Akira Ishimaru, Sermsak Jaruwatanadilok and Yasuo Kuga
|
|
- Owen Hodge
- 5 years ago
- Views:
Transcription
1 INSTITUTE OF PHYSICS PUBLISHING Waves Random Media 4 (4) WAVES IN RANDOMMEDIA PII: S (4)789- Multiple scattering effects on the radar cross section (RCS) of objects in a random medium including backscattering enhancement and shower curtain effects Akira Ishimaru, Sermsak Jaruwatanadilok and Yasuo Kuga Department of Electrical Engineering, Box 355, University of Washington, Seattle, WA 9895, USA ishimaru@ee.washington.edu Received 4 March 4, in final form 8 May 4 Published 4 June 4 Online at stacks.iop.org/wrm/4/499 doi:.88/ /4/4/ Abstract This paper presents a theory of the radar cross section (RCS) of objects in multiple scattering random media. The general formulation includes the fourthorder moments including the correlation between the forward and the backward waves. The fourth moments are reduced to the second-order moments by using the circular complex Gaussian assumption. The stochastic Green s functions are expressed in parabolic approximation, and the objects are assumed to be large in terms of wavelength; therefore, Kirchhoff approximations are applicable. This theory includes the backscattering enhancement and the shower curtain effects, which are not normally considered in conventional theory. Numerical examples of a conducting object in a random medium characterized by the Gaussian and Henyey Greenstein phase functions are shown to highlight the difference between the multiple scattering RCS and the conventional RCS in terms of optical depth, medium location and angular dependence. It shows the enhanced backscattering due to multiple scattering and the increased RCS if a random medium is closer to the transmitter.. Introduction There has been extensive research on the radar cross section of rough surfaces and the atmosphere [ 4]. However, if an object is located within a random medium, its cross section is affected by multiple scattering in the medium. The wave incident upon the object consists of coherent and incoherent waves, and the scattered wave also experiences multiple scattering. In addition, the induced current on the object has coherent and incoherent components. To /4/4499+3$3. 4 IOP Publishing Ltd Printed in the UK 499
2 5 A Ishimaru et al r r t G(r,r ) Random medium G(r s,r t ) T r Object r s Figure. The Dirichlet object is illuminated by a transmitter at r t through a random medium and the scattered wave is observed at r. obtain the stochastic induced current, we need to solve the stochastic surface integral equation for the object [5, 6]. In this paper, however, we assume that the object size and surface radius of curvature are much greater than the wavelength, and therefore, the Kirchhoff approximation is applicable to obtain the induced current. There are correlations between the incident and scattered waves in the medium. To take into account the above effects, we start with general formulations including stochastic Green s functions. We make use of the parabolic equation approximation for the stochastic Green s functions, which should be applicable to many practical problems in microwave and optical scattering in the atmosphere and the ocean, as well as optical scattering in biological media [4]. The formulation is based on the use of the circular complex Gaussian assumption. It includes two effects: backscattering enhancement [7, 8] and shower curtain effects [9]. Both phenomena have been discussed recently, but have not been included in most RCS studies. The backscattering enhancement effect is interesting as RCS can become higher than the conventional RCS, while the shower curtain effects give different RCS depending on whether the random medium is closer to or further from the transmitter.. Formulation of the problem Let us consider a Dirichlet object in a random medium. The scattered field at r is given by [] ψ s ( r) = G( r, r ) ψ( r ) ds, () s n where G is the stochastic Green s function. In general, the random surface field ψ( r )/n can be expressed using the random transition (scattering) operator T, ψ( r ) = T( r, r s ) ψ in( r s ) ds, () n n s where ψ in ( r s ) is the incident field at r s as shown in figure. Now, we assume that the object size and its surface radius of curvature are greater than the wavelength and the Kirchhoff approximation is applicable. Then, we get the well-known Kirchhoff surface field T( r, r s ) = δ( r r s ), ψ( r ) = ψ in( r ) (3). n n The scattered field is then given by ψ s ( r) = G( r, r ) ψ i( r ) ds. (4) n
3 Multiple scattering effects on the RCS of objects in a random medium 5 ˆ n r G G Random medium r s r Object nˆ Figure. Stochastic Green s functions G and G. The scattered power at r is given by ψ s ( r) =4 ds ds G ψ i n G ψi, (5) n where G = G ( r, r ), ψ i = ψ i ( r ), G = G ( r, r ), ψ i = ψ i ( r ) as shown in figure. This contains the fourth-order moment, and using the circular complex Gaussian assumption, we can express the fourth-order moment in terms of second-order moments [5]. (see the appendix) G ψ i G ψi = G G n n ψ i ψi n n G ψ i G n ψi n + ψ i G n G ψi n. (6) It is important to note that equation (6) includes the correlation between the incident ψ i and the scattered field G, and this gives the backscattering enhancement to be discussed in a later section. In contrast, it is often assumed that there is no correlation between the incident and the scattered field and with this assumption, we get G ψ i G ψi = G G n n ψ i ψi. (7) n n We will show later the difference between our formulation in equation (6) and the formulation in equation (7). The apparent RCS of the object is then given by RCS = 4πL ψ s, (8) ψ o where ψ o is the free space incident field at the object, ψ s is the scattered field at the observation point and L is the distance between the object and the observation point. We now examine equations (5) and (6). The incident wave ψ i is the stochastic Green s function and we write ψ i = ψ( r ) = G ( r, r ), (9) ψ i = ψ( r ) = G ( r, r ), where we used the reciprocity: G ( r, r ) = G ( r, r). Under the parabolic equation approximation [4, ], Green s function is given by G = (slowly varying function of z and ρ)exp(ikz). () Therefore, n = ikŝ ˆn = ik(ẑ ˆn ), ()
