossRef]

9. Y.-L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of
spheres: Theoretical and experimental study of the amplitude scattering
matrix,” Phys. Rev. E **58**, 3931–3948
(1998). [CrossRef]

## 2. Model aggregate

### 2.1. Choice of the model

*N*is the number of monomers in the aggregate,

*k*

_{0}is the prefactor,

*R*the gyration radius,

_{g}*a*the monomer radius and

*D*the fractal dimension.

_{f}*x*of the single spheres varies from 0.16 to 0.52 in the frequency interval [3 – 10] GHz), thus making it harder to carry out meaningful comparisons with the numerical methods. Over the interval [15 – 20] GHz, the refractive index can be considered constant and we took

*n*= √

*ε*= 1.688. This value is slightly different from the one we used in [12

12. O. Merchiers, J.-M. Geffrin, R. Vaillon, P. Sabouroux, and B. Lacroix, “Microwave analog to light scattering measurements on
a fully characterized complex aggregate,” Appl. Phys.
Lett. **94**, 181107-+ (2009). [CrossRef]

### 2.2. Target construction method

## 3. Experimental set-up: microwave scattering facility

13. C. Eyraud, J.-M. Geffrin, P. Sabouroux, P. C. Chaumet, H. Tortel, H. Giovannini, and A. Litman, “Validation of a 3D bistatic microwave scattering
measurement setup,” Radio Sci. **43**, 4018-+ (2008). [CrossRef]

14. J.-M. Geffrin and P. Sabouroux, “Continuing with the fresnel database: experimental
setup and improvements in 3d scattering measurements,”
Inverse Problems **25**, 024001 (18pp) (2009). [CrossRef]

15. C. Eyraud, J.-M. Geffrin, A. Litman, P. Sabouroux, and H. Giovannini, “Drift correction for scattering
measurements,” Appl. Phys. Lett. **89**, 244104-3 (2006). [CrossRef]

**55**, 663–672
(1996). [CrossRef]

12. O. Merchiers, J.-M. Geffrin, R. Vaillon, P. Sabouroux, and B. Lacroix, “Microwave analog to light scattering measurements on
a fully characterized complex aggregate,” Appl. Phys.
Lett. **94**, 181107-+ (2009). [CrossRef]

*k*=

*i, s*which stand for incident and scattered respectively.

**E**

^{k}is the incident/scattered field, and (

**e**

_{θ},

**e**

_{ϕ}) are the basis vectors of a spherical coordinate system, where the somewhat unusual convention for the polar and azimuthal angle notations (

*θ, ϕ*) is the one defined in [17

17. D. Zwillinger, *CRC Standard Mathematical Tables and Formulae*
(Chapman & Hall/CRC: 31st revised edition, 2002). [CrossRef]

## 4. Numerical analysis

### 4.1. Electromagnetic scattering codes: a short description

18. B. Draine and P. Flatau, “Discrete-dipole approximation for scattering
calculations,” J. Opt. Soc. Am. A **11**, 1491–1499
(1994). [CrossRef]

20. C. Eyraud, A. Litman, A. Herique, and W. Kofman, “Microwave imaging from experimental data within a
bayesian framework with realistic random noise,” Inverse
Problems **25**, 024005 (2009). [CrossRef]

21. D. Mackowski, “Calculation of total cross sections of
multiple-sphere clusters,” J. Opt. Soc. Am. A **11**, 2851 (1994). [CrossRef]

22. D. W. Mackowski and M. I. Mishchenko, “Calculation of the t matrix and the scattering
matrix for ensembles of spheres,” J. Opt. Soc. Am.
A **13**, 2266–2278
(1996). [CrossRef]

23. B. Stout, J. C. Auger, and A. Devilez, “Recursive t matrix algorithm for resonant multiple
scattering: applications to localized plasmon excitations,”
J. Opt. Soc. Am. A **25**, 2549 (2008). [CrossRef]

#### 4.1.1. Volume Integral methods

*ε*

_{0}is the vacuum permittivity,

*ε*(

_{r}**r**′) is the real part of the permittivity and

*ε*(

_{i}**r**′) is the imaginary part at point

**r**′. The scattered field

**E**on the receiver positions

^{s}**r**∈ Γ is obtained with the observation equation [24]:

