AIR FORCE INSTITUTE OF TECHNOLOGY

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1 THE CATTERING OF PARTIALLY COHERENT ELECTROMAGNETIC BEAM ILLUMINATION FROM TATITICALLY ROUGH URFACE DIERTATION Mark F. pencer AFIT-ENG-D-4-J-7 DEPARTMENT OF THE AIR FORCE AIR UNIVERITY AIR FORCE INTITUTE OF TECHNOLOGY Wrght-Patteron Ar Force Bae, Oho DITRIBUTION TATEMENT A. APPROVED FOR PUBLIC RELEAE; DITRIBUTION UNLIMITED

2 The vew expreed n th dertaton are thoe of the author and do not reflect the offcal polcy or poton of the Unted tate Ar Force, Department of Defene, or the Unted tate Government. Th materal declared a work of the U.. Government and not ubject to copyrght protecton n the Unted tate.

3 AFIT-ENG-D-4-J-7 THE CATTERING OF PARTIALLY COHERENT ELECTROMAGNETIC BEAM ILLUMINATION FROM TATITICALLY ROUGH URFACE DIERTATION Preented to the Faculty Graduate chool of Engneerng and Management Ar Force Inttute of Technology Ar Unverty Ar Educaton and Tranng Command In Partal Fulfllment of the Requrement for the Degree of Doctor of Phloophy Mark F. pencer, B, M June 4 DITRIBUTION TATEMENT A. APPROVED FOR PUBLIC RELEAE; DITRIBUTION UNLIMITED

4 AFIT-ENG-D-4-J-7 THE CATTERING OF PARTIALLY COHERENT ELECTROMAGNETIC BEAM ILLUMINATION FROM TATITICALLY ROUGH URFACE DIERTATION Mark F. pencer, B, M Approved: //gned// 9 Aprl 4 Mlo W. Hyde IV, Maj, UAF, PhD (Charman Date //gned// 9 Aprl 4 Mchael A. Marcnak, PhD (Member Date //gned// 9 Aprl 4 Tmothy W. Zen, Maj, UAF, PhD (Member Date //gned// 9 Aprl 4 Mchael J. Havrlla, PhD (Member Date Accepted: //gned// May 4 ADEDEJI B. BADIRU, PhD Date Dean, Graduate chool of Engneerng and Management

5 AFIT-ENG-D-4-J-7 Abtract Much of the rough urface catterng theory developed to date conder only the effect of fully coherent and fully ncoherent llumnaton n the formaton of oluton a problem tuded n earnet nce the late 8. In repone, th dertaton extend the theory currently avalable n modelng rough urface catterng to nclude the effect of partally coherent llumnaton. uch llumnaton play a pvotal role n our undertandng of actve-llumnaton ytem, mlar to thoe found n drected-energy and remote-enng applcaton, whch ue the lght cattered from dtant target for tactcal purpoe. pecfcally, th dertaton ue the phycal optc approxmaton (Krchhoff boundary condton to determne a 3D vector oluton for the far-feld catterng of electromagnetc beam llumnaton wth partal patal coherence from tattcally rough urface. The analy conder three dfferent materal ubtrate: delectrc, conductor, and a perfect electrcal conductor. It alo make ue of a Gauan chell-model form for the ncdent-feld cro-pectral denty matrx. In o dong, th dertaton develop cloed-form expreon for the cattered feld cropectral denty matrx wth two analytcal form one applcable to mooth-tomoderately rough urface and the other applcable to very rough urface. The analy how that thee cloed-form expreon are, n general, complcated functon of both the ource (ze and coherence properte and urface parameter (urface heght tandard devaton and correlaton length. Under approprate condton, the analy alo compare the 3D vector oluton to prevouly valdated oluton and emprcal meaurement. The reult how good agreement. v