4 5 A Ishimaru et al where ŝ ẑ is the direction of wave propagation. Using equations (9) and (), we get ψ s =4k (ds ds [ G G +4 G G G f G f + G f G f ](ẑ ˆn )(ẑ ˆn )), () where G = G + G f, G is the coherent component and G f is the incoherent component. Note that if we ignore the correlation between the incident and the scattered field as in equation (7), we have ψ s =4k (ds ds [ G G + G G G f G f + G f G f ](ẑ ˆn )(ẑ ˆn )). (3) The difference between equations () and (3) is that equation () includes the enhancement while equation (3) does not. We now need to evaluate equation () to obtain the RCS. 3. Stochastic Green s functions Let us now evaluate equation (). Under the parabolic approximation, we get G = ( ikρ 4πL exp +ikz τ ) o, z G = ( ikρ 4πL exp +ikz τ ) (4) o, z where τ o is the optical depth. We assume that the transmitter is in the far field of the object, and therefore kρ/ z and kρ/ z are negligibly small. Next we consider the incoherent mutual coherence function G f G f =Ɣ f (z, ρ ; z, ρ ). (5) The mutual coherence function Ɣ due to a point source has been obtained previously including the inhomogeneous random medium [4, 9, ] Ɣ = Ɣ(z, ρ ; z, ρ ) = G(z, ρ )G (z, ρ ) ( = (4πX) exp ik ρ ) d ρ c X H +ik(z z ), (6) where X = X(z) = z dz n(z ),H = z a dz + z b dz p(s)s ds[ J o(knsρ)]. The phase function p(s) is normalized so that p(s)s ds =, (7) ρ = X(z ) ρd X(z),n(z) is the refractive index, a is the absorption coefficient, b is the scattering coefficient and s = sin(θ/), θ = scattering angle. This is the mutual coherence function for a random medium consisting of a random distribution of particles. First, we note that the term k ρ d ρ c /X in the exponent is negligible when the transmitter is in the far field of the object. We then write the coherent Ɣ c and incoherent Ɣ i parts of Ɣ as Ɣ = Ɣ c + Ɣ i, Ɣ c = (4πX) exp( τ o +ik(z z )), (8) Ɣ i = (4πX) [exp( H) exp( τ o)]exp(ik(z z )),
5 Multiple scattering effects on the RCS of objects in a random medium 53 where H is given in equation (6) and H(ρ ) = τ o = d (a + b) dz = optical depth and d is the thickness of the random medium. Let us use the Henyey Greenstein (HG) phase function for the scattering medium. We have ( g) p(s) =, g = anisotropy factor, (9) (+g g cos θ) 3/ which satisfies the normalization p(s) d =. () 4π 4π It is more convenient to use s = sin(θ/). Then, we write p(s) = (+g) ( g) [ ( ) ], s ( g) 3/ o =, g s + s o () p(s)s ds =. Even though this is the HG phase function used for <s<(or<θ<π), it can be approximated by p(s) = [ s o ( ) ] 3/ s + s o p(s)s ds =, when g is close to one. Making use of this, and using p(s)j o (knsρ)s ds = exp( kns o ρ) (3) we get H(ρ) = z a dz + z () b dz [ exp( kns o ρ)]. (4) For large optical depth, the incoherent mutual coherence function Ɣ i in equation (8) is well approximated by expanding H(ρ) in equation (4) about ρ = and keeping the first term [ z z ] Ɣ i = (4πX) exp a dz bkns o ρ dz. (5) A better approximation which reduces to the proper limit as the optical scattering depth τ s and τ s, and as ρ and ρ is given by [] where Ɣ i = ρ o = ( (4πX) exp( τ a)[ exp( τ s )]exp ρ ) d ρ o +ik(z z ), (6) z b(z )kn(z )s o X(z ) X(z) dz [ exp( τ s )],
6 54 A Ishimaru et al τ a = d a dz is the optical absorption depth and τ s = d b dz is the optical scattering depth. The coherence length ρ o at the object is important to characterize RCS. If the uniform random medium with n(z) = is located from d to d + d, we get = ks [ ] o (d + d ) d τ s [ exp( τ s )]. (7) ρ o Ld If the phase function is given by the Gaussian function: p(s) = 4α p exp( α p s ) (8) with normalization p(s)s ds =, the incoherent mutual coherence function Ɣ i is given by Ɣ i = ( ) X exp( τ a)[ exp( τ s )]exp ρ d +ik(z ρo z ). (9) If the medium is uniform from d to d + d, we get = τ sk [ ] (d + d ) 3 d 3 α p L d [ exp( τ s )]. (3) ρ o 4. RCS of a large Dirichlet object Making use of the stochastic Green s functions and the mutual coherence functions, we can now evaluate the apparent RCS given in equation (8). Noting equation (), RCS is given by RCS = 4πL ψ s = I ψ o + I + I 3, (3) where I,I,I 3 are the three terms in equation (). We note that in equation (), we have ds (ẑ ˆn ) = dˆρ, ds (ẑ ˆn ) = dˆρ. (3) Therefore, we have where I = 4π λ exp( τ o)f, I = 4π λ (4exp( τ o))[exp( τ a )( exp( τ s ))]F, I 3 = 4π λ [exp( τ a)( exp( τ s ))] F 3, F = dˆρ dˆρ exp(ik(z z )), ( F = dˆρ dˆρ exp ik(z z ) ρ ) d ρ o ( ) = dˆρ dˆρ exp ik(z z ) ρ d ρo ( F 3 = dˆρ dˆρ exp ik(z z ) ρ ) d ρ o ( ) = dˆρ dˆρ exp ik(z z ) ρ d ρo (33) for HG medium, for Gaussian medium, for HG medium, for Gaussian medium,
7 Multiple scattering effects on the RCS of objects in a random medium 55 ρ Illuminated surface z(ρ) Shadow surface z Figure 3. Illuminated surface given by z = z( ρ). Tr Random medium nˆ ρ d d z θ z ρ z L Figure 4. Inclined square conducting plate illuminated by the source. The random medium with thickness d is located between the transmitter and the object. The plate size is a a, and ρ = x ˆx + y ŷ and ρ = x ˆx + y ŷ. (x,y) Shadow Illuminated z z Figure 5. Spherical object. and ρ d = ρ ρ, ρ = x ˆx + y ŷ, ρ = x ˆx + y ŷ. The illuminated surface of the object is given by z = z ( ˆρ ) and z = z ( ˆρ ) as shown in figure 3. The coherence length ρ o is given by equation (7) for a HG medium and by equation (3) for Gaussian medium. For a large conducting plate shown in figure 4, wehavez(ρ) = x tan θ, a cos θ a a cos θ a d ρ d ρ = dx dy dx dy, (34) a cos θ a a cos θ and z z = (x x ) tan θ,ρ d = (x x ) + (y y ). The integral in (34) can be calculated numerically. For a sphere with radius a shown in figure 5, weuse a π a π d ρ d ρ = ρ dρ dφ ρ dρ dφ, (35) and z = a a x y,z = a a x y, ρ = (x x ) + (y y ), x = ρ cos φ,y = ρ sin φ,x = ρ cos φ,y = ρ sin φ. a
8 56 A Ishimaru et al Case A: d /L= d /L=.5 Case B: d /L=.5 d /L=.5 Tr Tr Target size is a Figure 6. Geometry of RCS calculations. L Coherence length ( ρ o ) in λ (A) Case A Case B Coherence length ( ρ o ) in λ (B) Case A Case B Optical depth (τ ) o Optical depth (τ ) o Figure 7. Coherence length for (A) Henyey Greenstein phase function with albedo (W o ) =.9 and g =.85 and (B) Gaussian phase function with W o =.9 andα p which gives the same half-power beamwidth. 5. Coherence length and shower curtain effects Equations (7) and (3) give general expressions of coherence length ρ o for HG and Gaussian media. It is seen that ρ o in terms of the wavelength depends on the optical scattering depth τ s, the medium characteristics (s o and α p ) and the ratio of the distances (d /L and d /L). In figures 6 and 7, we show the geometry and the coherence length in wavelength for case A and case B. Note that if the medium is close to the transmitter (case A), the coherence length is greater than case B, showing the shower curtain effect. For a HG medium, we use W o =.9 and g =.85 where W o is the albedo defined by W o = b. For a Gaussian medium, we use a+b W o =.9 and α p which gives the same half-power beamwidth as HG with g = RCS of a square plate Let us examine the RCS of the square plate shown in figure 4. The plate size a is 3λ. In figure 8,weshowI,I,I 3 and I com = I +I +I 3 at θ =. Note that I is the conventional RCS