**G**is the free-space dyadic Green function between

_{0r}**r**′ ∈ Ω and

**r**∈ Γ, χ is the contrast (χ(

**r**′) =

*k*

^{2}(

**r**′) -

**k**

^{2}

_{0}) with

*k*the wavenumber in the zone Ω,

*k*

_{0}the wavenumber in vacuum and

**E**is the field inside the zone Ω.

**r**′ ∈ Ω):

**E**

^{0}is the incident field and

**G**is the free-space dyadic Green function.

_{00}25. H. van der Vorst, *Iterative Krylov Methods for Large Linear Systems*
(Cambridge University Press,
2003). [CrossRef]

26. F. M. Kahnert, “Numerical methods in electromagnetic scattering
theory,” J. Quant. Spectrosc. Radiat. Transf. **79–80**, 775–824
(2003). [CrossRef]

*O*(

*NlogN*) and the memory requirement is

*O*(

*N*), these methods allows the calculation of the scattered field for large objects. In our case, the aggregate is a relatively large target at the frequency of 20 GHz since the enclosing sphere has a volume of ≈ 150

*λ*

^{3}. Cubic cells with lattice spacing parameter

*d*are used in both methods for the computations presented here.

#### 4.1.2. T-matrix method

23. B. Stout, J. C. Auger, and A. Devilez, “Recursive t matrix algorithm for resonant multiple
scattering: applications to localized plasmon excitations,”
J. Opt. Soc. Am. A **25**, 2549 (2008). [CrossRef]

*N*-particle system, (labeled by

*j*= 1,..

*N*), completely characterizes its electromagnetic response by relating a column ‘matrix’,

*f*

^{(j)}, composed of the coefficients of the field scattered by a particle to a column ‘matrix’ ,

*e*

^{(j)}, containing the field coefficients of the field incident on the particle (called the

*excitation*field). In this convenient matrix notation [27], the T-matrix for the particle has the property,

*f*

^{(j)}=

*T*

^{(j)}

*e*

^{(j)}. For a sphere, this matrix is diagonal and its matrix elements can be expressed analytically in terms of the ‘Mie’ coefficients [29, 30].

31. M. Lax, “Multiple scattering of waves,”
Rev. Mod. Phys. **23**, 287 (1951). [CrossRef]

23. B. Stout, J. C. Auger, and A. Devilez, “Recursive t matrix algorithm for resonant multiple
scattering: applications to localized plasmon excitations,”
J. Opt. Soc. Am. A **25**, 2549 (2008). [CrossRef]

33. O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use
of the vector addition theorem,” J. Opt. Soc. Am.
B **22**, 1620–1631
(2005). [CrossRef]

34. J. Auger and B. Stout, “A recursive centered t-matrix algorithm to solve the
multiple scattering equation: numerical validation,” J.
Quant. Spectrosc. Radiat. Transf. **79**, 533–547
(2003). [CrossRef]

### 4.2. Calibration of experimental and simulated data

*C*is determined using a reference target in the

*ϕ*co-polarization case [14

14. J.-M. Geffrin and P. Sabouroux, “Continuing with the fresnel database: experimental
setup and improvements in 3d scattering measurements,”
Inverse Problems **25**, 024001 (18pp) (2009). [CrossRef]

35. P. van den Berg, M. Cote, and R. Kleinman, “;“blind” shape reconstruction from experimental
data,” Antennas and Propagation , IEEE
Transactions on **43**, 1389–1396
(1995). [CrossRef]

*E*(resp.

^{s}_{c,i}*E*) is the scattered field calculated (resp. measured),

^{s}_{m,i}*N*the number of receiver positions and

_{r}*x̄*is the complex conjugated of

*x*. A metallic sphere was chosen as a reference target as it can be accurately approximated by a perfect conductor in the microwave domain and as its scattered field can also be readily computed using the Mie series. All the experimental fields are then calibrated using this multiplicative coefficient

*C*.