6 AFIT-ENG-D-4-J-7 To all my nfluental teacher, you really made the dfference v

7 Acknowledgment Frt and foremot, I would lke to thank my dertaton advor, Maj Hyde. He took me on a an out of department advee and never complaned once, depte the extra paperwork! Furthermore, he beleved n me. H nght nto my dertaton problem wa the catalyt for my ucce, and depte the hand cramp from dong a theory-baed dertaton, I am extremely grateful for all the help and advce that I receved I look forward to our future collaboraton. I would alo lke to thank my commttee member for the tme commtted to th document. In partcular, I would lke to thank Dr. Marcnak. He wa my profeor for multple optc coure at the Ar Force Inttute of Technology (AFIT, and I am very apprecatve of h effort. Wth that ad, I would alo lke to thank Dr. Cox, Dr. DeWeerd, Dr. Hawk, Dr. Hll, and Dr. chmdt for ther teachng effort throughout my collegate career my reearch ablte would not have grown nto what they are today wthout ther dedcaton to the claroom. The upport that I receved from my fellow reearcher, both at AFIT and abroad, cannot go wthout prae a well. Thank you o much to Dr. Bau, Dr. Bo, M. Pouln- Grard, Dr. Louthan, Dr. McCrae, Dr. Merrtt, Dr. Nauyok, Dr. Perram, Mr. tenbock, and Capt Thornton for ervng a oundng board to an over-caffenated graduate tudent. I mut alo thank my fellow clamate n the PhD-4 ecton at AFIT. They provded the upport ytem needed for my ucce. Epecally the member of the PhD Outhoue Maj Crowe, Capt Patel, Mr. Pennngton, and Capt howalter thank you! I would alo lke to thank my ponor. My graduate career ndebted to the upport I receved early on from the Center for Drected Energy at AFIT. Th upport v

8 nclude the ponorhp of my B the reearch n tmulated Brlloun catterng wth Lt Col Ruell and Lt Col Maey; my M the reearch n adaptve optc and thermal bloomng wth Dr. Cuumano and Dr. Forno; and my mall Bune Technology Tranfer (TTR reearch n optcal phaed array wth Mr. Marker and Mr. Green of the Ar Force Reearch Laboratory Drected Energy Drectorate (AFRL/RD and Dr. Tyler and Dr. Mann of the Optcal cence Company. Thee experence/relatonhp haped me nto the centt I am today and gave me the kll et needed to take on th dertaton topc. I can honetly ay that wthout th upport, I would not have receved my cence, Mathematc, and Reearch for Tranformaton (MART cholarhp. The ongong upport my MART cholarhp provde truly aweome, and upon the completon of my doctoral tude, I am excted to work for my ponorng faclty, AFRL/RD. Workng for AFRL/RD n Mau durng my two MART Internhp wa a phenomenal experence. I truly enjoyed my reearch n nvere ynthetc aperture LADAR, and I would lke to thank Dr. Wllam, Lt Col Phllp, and my MART Mentor and frend, Capt Pellzzar, for the tme they pent mentorng a young centt. Latly, I would lke to thank PIE, the Internatonal ocety for Optc and Photonc, and the Drected Energy Profeonal ocety for ther fnancal upport n the form of addtonal cholarhp. I would alo lke to thank the Ar Force Offce for centfc Reearch for ther fnancal upport n the form of a reearch grant. Th upport of my graduate educaton/reearch wa much apprecated and I look forward to long-latng relatonhp wth each organzaton. For all the upport that I receved, I am extremely grateful. Mark F. pencer v

9 Table of Content Page Abtract... v Acknowledgment... v Table of Content... v Lt of Fgure... x Introducton.... Problem tatement.... Dertaton overvew...3 Background electromagnetc and optc theory revew urface equvalence Interor problem Exteror problem pecalzaton to a perfect electrcal conductor.... Integral equaton..... Electrc-feld ntegral equaton..... Magnetc-feld ntegral equaton Method of moment Phycal optc approxmaton General problem for delectrc pecalzaton to conductor pecalzaton to a perfect electrcal conductor Mathematcal technque Far-feld approxmaton Plane-wave pectrum repreentaton Method of tatonary phae Coherence elf-coherence functon Mutual-coherence functon...7 v