9 Multiple scattering effects on the RCS of objects in a random medium 57 (A) (B) I fs I fs 3 3 RCS (db) 4 RCS (db) 4 5 I 5 I I I I 6 3 I com I fs Optical Depth (τ ) o I 6 3 I com I fs Optical Depth (τ ) o Figure 8. Components of RCS. Phase function is Henyey Greenstein with albedo (W o ) =.9 and g =.85 in all case. (A) Case A geometry, (B) case B geometry. fs denotes free space. I fs denotes the free space RCS. 5 5 RCS(dB) 3 35 OD=. 4 OD= OD=5 OD= g Figure 9. Effect of asymmetry factor g. in a scattering medium and I com is increasingly larger than I as the optical depth increases. This indicates that the multiple scattering contributes significantly to the total RCS. The effects of asymmetry factor g and the albedo W o are shown in figures 9 and, respectively. It shows that as g increases, the scattering pattern becomes more peaked in the forward direction and as W o increases, the scattering increases, resulting in increased RCS. We also compare the RCS from the geometry of case A and case B, which are shown in figure. Note that if the medium is close to the transmitter (case A), the RCS is higher than that of case B where the random medium is away from the transmitter. This result illustrates the shower curtain effect. Figure shows the shower curtain effects as the random medium moves from d = tod = L/. Figure 3 compares RCS, which includes backscattering enhancement from equation () with RCS without enhancement from equation (3). Note that the enhancement approaches 3 db for large optical depth. Figure 4 shows the angular dependence of RCS at an optical depth of and 5. It is shown that RCS, as a function of angle, is very much affected by
10 58 A Ishimaru et al 3 RCS(dB) OD=. 7 OD= OD=5 8 OD= Wo Figure. Effect of albedo W o. RCS (db) I fs 3 Case A 34 Case B Optical Depth (τ ) o Figure. The shower curtain effect with Henyey Greenstein phase function for geometry in case A and case B geometry. I fs denotes the free space RCS. Normalized RCS(dB) 3 OD=. 4 OD= OD=5 OD= d /L Figure. The shower curtain effect as a function of d /L. the multiple scattering. This is more pronounced for large optical depth where RCS is quite different from the conventional RCS (I ).
11 Multiple scattering effects on the RCS of objects in a random medium 59 5 RCS (db) 5 3 I fs 35 Case A WE Case B WE Case A Case B Optical Depth (τ ) o Figure 3. RCS comparison for the cases with and without backscattering enhancement. WE denotes without enhancement. I fs denotes the free space RCS. (A ) (B ) I I I I 3 4 I 3 I com 3 4 I 3 I com RCS(dB) 5 6 RCS(dB) Angles (degrees) Angles (degrees) (A ) (B ) I I 4 6 I I 3 I com 4 6 I I 3 I com RCS(dB) 8 RCS(dB) Angles (degrees) Angles (degrees) Figure 4. Angular dependence of RCS with Henyey Greenstein phase function. (A-) Case A with an optical depth (OD) of, (A-) case A with an optical depth (OD) of 5, (B-) case B with an optical depth (OD) of, and (B-) case B with an optical depth (OD) of 5.
12 5 A Ishimaru et al 7. Conclusions A theory of RCS in a multiple scattering environment is given by equations (8) and (). This is applied to a flat conducting plate. The conventional RCS is given by I and the multiple scattering RCS is given by I + I + I 3 in equation (33). The shower curtain effect, the backscattering enhancement and the angular dependence are shown to highlight the difference between the conventional RCS and the multiple scattering RCS. Acknowledgments This work is supported by the National Science Foundation (grant ECS ) and the Office of Naval Research (grant N ). Appendix. Circular complex Gaussian random variable [, 3, 5] Consider a complex random function u and its fluctuation v u = u + v. (A.) Two circular complex Gaussian random variables u and u satisfy the following conditions: v v =, but v v (A.) and the odd moments of v are zero. Thus, the following relations hold v v v3 v 4 = v v3 v v4 + v v4 v v3 u u u 3 u 4 = u u 3 u u 4 + u u 4 u u 3 u u u 3 u 4 = u u 3 u u 4 + u u 3 v v4 + u u 4 v v3 + v v3 v v4 + u u 4 v v3 + u u 3 v v4 + v v4 v v3 u u = u u u u = u u +4 u u v v + v v u u = u u + v v = u u. We have applied this assumption to the wave propagation in a random medium. We found that the results reduce to the correct limits in the weak fluctuation as well as the strong fluctuation, and that the phenomena such as the backscattering enhancement result from the use of this assumption. Note that the circular complex Gaussian variables are similar to but different from the real Gaussian random variables [3]. References [] Kim H and Johnson J T Radar images of rough surface scattering: comparison of numerical and analytical models IEEE Trans. Antennas Propag [] Hesany V, Plant W J and Keller W C The normalized radar cross section of the sea at incidence IEEE Trans. Geosci. Remote. Sens [3] Bissonnette L R, Roy G, Poutier L, Cober S G and Isaac G A Multiple-scattering lidar retrieval method: tests on Monte Carlo simulations and comparisons with in situ measurements Appl. Opt [4] Ishimaru A 997 Wave Propagation and Scattering in Random Media (New York: IEEE) [5] Rockway J D Integral equation formulation for object scattering above a rough surface PhD Dissertation Dept. of Electrical Engineering, University of Washington, Seattle, WA [6] Ocla H E and Tateiba M The effect of H-polarization on backscattering enhancement for partially convex targets of large sizes in continuous random media Waves Random Media
13 Multiple scattering effects on the RCS of objects in a random medium 5 [7] Ishimaru A 99 Backscattering enhancement: from radar cross sections to electron and light localizations to rough surface scattering IEEE Antennas Propag. Mag [8] Kuga Y and Ishimaru A 984 Retroreflectance from a dense distribution of spherical particles A 83 5 [9] Kuga Y and Ishimaru A 986 Modulation transfer function of layered inhomogeneous random media using the small-angle approximation Appl. Opt [] Ishimaru A 99Electromagnetic Wave Propagation, Radiation, and Scattering (Englewood Cliffs, NJ: Prentice Hall) [] Ishimaru A 978 Limitation on image resolution imposed by a random medium Appl. Opt [] Goodman J W 985 Statistical Optics (New York: Wiley) [3] Middleton D 96 Introduction to Statistical Communication Theory (New York: McGraw-Hill)
Polarized light propagation and scattering in random media
Polarized light propagation and scattering in random media Arnold D. Kim a, Sermsak Jaruwatanadilok b, Akira Ishimaru b, and Yasuo Kuga b a Department of Mathematics, Stanford University, Stanford, CA
More informationWAVE PROPAGATION AND SCATTERING IN RANDOM MEDIA
WAVE PROPAGATION AND SCATTERING IN RANDOM MEDIA AKIRA ISHIMARU UNIVERSITY of WASHINGTON IEEE Antennas & Propagation Society, Sponsor IEEE PRESS The Institute of Electrical and Electronics Engineers, Inc.