### 4.3. Convention for the incident and scattered polarization vectors

*ϕ*. We say that

_{E}*ϕ*= 90° when it is located in the azimuthal plane and

_{E}*ϕ*= 0° when it is at the zenith point of the arch. When the emitter is placed at

_{E}*ϕ*= 90°, the azimuthal plane corresponds to the scattering plane (defined by incident and scattered wavevectors). We will call this configuration the ‘In Plane’ scattering configuration (IP). When the emitter is moved along the vertical arch out of the azimuthal plane (-11° ≤

_{E}*ϕ*≤ 179°), we have a scattering plane which is different from the azimuthal plane. We will call this situation ‘Out of Plane’ scattering configuration (OP). In this case, it should be noted that the receiver position no longer corresponds to the scattering angle. By scattering angle we mean the angle 6S defined by cos

_{E}*θ*=

_{S}**k**̂

*·*

_{i}**k̂**

_{s}where

**k̂**

_{i}and

**k̂**

_{s}are the unit vectors corresponding with incident and scattered directions respectively. We adopt the notations

*θ*for scattering angle and

_{S}*θ*for the receiver position angle. We nevertheless represent all the experimental data with respect to the receiver positions (

_{R}*θ*). In Fig. 3, we represented the scattering angle as a function of receiver position. For the IP configuration both are identical, but for the OP situation, we see that the scattering angle only spans the interval [30°, 123.83°] instead of [0°, 130°].

_{R}*S*

_{1}and

*S*

_{2}, the co-polarization elements and

*S*

_{3}and

*S*

_{4}, the cross-polarizations. The BH convention differs considerably from the one that is used in our experimental set-up. In the above conventions, incident and scattered polarization vectors are determined by the scattering plane as clearly expressed by the previous definition. For the In Plane measurements, the scattering plane remains the same for different scattering angles. For the Out of Plane measurements however, at each scattering angle, we have a different scattering plane. In order to measure the matrix elements, incident and scattered polarization should be changed accordingly. This is very unpractical with our set-up and would be a new source of measurement errors. It is for this reason that in the experimental set-up, fixed polarizations of emitter and receiver are chosen. Hence, the measured matrix elements are no longer identical to those given by Eq. (8). In order to distinguish between these two conventions, we use another notation for the experimentally obtained matrix elements:

*θ*and

*ϕ*indicate the polarization state as defined in section 3.

*M*

_{TP}is the amplitude scattering matrix in the ‘Theta-Phi’ convention as used for the raw measurements and

*M*

_{BH}is the one in ‘Bohren/Huffman’ convention. The matrices

*T*

_{i}and

*T*

_{s}are the rotation matrices which transform the incident and scattered field components from the TP- to the BH-convention. The TP-convention has the disadvantage that the off-diagonal terms include the contributions of the

*S*

_{1}and

*S*

_{2}matrix elements, which means that

*S*and

_{ϕθ}*S*can’t be interpreted as the pure cross-polarization signal. The

_{θϕ}*S*

_{3}and

*S*

_{4}elements on the the other hand, do offer such an interpretation. For this reason, and because the light scattering community is familiar with it, we will present the results in the BH-convention. For the In Plane measurements, both transformation matrices in Eq. (10) reduce to the identity matrix.

## 5. Results

*S*

_{1}and

*S*

_{3}) of the four matrix elements (both amplitude and phase) as a function of frequency for a single position (position (c) in Fig. 6). We concentrate efforts on those two elements since the discussion for the

*S*

_{2}and

*S*

_{4}elements is completely analogous. We then consider a single frequency and analyze the scattering matrix as a function of the aggregate orientation. Finally, we compare the different computational methods for one frequency and one orientation.

### 5.1. Measurements as a function of frequency for a single position

*S*

_{1}matrix element is depicted in Fig. 4(a). For each frequency, we show the measured amplitudes for both In and Out of azimuthal Plane configurations. Furthermore, in this series of results, we compared the experimental amplitudes with those obtained by Mackowski’s T-matrix code. Here we chose only to compare with one numerical method in order to make interpretation easier. In Fig. 4(b) we did the same for the phase of

*S*

_{1}. We will start by discussing the amplitude. As can be seen, the In Plane configuration (IP) simulations compare very well to the experiment. For the Out of Plane configuration (OP), differences can be observed especially when the receiver position is close to 60° for 16, 17 and 18 GHz. Those differences are most probably due to tiny geometrical inconsistencies between the input coordinates of the spheres used by the code and the real positions of the spheres in the realized aggregate.

*q*

^{-1}where

*q*is the amplitude of the scattering wavevector defined by:

*θ*is the scattering angle and

_{S}*λ*is the wavelength of the incident radiation. All scatterers inside a region with radius equal to

*q*

^{-1}can be considered to scatter in phase.

*qR*< 1:

_{g}*I*(

*q*) is the co-polarized intensity. This equation is often called the Guinier equation [36] and describes the region of the forward scattering lobe. It depends only on the size of the object and is valid for all types of aggregates (not only fractal ones).