10 .5.3 Complex degree of coherence Cro-pectral denty pectral degree of coherence Gauan chell-model ource Cro-pectral denty matrx Polarzaton Background rough urface catterng lterature revew Fully coherent and fully ncoherent llumnaton Geometrcal-optc method Lnear-ytem method Perturbaton method Phycal-optc method Full-wave method Partally coherent llumnaton Phae-creen method ABCD-matrx method Coherent-mode method Methodology for the 3D vector oluton Incdent feld cro-pectral denty matrx cattered feld cattered feld cro-pectral denty matrx mooth-to-moderately rough urface Very rough urface Exploraton of the 3D vector oluton Comparon wth the D calar-equvalent oluton Angular pectral degree of coherence radu Angular pectral denty radu Fully coherent llumnaton valdaton Partally coherent llumnaton valdaton Comparon to a polarmetrc bdrectonal dtrbuton functon...87 x

11 5.. Normalzed pectral denty valdaton Degree of polarzaton valdaton Comparon to emprcal meaurement In-plane meaurement Out-of-plane meaurement Comparon to a paraxal oluton Concluon Contrbuton Future area of reearch... Appendx A. Ung the geometrcal optc approxmaton to relate the tangental feld at the catterng urface... 3 Appendx B. Ung the phycal optc approxmaton to mplfy the equvalent urface current dente... 6 Appendx C. Ung the method of tatonary phae to olve the ntegral wth repect to the plane-wave pectrum repreentaton... Appendx D. Defnng the dyadc that contan all of the ampltude and phae term evaluated at the crtcal pont of the frt knd... 7 General problem for delectrc...7 pecalzaton to conductor... pecalzaton to a perfect electrcal conductor...3 Appendx E. Examnng the valdty of the approxmaton ued when pecalzng to conductor... 5 Reference... 8 x

12 Lt of Fgure Page Fgure. A generc electromagnetc catterng problem. Here, a cloed urface urround a catterng object creatng an nteror and exteror regon of nteret Fgure. The nteror problem aocated wth ung urface equvalence. Here, the exteror regon contan null feld; conequently, equvalent ource n the form of urface current dente replcate the nteror feld. Thee equvalent ource effectvely radate n an nfnte homogeneou pace Fgure 3. The exteror problem aocated wth ung urface equvalence. Here, the nteror regon contan null feld; thu, equvalent ource n the form of urface current dente radate the cattered feld. Thee equvalent ource radate n the preence of the prmary ource and ncdent feld. uperpoton of the ncdent and cattered feld replcate the exteror feld whch propagate throughout free pace.... Fgure 4. The nteror regon wthn a perfect electrcal conductor contan null feld. A uch, only an exteror problem ext when ung urface equvalence. The reultng electrc current denty radate the cattered feld and uperpoton wth the known ncdent feld replcate the exteror feld whch propagate throughout free pace.... Fgure 5. A macro-cale decrpton of the phycal optc (PO approxmaton. Here, a pont ource llumnate a catterng object o that no current dente ext n the hadow regon predcted by the geometrcal optc approxmaton Fgure 6. A generc etup for 3D electromagnetc catterng problem. Here, the x axe algn n both the ource plane and the urface plane. Th aume otropy n the materal ubtrate Fgure 7. The mcro-cale geometry decrbng far-feld electromagnetc catterng. Here, the obervaton pont extend well pat what depcted and only a mall porton of the cloed urface dplayed. Th correpond to a zoomed-n decrpton of Fgure Fgure 8. A decrpton of the polarzaton geometry ued n the analy Fgure 9. The macro-cale (a and mcro-cale (b catterng geometry of a D tattcally rough urface of length L and wdth L x