More informationA Statistical Kirchhoff Model for EM Scattering from Gaussian Rough Surface
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 187 A Statistical Kirchhoff Model for EM Scattering from Gaussian Rough Surface Yang Du 1, Tao Xu 1, Yingliang Luo 1,
More informationI. Rayleigh Scattering. EE Lecture 4. II. Dipole interpretation
I. Rayleigh Scattering 1. Rayleigh scattering 2. Dipole interpretation 3. Cross sections 4. Other approximations EE 816 - Lecture 4 Rayleigh scattering is an approximation used to predict scattering from
More informationAN EXPRESSION FOR THE RADAR CROSS SECTION COMPUTATION OF AN ELECTRICALLY LARGE PERFECT CONDUCTING CYLINDER LOCATED OVER A DIELECTRIC HALF-SPACE
Progress In Electromagnetics Research, PIER 77, 267 272, 27 AN EXPRESSION FOR THE RADAR CROSS SECTION COMPUTATION OF AN ELECTRICALLY LARGE PERFECT CONDUCTING CYLINDER LOCATED OVER A DIELECTRIC HALF-SPACE
More informationGENERALIZED SURFACE PLASMON RESONANCE SENSORS USING METAMATERIALS AND NEGATIVE INDEX MATERIALS
Progress In Electromagnetics Research, PIER 5, 39 5, 005 GENERALIZED SURFACE PLASMON RESONANCE SENSORS USING METAMATERIALS AND NEGATIVE INDEX MATERIALS A. Ishimaru, S. Jaruwatanadilok, and Y. Kuga Box
More informationEITN90 Radar and Remote Sensing Lecture 5: Target Reflectivity
EITN90 Radar and Remote Sensing Lecture 5: Target Reflectivity Daniel Sjöberg Department of Electrical and Information Technology Spring 2018 Outline 1 Basic reflection physics 2 Radar cross section definition
More informationAPPLICATION OF THE MAGNETIC FIELD INTEGRAL EQUATION TO DIFFRACTION AND REFLECTION BY A CONDUCTING SHEET
In: International Journal of Theoretical Physics, Group Theory... ISSN: 1525-4674 Volume 14, Issue 3 pp. 1 12 2011 Nova Science Publishers, Inc. APPLICATION OF THE MAGNETIC FIELD INTEGRAL EQUATION TO DIFFRACTION
More informationMCRT L10: Scattering and clarification of astronomy/medical terminology
MCRT L10: Scattering and clarification of astronomy/medical terminology What does the scattering? Shape of scattering Sampling from scattering phase functions Co-ordinate frames Refractive index changes
More informationX. Wang Department of Electrical Engineering The Pennsylvania State University University Park, PA 16802, USA
Progress In Electromagnetics Research, PIER 91, 35 51, 29 NUMERICAL CHARACTERIZATION OF BISTATIC SCAT- TERING FROM PEC CYLINDER PARTIALLY EMBED- DED IN A DIELECTRIC ROUGH SURFACE INTERFACE: HORIZONTAL
More informationLaser Cross Section (LCS) (Chapter 9)
Laser Cross Section (LCS) (Chapter 9) EC4630 Radar and Laser Cross Section Fall 010 Prof. D. Jenn jenn@nps.navy.mil www.nps.navy.mil/jenn November 011 1 Laser Cross Section (LCS) Laser radar (LADAR), also
More informationUniversity of Illinois at Chicago Department of Physics
University of Illinois at Chicago Department of Physics Electromagnetism Qualifying Examination January 4, 2017 9.00 am - 12.00 pm Full credit can be achieved from completely correct answers to 4 questions.
More information1 Electromagnetic concepts useful for radar applications
Electromagnetic concepts useful for radar applications The scattering of electromagnetic waves by precipitation particles and their propagation through precipitation media are of fundamental importance
More informationMie theory for light scattering by a spherical particle in an absorbing medium
Mie theory for light scattering by a spherical particle in an absorbing medium Qiang Fu and Wenbo Sun Analytic equations are developed for the single-scattering properties of a spherical particle embedded
More informationFrequency and Spatial Features of Waves Scattering on Fractals
nd Chaotic Modeling and Simulation International Conference, -5 June 009, Chania Crete Greece Frequency and Spatial Features of Waves Scattering on Fractals A.V. Laktyunkin, A.A. Potapov V.A. Kotelinikov
More informationPC4262 Remote Sensing Scattering and Absorption
PC46 Remote Sensing Scattering and Absorption Dr. S. C. Liew, Jan 003 crslsc@nus.edu.sg Scattering by a single particle I(θ, φ) dφ dω F γ A parallel beam of light with a flux density F along the incident
More informationSEAFLOOR MAPPING MODELLING UNDERWATER PROPAGATION RAY ACOUSTICS
3 Underwater propagation 3. Ray acoustics 3.. Relevant mathematics We first consider a plane wave as depicted in figure. As shown in the figure wave fronts are planes. The arrow perpendicular to the wave
More informationPolarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere
INSTITUTE OF PHYSICS PUBLISHING Waves Random Media 14 (2004) 513 523 WAVES IN RANDOMMEDIA PII: S0959-7174(04)78752-2 Polarization changes in partially coherent electromagnetic beams propagating through
More informationNotes 19 Gradient and Laplacian
ECE 3318 Applied Electricity and Magnetism Spring 218 Prof. David R. Jackson Dept. of ECE Notes 19 Gradient and Laplacian 1 Gradient Φ ( x, y, z) =scalar function Φ Φ Φ grad Φ xˆ + yˆ + zˆ x y z We can
More informationVector diffraction theory of refraction of light by a spherical surface
S. Guha and G. D. Gillen Vol. 4, No. 1/January 007/J. Opt. Soc. Am. B 1 Vector diffraction theory of refraction of light by a spherical surface Shekhar Guha and Glen D. Gillen* Materials and Manufacturing
More informationOn Electromagnetic-Acoustic Analogies in Energetic Relations for Waves Interacting with Material Surfaces
Vol. 114 2008) ACTA PHYSICA POLONICA A No. 6 A Optical and Acoustical Methods in Science and Technology On Electromagnetic-Acoustic Analogies in Energetic Relations for Waves Interacting with Material
More informationCurvilinear Coordinates
University of Alabama Department of Physics and Astronomy PH 106-4 / LeClair Fall 2008 Curvilinear Coordinates Note that we use the convention that the cartesian unit vectors are ˆx, ŷ, and ẑ, rather than
More information10. Atmospheric scattering. Extinction ( ε ) = absorption ( k ) + scattering ( m ): ε. = = single scattering albedo (SSA).