*q*is never zero since

**k**

_{i}and

**k**

_{s}are never colinear. For this aggregate, the minimum value of

*qR*is attained for the scattering direction which lies in the plane defined by the vertical arch. In the frequency range [15–20

_{g}15. C. Eyraud, J.-M. Geffrin, A. Litman, P. Sabouroux, and H. Giovannini, “Drift correction for scattering
measurements,” Appl. Phys. Lett. **89**, 244104-3 (2006). [CrossRef]

*qR*< 9.09 which clearly does not satisfy the Guinier condition, and hence, no forward scattering lobe is observed.

_{g}*S*

_{1}as a function of frequency. In the region corresponding with the forward scattering lobe, we see a constant value of the phase. As we have written before, we have here

*qR*< 1 such that the whole aggregate lies in one

_{g}*q*-region and hence all constituent spheres scatter in phase. When the scattering angle is increased, higher

*q*-values are scanned, which corresponds to smaller spatial regions of the object and we are able to resolve smaller details of the object. For our aggregate, this implies that we gain insight of its internal anisotropy. We are sure this internal structure is not due to the excitation of higher multipole moments of the single particles, since in the considered frequency range, the scattering pattern is very close to the dipolar one except for a slight asymmetry between forward- and backscattering hemispheres.

*qR*is always larger than one, no constant phase region is observed around

_{g}*θR*≈ 0°.

*S*

_{3}matrix element is represented, again as a function of frequency.

*S*

_{3}(and

*S*

_{4}) are different from zero due to multiple scattering, no forward scattering lobe is observed and the scattered cross-polarized amplitude is almost isotropic [36]. The presence of a forward lobe is a single scattering effect and gives direct size information. Since for the cross-polarizations, no such lobe is present, a model should be built in order to extract the relevant information. While for the co-polarized amplitude, it is quite easy to develop such a model, it is a much harder task for the cross-polarized intensities. This is an unsatisfactory situation since those polarizations contain information about the system shape and anisotropy.

*S*

_{3}and

*S*

_{4}. Before continuing our analysis, we recall that the phase for these elements does not have the same interpretation as for the diagonal elements. The scaling approach is no longer useful here. If we consider the previous situation of

*qR*< 1, then the scattered waves from each scatterer are by definition not in phase since they are the result of multiple scattering. In this sense every scattering direction is equivalent. Hence, the interpretation of

_{g}*q*as a measure of the resolution is no longer valid. This explains why both In and Out of Plane results of amplitude and phase as a function of the receiver position do not exhibit qualitative differences between them.

### 5.2. Measurements for different orientations at a single frequency

### 5.3. Comparison of numerical methods at a single frequency

#### 5.3.1. Extremal curves

#### 5.3.2. Comparison of the methods

*S*

_{1}element is reproduced almost within experimental errors. For the phase, differences cannot be explained by the experimental noise only. In order to estimate correctly the degree of confidence with which the computational methods reproduce the experimental results and the importance of each error source in the final result, a much more detailed study should be realized. With the presented results, we can roughly estimate the positioning error and indicate the presence of the geometrical inconsistencies in the model. It is nevertheless possible and useful here to make a more quantitative analysis. Therefore we shall use an error function defined as:

*E*

^{s}_{m,1},…,

*E*) =

^{s}_{m,Nr}**E**

*and analogously for the simulated fields) and applied the definition of the quadratic vector norm. The values of*

^{s}_{m}*f*

^{err}for the four matrix elements and for each method are given in table 1. We see immediately that the values of the error function for the elements

*S*

_{1}and

*S*

_{2}are an order of magnitude lower than the ones for

*S*

_{3}and

*S*

_{4}. This is simply due to the fact that the signal level is 10 times lower for

*S*

_{3}and

*S*

_{4}than for

*S*

_{1}and

**S**

_{2}. Hence for a constant noise level (∥

**E**

^{exp}-

**E**

^{sim}∥

^{2}is approximately constant for the co- and cross-terms), we have an error function which is ten times larger. On the whole we see that the

*f*

^{err}-values are of the same order of magnitude for each of the methods and matrix elements.