13 Fgure. Comparon between a full-wave D method of moment (MoM oluton, the D calar-equvalent oluton, and the 3D vector oluton for fully coherent llumnaton at normal ncdence of a very rough conductng urface. (a how the magntude of the cattered pectral degree of coherence a a functon of the dfference between two polar angle, wherea (b how the normalzed cattered pectral denty a a functon of a ngle polar angle Fgure. Comparon between the D calar-equvalent and 3D vector oluton for fully coherent llumnaton at normal ncdence of mooth-to-moderately rough conductng urface. (a how the magntude of the cattered pectral degree of coherence a a functon of the dfference between two polar angle. (b how the normalzed cattered pectral denty a a functon of a ngle polar angle Fgure. Comparon between the D oluton (crcle and 3D oluton (lne for partally coherent llumnaton at normal ncdence of mooth-to-very rough conductng urface. (a-(d how the magntude of the cattered pectral degree of coherence a a functon of the dfference between two polar angle for varyng ource parameter rato and urface lope tandard devaton Fgure 3. Comparon between the D oluton (crcle and 3D oluton (lne for partally coherent llumnaton at normal ncdence of mooth-to-very rough conductng urface. (a-(d how the normalzed cattered pectral denty a a functon of a ngle polar angle for varyng ource parameter rato and urface lope tandard devaton Fgure 4. Comparon between the D oluton (crcle and 3D oluton (lne for partally coherent llumnaton at non-normal ncdence of mooth-to-very rough conductng urface. (a-(d how the normalzed cattered pectral denty a a functon of a ngle polar angle for varyng ource parameter rato and urface lope tandard devaton Fgure 5. Comparon of the normalzed cattered pectral dente obtaned from a polarmetrc bdrectonal dtrbuton functon (pbrdf and the 3D vector oluton for unpolarzed llumnaton at non-normal ncdence and a very rough conductng urface. (a depct an n-plane catterng geometry, wherea (b depct an out-of-plane catterng geometry wth reult a a functon of a ngle polar angle. Converely, (c and (d depct b-tatc catterng geometre a a functon of a ngle azmuth angle. Note that the mnmum occur at the mono-tatc obervaton pont n both (c and (d Fgure 6. Comparon of the cattered degree of polarzaton obtaned from a polarmetrc bdrectonal dtrbuton functon (pbrdf and the 3D vector oluton for unpolarzed llumnaton at non-normal ncdence and a very rough conductng urface. (a depct an n-plane catterng geometry, wherea (b depct an out of plane catterng geometry wth reult a a functon of a ngle polar angle. Converely, (c and (d depct b-tatc catterng geometre a a functon of a ngle azmuth angle. Note that the mnmum occur at the mono-tatc obervaton pont n both (c and (d x

14 Fgure 7. Comparon of the cattered degree of polarzaton for unpolarzed llumnaton at Brewter angle of a delectrc urface wth varyng roughne condton. (a how the reult obtaned from a polarmetrc bdrectonal dtrbuton functon (pbrdf and the 3D vector oluton for non-normal ncdence at Brewter angle wth very rough urface condton. (b how the reult from the 3D vector oluton for both very rough and mooth-to-moderately rough urface condton... 9 Fgure 8. Decrpton of the Complete Angle catter Intrument at the Ar Force Inttute of Technology. (a how the catterng geometry ued to collect n-plane meaurement [4], wherea (b how the catterng geometry ued to collect out-ofplane meaurement [48] Fgure 9. Comparon between n-plane meaurement obtaned wth the Complete Angle catter Intrument (CAI at the Ar force Inttute of Technology, a full-wave D method of moment (MoM oluton wth two dfferent urface model, and the 3D vector oluton for unpolarzed llumnaton at non-normal ncdence of a very rough conductng urface. (a and (c how the normalzed cattered pectral denty and the cattered degree of polarzaton a a functon of a ngle polar angle. (b and (d how the ame reult n log cale Fgure. Comparon between out-of-plane meaurement obtaned wth the Complete Angle catter Intrument (CAI at the Ar force Inttute of Technology and the 3D vector oluton for unpolarzed llumnaton at non-normal ncdence of a very rough conductng urface. (a and (c how the normalzed cattered pectral denty and the cattered degree of polarzaton a a functon of a ngle tranformaton polar angle. (b and (d how the ame reult n log cale Fgure. Comparon between an ABCD paraxal oluton and the 3D vector oluton for partally coherent llumnaton at normal ncdence of a very rough perfectly reflectng urface. (a how the magntude of the cattered pectral degree of coherence a a functon of the dtance between two value n the x drecton, wherea (b how the cattered normalzed pectral denty and (c how the cattered degree of polarzaton a a functon of a ngle value n the x drecton Fgure : The mcro-cale geometry decrbng how to relate the ncdent feld to the reflected feld ung the geometrcal optc approxmaton Fgure 3: The mcro-cale geometry decrbng how to relate the ncdent feld to the reflected and tranmtted feld ung the geometrcal optc approxmaton Fgure 4. Aement of the approxmaton ued wth the parallel and perpendcular Frenel reflecton coeffcent for ttanum (T, alumnum (Al, nckel (N, and lver (Ag. (a depct the ampltude, wherea (b depct the phae a a functon of a ngle polar angle x