1. Atmospheric scattering Extinction ( ε ) = absorption ( k ) + scattering ( m ): ε σ = kσ + mσ, mσ mσ = = single scattering albedo (SSA). k + m ε σ σ σ As before, except for polarization (which is quite
More informationAnalysis of second-harmonic generation microscopy under refractive index mismatch
Vol 16 No 11, November 27 c 27 Chin. Phys. Soc. 19-1963/27/16(11/3285-5 Chinese Physics and IOP Publishing Ltd Analysis of second-harmonic generation microscopy under refractive index mismatch Wang Xiang-Hui(
More informationLecture 11 March 1, 2010
Physics 54, Spring 21 Lecture 11 March 1, 21 Last time we mentioned scattering removes power from beam. Today we treat this more generally, to find the optical theorem: the relationship of the index of
More informationEvaluation of the Sacttering Matrix of Flat Dipoles Embedded in Multilayer Structures
PIERS ONLINE, VOL. 4, NO. 5, 2008 536 Evaluation of the Sacttering Matrix of Flat Dipoles Embedded in Multilayer Structures S. J. S. Sant Anna 1, 2, J. C. da S. Lacava 2, and D. Fernandes 2 1 Instituto
More informationSignal Loss. A1 A L[Neper] = ln or L[dB] = 20log 1. Proportional loss of signal amplitude with increasing propagation distance: = α d
Part 6 ATTENUATION Signal Loss Loss of signal amplitude: A1 A L[Neper] = ln or L[dB] = 0log 1 A A A 1 is the amplitude without loss A is the amplitude with loss Proportional loss of signal amplitude with
More informationLecture 19 Optical MEMS (1)
EEL6935 Advanced MEMS (Spring 5) Instructor: Dr. Huikai Xie Lecture 19 Optical MEMS (1) Agenda: Optics Review EEL6935 Advanced MEMS 5 H. Xie 3/8/5 1 Optics Review Nature of Light Reflection and Refraction
More informationPHYS 110B - HW #5 Fall 2005, Solutions by David Pace Equations referenced equations are from Griffiths Problem statements are paraphrased
PHYS 0B - HW #5 Fall 005, Solutions by David Pace Equations referenced equations are from Griffiths Problem statements are paraphrased [.] Imagine a prism made of lucite (n.5) whose cross-section is a
More informationProblem Solving 1: Line Integrals and Surface Integrals
A. Line Integrals MASSACHUSETTS INSTITUTE OF TECHNOLOY Department of Physics Problem Solving 1: Line Integrals and Surface Integrals The line integral of a scalar function f ( xyz),, along a path C is
More informationTHE PARAXIAL WAVE EQUATION GAUSSIAN BEAMS IN UNIFORM MEDIA:
THE PARAXIAL WAVE EQUATION GAUSSIAN BEAMS IN UNIFORM MEDIA: In point-to-point communication, we may think of the electromagnetic field as propagating in a kind of "searchlight" mode -- i.e. a beam of finite
More informationSome negative results on the use of Helmholtz integral equations for rough-surface scattering
In: Mathematical Methods in Scattering Theory and Biomedical Technology (ed. G. Dassios, D. I. Fotiadis, K. Kiriaki and C. V. Massalas), Pitman Research Notes in Mathematics 390, Addison Wesley Longman,
More informationAppendix A Vector Analysis
Appendix A Vector Analysis A.1 Orthogonal Coordinate Systems A.1.1 Cartesian (Rectangular Coordinate System The unit vectors are denoted by x, ŷ, ẑ in the Cartesian system. By convention, ( x, ŷ, ẑ triplet
More informationTHE SCATTERING FROM AN ELLIPTIC CYLINDER IRRADIATED BY AN ELECTROMAGNETIC WAVE WITH ARBITRARY DIRECTION AND POLARIZATION
Progress In Electromagnetics Research Letters, Vol. 5, 137 149, 2008 THE SCATTERING FROM AN ELLIPTIC CYLINDER IRRADIATED BY AN ELECTROMAGNETIC WAVE WITH ARBITRARY DIRECTION AND POLARIZATION Y.-L. Li, M.-J.