*d*. Its value is imposed by the size of the scatterer and the refractive index. The most commonly used discretization parameter that takes into account both physical variables is

*y*= |

*n*|

*kd*where

*n*is the refractive index and

*k*the magnitude of the wavevector in vacuum [38, 39

39. M. Yurkin and A. Hoekstra, “The discrete dipole approximation: An overview and
recent developments,” J. Quant. Spectrosc. Radiat.
Transf. **106**, 558–589
(2007). [CrossRef]

*n*|

*kd*≲ 0.5 [40

40. B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti - wave propagation on a
polarizable point lattice and the discrete dipole approximation,”
Astrophys. J. **405**, 685–697
(1993). [CrossRef]

*d*around 0.71 mm. In order not to be just on the upper limit, we chose

*d*= 0.6 mm for both ddscat7.0 and MoM calculations.

39. M. Yurkin and A. Hoekstra, “The discrete dipole approximation: An overview and
recent developments,” J. Quant. Spectrosc. Radiat.
Transf. **106**, 558–589
(2007). [CrossRef]

40. B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti - wave propagation on a
polarizable point lattice and the discrete dipole approximation,”
Astrophys. J. **405**, 685–697
(1993). [CrossRef]

42. W. J. Wiscombe, “Improved mie scattering algorithms,”
Appl. Opt. **19**, 1505–1509
(1980). [CrossRef] [PubMed]

*x*is the size parameter of the circumscribing sphere of the cluster. However, both T-matrix codes have more sophisticated tests in order to determine

_{e}*N*. Since all those estimations rely on the internal convergence tests, it would be highly desirable to have another independent criterion based on comparison with experiment. Such a criterion would have a more objective value but is clearly harder to establish. Indeed, it is much easier to cover a wide parameter space by means of numerical simulations than by experiments. Moreover, different kinds of errors have to be corrected before the experiments can be used properly. However, in our opinion, experiments remain the only real test of a theory. Hence, the attainable precision of experimental studies gives a quantitative estimation of the precision limit to which simulations have to be compared.

_{O,c}#### 6. Conclusion

## Acknowledgments

## References and links

1. | M. I. Mishchenko, “Scale invariance rule in electromagnetic
scattering,” J. Quant. Spectrosc. Radiat. Transf. |

2. | J. M. Greenberg, N. E. Pedersen, and J. C. Pedersen, “Microwave analog to the scattering of light by
nonspherical particles,” J. Appl. Phys. |

3. | B. Gustafson, |

4. | B. A. S. Gustafson, “Microwave analog to light scattering measurements: a
modern implementation of a proven method to achieve precise
control.” J. Quant. Spectrosc. Radiat. Transf. |

5. | R. H. Zerull, B. Gustafson, K. Schulz, and E. Thiele-Corbach, “Scattering by aggregates with and without an
absorbing mantle: Microwave analog experiments,” Appl.
Opt. |

6. | Y.-L. Xu and B. A. S. Gustafson, “A generalized multiparticle mie-solution: further
experimental verification,” J. Quant. Spectrosc. Radiat.
Transf. |

7. | L. Kolokolova and B. A. S. Gustafson, “Scattering by inhomogeneous particles: microwave
analog experiments and comparison to effective medium theories,”
J. Quant. Spectrosc. Radiat. Transf. |

8. | J. E. Thomas-Osip, B. A. S. Gustafson, L. Kolokolova, and Y.-L. Xu, “An investigation of titan’s aerosols using microwave
analog measurements and radiative transfer modeling,”
Icarus |

9. | Y.-L. Xu and R. T. Wang, “Electromagnetic scattering by an aggregate of
spheres: Theoretical and experimental study of the amplitude scattering
matrix,” Phys. Rev. E |

10. | R. A. Dobbins and C. M. Megaridis, “Morphology of flame-generated soot as determined by
thermophoretic sampling,” Langmuir |

11. | I. A. Kilin, “A nonintrusive diagnostics technique for flame soot based on near-infrared emission spectrometry,” Ph.D. thesis, Middle-East Technical University, Ankara, Turkey and INSA Lyon, Villeurbanne, France (2007). |

12. | O. Merchiers, J.-M. Geffrin, R. Vaillon, P. Sabouroux, and B. Lacroix, “Microwave analog to light scattering measurements on
a fully characterized complex aggregate,” Appl. Phys.
Lett. |

13. | C. Eyraud, J.-M. Geffrin, P. Sabouroux, P. C. Chaumet, H. Tortel, H. Giovannini, and A. Litman, “Validation of a 3D bistatic microwave scattering
measurement setup,” Radio Sci. |