15 Fgure 5. Aement of the approxmaton ued when pecalzng to conductor. (a how the magntude of the cattered pectral degree of coherence a a functon of the dfference between two polar angle, wherea (b how the cattered normalzed pectral denty and (c how the cattered degree of polarzaton a a functon of a ngle polar angle. Note that the analy nclude reult for a perfect electrcal conductor (PEC for comparon purpoe xv

16 THE CATTERING OF PARTIALLY COHERENT ELECTROMAGNETIC BEAM ILLUMINATION FROM TATITICALLY ROUGH URFACE Introducton In 96, Theodore Maman nvented the frt workng laer [], whch orgnally boated only a few mllwatt of power. Yet, by the 97, laer power reached the megawatt level and the drected-energy (DE reearch communty came to lfe []. The technology found n DE applcaton nprng n that t preent game-changng capablte by offerng ytem wth varyng lethalty, peed-of-lght delvery, and unparalleled precon [3-5]. Wth th n mnd, the analy preented n th dertaton hope to ad the burgeonng DE reearch communty and brng u one tep cloer to feldng an operatonal ytem [6, 7].. Problem tatement When ung actve-llumnaton ytem, more often than not a hghly coherent laer beam propagate from the ource through the atmophere reultng n partally coherent beam llumnaton on the target. Th topc play a key role n DE and remoteenng applcaton whch ue the lght cattered from dtant target for tactcal purpoe [8, 9]. Interetngly enough, not much lterature ext pertanng to the catterng of partally coherent lght from rough urface. In an effort to brdge th gap, recent publcaton derved a D calar-equvalent oluton for the catterng of partally coherent beam from tattcally rough urface

17 ung the phycal-optc (PO approxmaton (Krchhoff boundary condton [, ]. pecfcally, the analy made ue of a Gauan chell-model (GM form n creatng the ncdent feld cro-pectral denty functon (CDF. Th allow one to vary the ze and patal coherence properte of the ncdent radaton. In o dong, the analy developed cloed-form expreon for the cattered feld CDF to oberve the ze and patal coherence properte of the cattered radaton n the far feld. The analy alo valdated thee analytcal expreon through computatonal mulaton and howed good agreement between the theoretcal predcton and the numercal reult. Whle the D calar-equvalent oluton a convenent tool for ganng nght nto rough urface catterng, a complete undertandng of the problem requre a 3D vector oluton. Wth that ad, th dertaton make ue of the PO approxmaton to determne a 3D vector oluton for the far-feld catterng of electromagnetc beam llumnaton wth partal patal coherence from tattcally rough urface. By formulatng the analy n a manner content wth Wolf unfed theory of coherence and polarzaton [, 3], all phycal mplcaton nherent n Wolf work apply here. The 3D vector oluton developed n th dertaton conder three dfferent materal ubtrate: delectrc, conductor, and a perfect electrcal conductor. In addton, t ue a GM form n creatng the ncdent feld cro-pectral denty matrx (CDM. Th allow for the formulaton of cloed-form expreon for the cattered feld CDM. A uch, the analy how that two analytcal form reult for the cattered feld CDM one applcable to mooth-to-moderately rough urface and the other applcable to very rough urface.