More informationFundamentals on light scattering, absorption and thermal radiation, and its relation to the vector radiative transfer equation
Fundamentals on light scattering, absorption and thermal radiation, and its relation to the vector radiative transfer equation Klaus Jockers November 11, 2014 Max-Planck-Institut für Sonnensystemforschung
More informationComparison of Travel-time statistics of Backscattered Pulses from Gaussian and Non-Gaussian Rough Surfaces
Comparison of Travel-time statistics of Backscattered Pulses from Gaussian and Non-Gaussian Rough Surfaces GAO Wei, SHE Huqing Yichang Testing Technology Research Institute No. 55 Box, Yichang, Hubei Province
More informationANALYTICAL ONE-PARAMETER SEA WATER LIGHT SCATTERING PHASE FUNCTION WITH INTEGRAL PROPERTIES OF EXPERIMENTAL PHASE FUNCTIONS
ANALYTICAL ONE-PARAMETER SEA WATER LIGHT SCATTERING PHASE FUNCTION WITH INTEGRAL PROPERTIES OF EXPERIMENTAL PHASE FUNCTIONS Vladimir I. Haltrin Naval Research Laboratory, Ocean Optics Section, Code 7333,
More informationPreface to the Second Edition. Preface to the First Edition
Contents Preface to the Second Edition Preface to the First Edition iii v 1 Introduction 1 1.1 Relevance for Climate and Weather........... 1 1.1.1 Solar Radiation.................. 2 1.1.2 Thermal Infrared
More informationThermal Emission from a Layered Medium Bounded by a Slightly Rough Interface
368 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 39, NO. 2, FEBRUARY 2001 Thermal Emission from a Layered Medium Bounded by a Slightly Rough Interface Joel T. Johnson, Member, IEEE Abstract
More informationTypical anisotropies introduced by geometry (not everything is spherically symmetric) temperature gradients magnetic fields electrical fields
Lecture 6: Polarimetry 1 Outline 1 Polarized Light in the Universe 2 Fundamentals of Polarized Light 3 Descriptions of Polarized Light Polarized Light in the Universe Polarization indicates anisotropy
More informationYell if you have any questions
Class 36: Outline Hour 1: Concept Review / Overview PRS Questions Possible Exam Questions Hour : Sample Exam Yell if you have any questions P36-1 Before Starting All of your grades should now be posted
More informationEvaluating Realistic Volume Scattering Functions on Underwater Imaging System Performance
Evaluating Realistic Volume Scattering Functions on Underwater Imaging System Performance Deric J Gray Ocean Optics Section Code 7333, Naval Research Laboratory Stennis Space Center, MS 39529 phone: (228)
More informationElectromagnetism Answers to Problem Set 9 Spring 2006
Electromagnetism 70006 Answers to Problem et 9 pring 006. Jackson Prob. 5.: Reformulate the Biot-avart law in terms of the solid angle subtended at the point of observation by the current-carrying circuit.
More informationScattering of Electromagnetic Radiation. References:
Scattering of Electromagnetic Radiation References: Plasma Diagnostics: Chapter by Kunze Methods of experimental physics, 9a, chapter by Alan Desilva and George Goldenbaum, Edited by Loveberg and Griem.
More informationModelling Microwave Scattering from Rough Sea Ice Surfaces
Modelling Microwave Scattering from Rough Sea Ice Surfaces Xu Xu 1, Anthony P. Doulgeris 1, Frank Melandsø 1, Camilla Brekke 1 1. Department of Physics and Technology, UiT The Arctic University of Norway,
More informationI UNCLASSIFIED ONR-TR-2 NL
I AD-A083 615 WASHINGTON UNIV SEATTLE DEPT OF ELECTRICAL ENGINEERING F/G 4/1 I MULTIPLE SC ATTERING EFFECTS ON TRANSMISSION THROUGH THE ATMOSPH-ETC(U) IAPR GO0 A I SHIMARU NOOGI#-78C50723 I UNCLASSIFIED
More information6. LIGHT SCATTERING 6.1 The first Born approximation
6. LIGHT SCATTERING 6.1 The first Born approximation In many situations, light interacts with inhomogeneous systems, in which case the generic light-matter interaction process is referred to as scattering
More informationAnalysis of a point-source integrating-cavity absorption meter
Analysis of a point-source integrating-cavity absorption meter Robert A. Leathers, T. Valerie Downes, and Curtiss O. Davis We evaluate the theoretical performance of a point-source integrating-cavity absorption
More informationLecture 06. Fundamentals of Lidar Remote Sensing (4) Physical Processes in Lidar
Lecture 06. Fundamentals of Lidar Remote Sensing (4) Physical Processes in Lidar Light interactions with objects (continued) Resonance fluorescence Laser induced fluorescence Doppler effect (Doppler shift
More informationxy 2 e 2z dx dy dz = 8 3 (1 e 4 ) = 2.62 mc. 12 x2 y 3 e 2z 2 m 2 m 2 m Figure P4.1: Cube of Problem 4.1.
Problem 4.1 A cube m on a side is located in the first octant in a Cartesian coordinate system, with one of its corners at the origin. Find the total charge contained in the cube if the charge density
More informationOptimized diffusion approximation
356 Vol. 35, No. 2 / February 208 / Journal of the Optical Society of America A Research Article Optimized diffusion approximation UGO TRICOLI, CALLUM M. MACDONALD, ANABELA DA SILVA, AND VADIM A. MARKEL*
More informationScattering of EM waves by spherical particles: Overview of Mie Scattering
ATMO 551a Fall 2010 Scattering of EM waves by spherical particles: Overview of Mie Scattering Mie scattering refers to scattering of electromagnetic radiation by spherical particles. Under these conditions
More informationSupporting Information
Supporting Information A: Calculation of radial distribution functions To get an effective propagator in one dimension, we first transform 1) into spherical coordinates: x a = ρ sin θ cos φ, y = ρ sin
More informationDesign of one-dimensional Lambertian diffusers of light
INSTITUTE OF PHYSICS PUBLISHING Waves Random Media 11 (2001) 529 533 WAVES IN RANDOM MEDIA PII: S0959-7174(01)18801-4 Design of one-dimensional Lambertian diffusers of light A A Maradudin 1, I Simonsen
More informationDEPOLARIZED UPWARD AND DOWNWARD MULTIPLE SCATTERING FROM A VERY ROUGH SURFACE
Progress In Electromagnetics Research, PIER 54, 199 220, 2005 DEPOLARIZED UPWARD AND DOWNWARD MULTIPLE SCATTERING FROM A VERY ROUGH SURFACE C.-Y. Hsieh Electrical and Computer Engineering Department National
More informationFUNDAMENTALS OF POLARIZED LIGHT
FUNDAMENTALS OF POLARIZED LIGHT A STATISTICAL OPTICS APPROACH Christian Brosseau University of Brest, France A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY & SONS, INC. New York - Chichester. Weinheim. Brisbane
More informationDouble-distance propagation of Gaussian beams passing through a tilted cat-eye optical lens in a turbulent atmosphere
Double-distance propagation of Gaussian beams passing through a tilted cat-eye optical lens in a turbulent atmosphere Zhao Yan-Zhong( ), Sun Hua-Yan( ), and Song Feng-Hua( ) Department of Photoelectric
More informationScattering of EM waves by spherical particles: Mie Scattering
ATMO/OPTI 656b Spring 2010 Scattering of EM waves by spherical particles: Mie Scattering Mie scattering refers to scattering of electromagnetic radiation by spherical particles. Under these conditions
More informationThree-scale Radar Backscattering Model of the Ocean Surface Based on Second-order Scattering