14. | J.-M. Geffrin and P. Sabouroux, “Continuing with the fresnel database: experimental
setup and improvements in 3d scattering measurements,”
Inverse Problems |

15. | C. Eyraud, J.-M. Geffrin, A. Litman, P. Sabouroux, and H. Giovannini, “Drift correction for scattering
measurements,” Appl. Phys. Lett. |

16. | “Agilent 85301b/c antenna measurement systems 45 mhz to 110 ghz configuration guide,”. |

17. | D. Zwillinger, |

18. | B. Draine and P. Flatau, “Discrete-dipole approximation for scattering
calculations,” J. Opt. Soc. Am. A |

19. | B. T. Draine and P. J. Flatau, |

20. | C. Eyraud, A. Litman, A. Herique, and W. Kofman, “Microwave imaging from experimental data within a
bayesian framework with realistic random noise,” Inverse
Problems |

21. | D. Mackowski, “Calculation of total cross sections of
multiple-sphere clusters,” J. Opt. Soc. Am. A |

22. | D. W. Mackowski and M. I. Mishchenko, “Calculation of the t matrix and the scattering
matrix for ensembles of spheres,” J. Opt. Soc. Am.
A |

23. | B. Stout, J. C. Auger, and A. Devilez, “Recursive t matrix algorithm for resonant multiple
scattering: applications to localized plasmon excitations,”
J. Opt. Soc. Am. A |

24. | J. Kong, |

25. | H. van der Vorst, |

26. | F. M. Kahnert, “Numerical methods in electromagnetic scattering
theory,” J. Quant. Spectrosc. Radiat. Transf. |

27. | W. C. Chew, |

28. | L. Tsang, J. A. Kong, and R. T. Shin, |

29. | M. Mishchenko, L. Travis, and A. Lacis, |

30. | G. Bohren and D. Huffman, |

31. | M. Lax, “Multiple scattering of waves,”
Rev. Mod. Phys. |

32. | M. Mishchenko, G. Videen, V. Babenko, N. Khlebtsov, and T. Wriedt, “T-matrix theory of electromagnetic scattering by
partciles and its applications: a comprehensive reference
database,” J. Quant. Spectrosc. Radiat. Transf.. |

33. | O. Moine and B. Stout, “Optical force calculations in arbitrary beams by use
of the vector addition theorem,” J. Opt. Soc. Am.
B |

34. | J. Auger and B. Stout, “A recursive centered t-matrix algorithm to solve the
multiple scattering equation: numerical validation,” J.
Quant. Spectrosc. Radiat. Transf. |

35. | P. van den Berg, M. Cote, and R. Kleinman, “;“blind” shape reconstruction from experimental
data,” Antennas and Propagation , IEEE
Transactions on |

36. | C. M. Sorensen, “Light scattering by fractal aggregates: A
review,” Aerosol Sci. Technol. |

37. | J.-M. Geffrin, C. Eyraud, A. Litman, and P. Sabouroux, “Optimization of a bistatic microwave scattering
measurement setup: From high to low scattering targets,”
Radio Sci. |

38. | B. Draine, |

39. | M. Yurkin and A. Hoekstra, “The discrete dipole approximation: An overview and
recent developments,” J. Quant. Spectrosc. Radiat.
Transf. |

40. | B. T. Draine and J. Goodman, “Beyond Clausius-Mossotti - wave propagation on a
polarizable point lattice and the discrete dipole approximation,”
Astrophys. J. |

41. | D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” astro-ph/0403082 (2004). |

42. | W. J. Wiscombe, “Improved mie scattering algorithms,”
Appl. Opt. |

**OCIS Codes**

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

(290.5820) Scattering : Scattering measurements

(350.4010) Other areas of optics : Microwaves

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Scattering

**History**

Original Manuscript: December 10, 2009

Manuscript Accepted: January 11, 2010

Published: January 19, 2010

**Citation**

Olivier Merchiers, Christelle Eyraud, Jean-Michel Geffrin, Rodolphe Vaillon, Brian Stout, Pierre Sabouroux, and Bernard Lacroix, "Microwave measurements of the full amplitude scattering matrix of a
complex aggregate: a database for the assessment of light scattering codes," Opt. Express **18**, 2056-2075 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-3-2056

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