18 Both analytcal form for the cattered feld CDM contan complcated functon of the ource parameter (ze and coherence properte and the urface parameter (urface heght tandard devaton and correlaton length. In partcular, the cloed-form expreon applcable to mooth-to-moderately rough urface expreed a an nfnte ere. Th nfnte ere lowly convergent; however, one can tll gather phycal ntuton from t analytcal form. On the other hand, the cloed-form expreon applcable to very rough urface ncredbly phycal, and under certan crcumtance, mantan a GM form. Baed on thee crcumtance, the analy develop cloed-form expreon for the angular pectral degree of coherence (DoC and pectral denty (D rad. Thee analytcal expreon alo contan complcated functon of both the ource and the urface parameter. The analy demontrate that for many cenaro of nteret, one can approxmate/mplfy the DoC radu a a functon of jut the ource parameter and the D radu a a functon of jut the urface parameter.. Dertaton overvew Chapter and 3 of th dertaton provde background nformaton n the form of theory and lterature revew, repectvely. The goal here to provde future reearch effort wth a thorough nvetgaton of the problem at hand. Chapter 4 provde the methodology ued to obtan the 3D vector oluton propoed above. Here, the analy tate all mplfyng aumpton and explan ther phycal mplcaton. Chapter 5 provde an exploraton of the 3D vector oluton. The analy gven here vually demontrate apect of the cloed-form expreon and how that the reult are content wth prevouly valdated oluton and emprcal meaurement. Chapter 6 provde a concluon for th dertaton wth a roadmap for future reearch effort. 3

19 Background electromagnetc and optc theory revew Electromagnetc theory and ubequently optc theory nvolve the applcaton of Maxwell equaton to the geometry of a pecfc problem. A uch, the oluton obtaned from Maxwell equaton determne the electromagnetc vector feld preent wthn a regon of nteret. Wth thee vector feld, one can then calculate quantte of mportance to ther work. The followng chapter revew the neceary electromagnetc and optc theory needed to undertake the problem propoed above n Chapter.. urface equvalence There are many approache to olvng Maxwell equaton for the vector feld preent n an electromagnetc catterng problem. One robut approach ue urface equvalence. Accordng to Balan [4], chelkunoff wa the frt to ntroduce urface equvalence n 936 [5]. In eence, urface equvalence a more rgorou extenon of Huygen prncple [6], whch, accordng to Hecht [7], tate that Every pont on a propagatng wavefront erve a the ource of phercal econdary wavelet, uch that the wavefront at ome later tme the envelope of thee wavelet. Wth th n mnd, urface equvalence effectvely defne equvalent ource n the form of urface current dente on a catterng object. Approprately defned, thee urface current dente, n addton to other ource, replcate the feld preent n a regon of nteret. To make th concept manfet, frt conder the generc electromagnetc catterng problem depcted n Fgure [8]. A hown, a prmary ource wth current dente, pr J and pr M, radate ncdent feld, nc E and nc H, whch propagate n free pace wth ndex of refracton n = and mpedance η = μ ε, where ε and μ are 4

20 the free-pace permttvty and permeablty, repectvely. Thee known ncdent feld llumnate a homogeneou, penetrable catterng object of volume V wth ndex of refracton n and mpedance η. Th llumnaton produce econdary ource n the form of current dente, ec J and ec M, whch radate cattered feld, ct E and mportant to note that thee cattered feld are unknown. Furthermore, a cloed ct H. It urface wth an outward pontng unt-normal vector ˆn create both an nteror and exteror regon of nteret. The nteror regon contan the feld, nt E and nde the cloed urface, wherea the exteror regon contan the feld, nt H, found ext E and ext H, found outde the cloed urface. Thee two regon of nteret create an nteror and exteror problem when ung urface equvalence.. The analy preented throughout th dertaton ue the MK ytem of unt o 7 that ε = (farad per meter and μ = 4π (henre per meter [4]. In addton, the analy ue the engneerng gn conventon for the tme-harmonc varaton,.e., (, (, exp ( (, e j ω u r t = U r ω jωt = U r ω t, where (,ω U r a poton r and angular frequency ω dependent vector feld of nteret wthn the analy. Note that ometme the analy omt the r and ω dependence n wrtng the vector feld. Th done for brevty n the notaton. Alo note that formulaton whch ue the phyc gn conventon,.e., ( t = ( ( t = (,, exp, e ω u r U r ω ω U r ω t, relate to th work by a complex conjugate, where j = =.. Wthn a homogeneou pace, the ndex of refracton n and mpedance η relate to the permttvty ε and permeablty μ, where n εμ ( ε μ = and η = μ ε [4]. 5