PIERS ONLINE, VOL. 4, NO. 2, 2008 171 Three-scale Radar Backscattering Model of the Ocean Surface Based on Second-order Scattering Ying Yu 1, 2, Xiao-Qing Wang 1, Min-Hui Zhu 1, and Jiang Xiao 1, 1 National
More informationATMO/OPTI 656b Spring Scattering of EM waves by spherical particles: Mie Scattering
Scattering of EM waves by spherical particles: Mie Scattering Why do we care about particle scattering? Examples of scattering aerosols (note: ugly looking air when the relative humidity > 80%) clouds,
More informationFRACTIONAL DUAL SOLUTIONS AND CORRESPONDING SOURCES
Progress In Electromagnetics Research, PIER 5, 3 38, 000 FRACTIONAL DUAL SOLUTIONS AND CORRESPONDING SOURCES Q. A. Naqvi and A. A. Rizvi Communications Lab. Department of Electronics Quaidi-i-Azam University
More informationReflection and Transmission of Light in Structures with Incoherent Anisotropic Layers
Optics and Spectroscopy, Vol. 87, No., 999, pp. 5. Translated from Optika i Spektroskopiya, Vol. 87, No., 999, pp. 2 25. Original Russian Text Copyright 999 by Ivanov, Sementsov. PHYSICAL AND QUANTUM OPTICS
More informationLaser Optics-II. ME 677: Laser Material Processing Instructor: Ramesh Singh 1
Laser Optics-II 1 Outline Absorption Modes Irradiance Reflectivity/Absorption Absorption coefficient will vary with the same effects as the reflectivity For opaque materials: reflectivity = 1 - absorptivity
More informationGreen's functions for the two-dimensional radiative transfer equation in bounded media
Home Search Collections Journals About Contact us My IOPscience Green's functions for the two-dimensional radiative transfer equation in bounded media This article has been downloaded from IOPscience Please
More informationRichard Miles and Arthur Dogariu. Mechanical and Aerospace Engineering Princeton University, Princeton, NJ 08540, USA
Richard Miles and Arthur Dogariu Mechanical and Aerospace Engineering Princeton University, Princeton, NJ 08540, USA Workshop on Oxygen Plasma Kinetics Sept 20, 2016 Financial support: ONR and MetroLaser
More informationMULTIVARIABLE INTEGRATION
MULTIVARIABLE INTEGRATION (SPHERICAL POLAR COORDINATES) Question 1 a) Determine with the aid of a diagram an expression for the volume element in r, θ, ϕ. spherical polar coordinates, ( ) [You may not
More informationNote: Each problem is worth 14 points except numbers 5 and 6 which are 15 points. = 3 2
Math Prelim II Solutions Spring Note: Each problem is worth points except numbers 5 and 6 which are 5 points. x. Compute x da where is the region in the second quadrant between the + y circles x + y and
More informationA Quantum Mechanical Model for the Vibration and Rotation of Molecules. Rigid Rotor
A Quantum Mechanical Model for the Vibration and Rotation of Molecules Harmonic Oscillator Rigid Rotor Degrees of Freedom Translation: quantum mechanical model is particle in box or free particle. A molecule
More informationp(θ,φ,θ,φ) = we have: Thus:
1. Scattering RT Calculations We come spinning out of nothingness, scattering stars like dust. - Jalal ad-din Rumi (Persian Poet, 1207-1273) We ve considered solutions to the radiative transfer equation
More informationSimulation of partially spatially coherent laser beam and comparison with field test data for both terrestrial and maritime environments
Simulation of partially spatially coherent laser beam and comparison with field test data for both terrestrial and maritime environments N. Mosavi * a,b, C. Nelson c, B. S. Marks a, B. G. Boone a, and
More informationThe mathematics of scattering and absorption and emission
The mathematics of scattering and absorption and emission The transmittance of an layer depends on its optical depth, which in turn depends on how much of the substance the radiation has to pass through,
More informationLaser Beam Interactions with Solids In absorbing materials photons deposit energy hc λ. h λ. p =
Laser Beam Interactions with Solids In absorbing materials photons deposit energy E = hv = hc λ where h = Plank's constant = 6.63 x 10-34 J s c = speed of light Also photons also transfer momentum p p
More informationKirchhoff, Fresnel, Fraunhofer, Born approximation and more
Kirchhoff, Fresnel, Fraunhofer, Born approximation and more Oberseminar, May 2008 Maxwell equations Or: X-ray wave fields X-rays are electromagnetic waves with wave length from 10 nm to 1 pm, i.e., 10
More informationINTRODUCTION TO MICROWAVE REMOTE SENSING. Dr. A. Bhattacharya
1 INTRODUCTION TO MICROWAVE REMOTE SENSING Dr. A. Bhattacharya Why Microwaves? More difficult than with optical imaging because the technology is more complicated and the image data recorded is more varied.
More informationGeneral Properties for Determining Power Loss and Efficiency of Passive Multi-Port Microwave Networks
University of Massachusetts Amherst From the SelectedWorks of Ramakrishna Janaswamy 015 General Properties for Determining Power Loss and Efficiency of Passive Multi-Port Microwave Networks Ramakrishna
More informationArc Length and Riemannian Metric Geometry
Arc Length and Riemannian Metric Geometry References: 1 W F Reynolds, Hyperbolic geometry on a hyperboloid, Amer Math Monthly 100 (1993) 442 455 2 Wikipedia page Metric tensor The most pertinent parts
More information1. Propagation Mechanisms
Contents: 1. Propagation Mechanisms The main propagation mechanisms Point sources in free-space Complex representation of waves Polarization Electric field pattern Antenna characteristics Free-space propagation
More informationFiber Optics. Equivalently θ < θ max = cos 1 (n 0 /n 1 ). This is geometrical optics. Needs λ a. Two kinds of fibers:
Waves can be guided not only by conductors, but by dielectrics. Fiber optics cable of silica has nr varying with radius. Simplest: core radius a with n = n 1, surrounded radius b with n = n 0 < n 1. Total
More informationSimplified Collector Performance Model
Simplified Collector Performance Model Prediction of the thermal output of various solar collectors: The quantity of thermal energy produced by any solar collector can be described by the energy balance
More informationScattering of light from quasi-homogeneous sources by quasi-homogeneous media
Visser et al. Vol. 23, No. 7/July 2006/J. Opt. Soc. Am. A 1631 Scattering of light from quasi-homogeneous sources by quasi-homogeneous media Taco D. Visser* Department of Physics and Astronomy, University
More informationThe Factorization Method for Inverse Scattering Problems Part I
The Factorization Method for Inverse Scattering Problems Part I Andreas Kirsch Madrid 2011 Department of Mathematics KIT University of the State of Baden-Württemberg and National Large-scale Research Center
More informationA Discussion on the Applicable Condition of Rayleigh Scattering
www.ijrsa.org International Journal of Remote Sensing Applications (IJRSA) Volume 5, 015 doi: 10.14355/ijrsa.015.05.007 A Discussion on the Applicable Condition of Rayleigh Scattering Nan Li *1, Yiqing
More informationLECTURE 18: Horn Antennas (Rectangular horn antennas. Circular apertures.)