21 pr pr ext J, M E ¹ E nc, H nc H ext ¹ E ct, H ct V J ec, M ec ( n, h ( n, h E H nt nt ¹ ¹ ˆn Fgure. A generc electromagnetc catterng problem. Here, a cloed urface urround a catterng object creatng an nteror and exteror regon of nteret... Interor problem Fgure decrbe the nteror problem when ung urface equvalence [8]. A hown, electrc and magnetc urface current dente, nt J and nt M, ext jut on the nde of the cloed urface and atfy the followng defnton [4, 9]: nt nt J ˆ = n H ( and nt nt M ˆ = n E. ( Thee equvalent ource radate n the abence of the prmary ource and the exteror ext ext feld,.e., E = H =. nce the exteror feld equate to zero, th allow the nteror regon to extend throughout the exteror regon creatng an unbounded homogeneou pace wth ndex of refracton n and mpedance η. In addton, the nteror feld, and nt H, atfy Maxwell equaton, uch that [9] nt E nt nt η nt E = F j ( n k + A (3 nk 6

22 nt nt η nt H A F, (4 nk and = j ( n k + where k nt A and nt F are the nteror magnetc and electrc vector potental, repectvely, = π λ the free-pace wavenumber, and λ the free-pace wavelength 3. nce nt J and nt M rede on a cloed urface, the vector potental n Eq. (3 and (4 atfy the followng convoluton ntegral [9]: ( ( G( ; A A r J r r r d (5 = = nt nt nt F F r M r r r d, (6 nt nt nt and ( ( G( ; where = = r the ource vector, r the obervaton vector, and G ( ; Green functon, uch that G ( rr ; ( jnk r r exp = 4π r r A a reult, the nteror problem nvolve equvalent ource, replcate the nteror feld, nt E and nt H rr the unbounded. (7 nt J and nt M, whch, n an unbounded homogeneou pace. 3. Free-pace wavelength n the optcal regme typcally range from λ = μm n the extreme ultravolet to λ = 3 μm n the far nfrared []; thu, the analy preented throughout th dertaton aume that the free-pace wavenumber much, much greater than one, k = π λ. 7

23 pr pr ext J, M E ¹ J pr, M pr E ext = E nc, H nc H ext ¹ E ct, H ct E nc, H nc H ext = E ct, H ct V J ec, M ec V nt M nt J ( n, h ( n, h E H nt nt ¹ ¹ ˆn ( n, h ( n, h E H nt nt ¹ ¹ ˆn Fgure. The nteror problem aocated wth ung urface equvalence. Here, the exteror regon contan null feld; conequently, equvalent ource n the form of urface current dente replcate the nteror feld. Thee equvalent ource effectvely radate n an nfnte homogeneou pace... Exteror problem mlar to the analy preented for the nteror problem, Fgure 3 decrbe the exteror problem ung urface equvalence [8]. Here, the electrc and magnetc current dente, ext J and followng relatonhp [4, 9]: ext M, ext jut on the outde of the cloed urface and atfy the ext ext J ˆ = n H (8 and ext ext M ˆ = n E. (9 Thee equvalent ource radate n the preence of the prmary ource whle the feld n nt nt the nteror regon are nulled,.e., E = H =. nce the nteror feld equate to zero, th allow the exteror regon to extend throughout the nteror regon creatng an unbounded free pace. Moreover, the exteror feld, uperpoton relatonhp: ext E and ext H, atfy the followng ext nc ct E = E + E ( and ext nc ct H = H + H. ( 8

24 olvng Maxwell equaton for the unknown cattered feld, and (, provde [9] ct E and ct H, n Eq. ( ext nc ext η ext E = E F j ( k + A ( k ext nc ext η ext H H A F, (3 k and = + j ( k + where nce ext A and ext J and ext F are the exteror magnetc and electrc vector potental, repectvely. ext M rede on a cloed urface, the vector potental n Eq. ( and (3 atfy the followng convoluton ntegral [9]: ( ( G ( A A r J r r r d (4 = = ext ext ext ; F F r M r r r d, (5 ext ext ext and ( ( G ( where here, G ( ; = = ; rr the free-pace Green functon, uch that G ( rr ; ( jk r r exp = 4π r r Conequently, the exteror problem nvolve equvalent ource, radate the cattered feld, nc E and ct E and. (6 ext J and ext M, whch ct H, n the preence of the known ncdent feld, nc H. Together thee feld replcate the exteror feld, propagate throughout free pace. ext E and ext H, whch 9