LCTUR 18: Horn Antennas (Rectangular horn antennas. Circular apertures.) 1 Rectangular Horn Antennas Horn antennas are popular in the microwave bands (above 1 GHz). Horns provide high gain, low VSWR (with
More informationReview Article Mathematical Methods in Biomedical Optics
ISRN Biomedical Engineering Volume 2013, Article ID 464293, 8 pages http://dx.doi.org/10.1155/2013/464293 Review Article Mathematical Methods in Biomedical Optics Macaveiu Gabriela Academic Center of Optical
More informationELECTROMAGNETIC WAVE SCATTERING FROM CY- LINDRICAL STRUCTURE WITH MIXED-IMPEDANCE BOUNDARY CONDITIONS
Progress In Electromagnetics Research M, Vol. 9, 7, 13 ELECTROMAGNETIC WAVE SCATTERING FROM CY- LINDRICAL STRUCTURE WITH MIXED-IMPEDANCE BOUNDARY CONDITIONS Mostafa Mashhadi *, Ali Abdolali, and Nader
More informationSCALAR RADIATIVE TRANSFER IN DISCRETE MEDIA WITH RANDOM ORIENTED PROLATE SPHEROIDS PARTICLES
Progress In Electromagnetics Research B, Vol. 33, 21 44, 2011 SCALAR RADIATIVE TRANSFER IN DISCRETE MEDIA WITH RANDOM ORIENTED PROLATE SPHEROIDS PARTICLES L. Bai, Z. S. Wu *, H. Y. Li, and T. Li Physics
More informationNotes 3 Review of Vector Calculus
ECE 3317 Applied Electromagnetic Waves Prof. David R. Jackson Fall 2018 A ˆ Notes 3 Review of Vector Calculus y ya ˆ y x xa V = x y ˆ x Adapted from notes by Prof. Stuart A. Long 1 Overview Here we present
More informationGeneral review: - a) Dot Product
General review: - a) Dot Product If θ is the angle between the vectors a and b, then a b = a b cos θ NOTE: Two vectors a and b are orthogonal, if and only if a b = 0. Properties of the Dot Product If a,
More informationAnalysis of Scattering of Radiation in a Plane-Parallel Atmosphere. Stephanie M. Carney ES 299r May 23, 2007
Analysis of Scattering of Radiation in a Plane-Parallel Atmosphere Stephanie M. Carney ES 299r May 23, 27 TABLE OF CONTENTS. INTRODUCTION... 2. DEFINITION OF PHYSICAL QUANTITIES... 3. DERIVATION OF EQUATION
More informationNumerical Studies of Backscattering from Time Evolving Sea Surfaces: Comparison of Hydrodynamic Models
Numerical Studies of Backscattering from Time Evolving Sea Surfaces: Comparison of Hydrodynamic Models J. T. Johnson and G. R. Baker Dept. of Electrical Engineering/ Mathematics The Ohio State University
More informationDigital Holographic Measurement of Nanometric Optical Excitation on Soft Matter by Optical Pressure and Photothermal Interactions
Ph.D. Dissertation Defense September 5, 2012 Digital Holographic Measurement of Nanometric Optical Excitation on Soft Matter by Optical Pressure and Photothermal Interactions David C. Clark Digital Holography
More informationAPPLICATION OF ICA TECHNIQUE TO PCA BASED RADAR TARGET RECOGNITION
Progress In Electromagnetics Research, Vol. 105, 157 170, 2010 APPLICATION OF ICA TECHNIQUE TO PCA BASED RADAR TARGET RECOGNITION C.-W. Huang and K.-C. Lee Department of Systems and Naval Mechatronic Engineering
More informationEnvelope PDF in Multipath Fading Channels with Random Number of Paths and Nonuniform Phase Distributions
Envelope PDF in Multipath Fading Channels with andom umber of Paths and onuniform Phase Distributions ALI ABDI AD MOSTAFA KAVEH DEPT. OF ELEC. AD COMP. EG., UIVESITY OF MIESOTA 4-74 EE/CSCI BLDG., UIO
More informationTHE OPTICAL SCIENCES COMPANY P.O. Box 446, Placentia, California Phone: (714) Pulsed Laser Backscatter. Glenn A. Tyler.
THE OPTICAL SCIENCES COMPANY P.O. Box 446, Placentia, California 92670 Phone: (714) 524-3622 Report No. TR-400 (v \P Pulsed Laser Backscatter Glenn A. Tyler and David L. Fried November 1980 WBTmiüTIÖlf
More informationPower Absorption of Near Field of Elementary Radiators in Proximity of a Composite Layer
Power Absorption of Near Field of Elementary Radiators in Proximity of a Composite Layer M. Y. Koledintseva, P. C. Ravva, J. Y. Huang, and J. L. Drewniak University of Missouri-Rolla, USA M. Sabirov, V.
More informationScintillation characteristics of cosh-gaussian beams
Scintillation characteristics of cosh-gaussian beams Halil T. Eyyuboǧlu and Yahya Baykal By using the generalized beam formulation, the scintillation index is derived and evaluated for cosh- Gaussian beams
More informationPECULIARITIES OF SPATIAL SPECTRUM OF SCATTERED ELECTROMAGNETIC WAVES IN ANISOTROPIC INHOMOGENEOUS MEDIUM
Progress In Electromagnetics Research B, Vol. 7, 191 28, 28 PECULIARITIES OF SPATIAL SPECTRUM OF SCATTERED ELECTROMAGNETIC WAVES IN ANISOTROPIC INHOMOGENEOUS MEDIUM V. G. Gavrilenko Department of Radiowave
More informationLecture 3: Asymptotic Methods for the Reduced Wave Equation
Lecture 3: Asymptotic Methods for the Reduced Wave Equation Joseph B. Keller 1 The Reduced Wave Equation Let v (t,) satisfy the wave equation v 1 2 v c 2 = 0, (1) (X) t2 where c(x) is the propagation speed
More informationMahamadou Tembely, Matthew N. O. Sadiku, Sarhan M. Musa, John O. Attia, and Pamela Obiomon
Journal of Multidisciplinar Engineering Science and Technolog (JMEST) Electromagnetic Scattering B Random Two- Dimensional Rough Surface Using The Joint Probabilit Distribution Function And Monte Carlo
More information14.1. Multiple Integration. Iterated Integrals and Area in the Plane. Iterated Integrals. Iterated Integrals. MAC2313 Calculus III - Chapter 14
14 Multiple Integration 14.1 Iterated Integrals and Area in the Plane Objectives Evaluate an iterated integral. Use an iterated integral to find the area of a plane region. Copyright Cengage Learning.
More information