25 J pr, M pr E ext ¹ J pr, M pr E ext ¹ E nc, H nc H ext ¹ E ct, H ct E nc, H nc H ext M ext ¹ ext J E ct, H ct V J ec, M ec V ( n, h ( n, h E H nt nt ¹ ¹ ˆn ( n, h ( n, h E H nt nt = = ˆn Fgure 3. The exteror problem aocated wth ung urface equvalence. Here, the nteror regon contan null feld; thu, equvalent ource n the form of urface current dente radate the cattered feld. Thee equvalent ource radate n the preence of the prmary ource and ncdent feld. uperpoton of the ncdent and cattered feld replcate the exteror feld whch propagate throughout free pace...3 pecalzaton to a perfect electrcal conductor Fgure 4 decrbe the ue of urface equvalence wth a perfect electrcal conductor (PEC [8]. Wthn a perfectly conductng materal the nteror feld vanh, nt nt E = H =, o that only an exteror problem ext [9]. In general, a PEC ha nfnte conductvty, σ =, and the tangental electrc feld goe to zero all along t ext urface,.e., nˆ E = [4]. Baed on th knowledge, the relatonhp gven n Eq. (8 and (9 mplfy to the followng expreon for a PEC: ext ext J ˆ = n H (7 ext and M. (8 = Furthermore, Eq. ( and (3 mplfy, uch that ext nc η ext E = E j ( k + A (9 k and ext nc ext H = H + A. (

26 Provded Eq. (7-(, only an equvalent ource feld, ct E and replcate the exteror feld, ct H. uperpoton wth the known ncdent feld, ext E and ext J needed to radate the cattered nc E and nc H, ext H, whch propagate throughout free pace. J pr, M pr E ext ¹ J pr, M pr E ext ¹ E nc, H nc H ext ¹ ext nˆ E = E ct, H ct E nc, H nc H ext ¹ ext J E ct, H ct V V ( n, h ( = E H nt nt = = ˆn ( n, h ( n, h E H nt nt = = ˆn Fgure 4. The nteror regon wthn a perfect electrcal conductor contan null feld. A uch, only an exteror problem ext when ung urface equvalence. The reultng electrc current denty radate the cattered feld and uperpoton wth the known ncdent feld replcate the exteror feld whch propagate throughout free pace.. Integral equaton The contnuty of the tangental feld at the nterface between the exteror and nteror regon dctate that [9] ( and ( ext nt ext nt ext nt eq nˆ E E = nˆ E = nˆ E J = J = J ( ext nt ext nt ext nt eq nˆ H H = nˆ H = nˆ H M = M = M. ( A uch, n an electromagnetc catterng problem ung urface equvalence, eq J and eq M readly become the prmary unknown and ntegral equaton reult. Numercal technque help n olvng thee ntegral equaton for the unknown equvalent urface current dente, eq J and eq M.

27 .. Electrc-feld ntegral equaton Provded the contnuty relatonhp found n Eq. (, the tangental component of Eq. (3 and ( mplfy, uch that η M eq ˆ ext ( ext ˆ nc n F j k + A = n E (3 k eq nt η nt M nˆ F + A =. (4 nk and j ( n k Thee coupled ntegro-dfferental equaton erve a the electrc-feld ntegral equaton (EFIE for the unknown equvalent urface current dente, eq J and eq M. Together, Eq. (3 and (4 repreent a lnear ytem of equaton wth two equaton and two unknown. It mportant to note that for a PEC, the tangental electrc feld goe to zero ext all along t urface,.e., nˆ E = ; thu, the followng EFIE reult from Eq. (9: nc η ext nˆ E = nˆ j ( k + A. (5 k Th EFIE an ntegro-dfferental equaton for the unknown equvalent urface current denty eq J... Magnetc-feld ntegral equaton Magnetc-feld ntegral equaton (MFIE reult from the contnuty relatonhp gven n Eq. (. pecfcally, the tangental component of Eq. (4 and (3 mplfy, o that η J eq ˆ ext ( ext ˆ nc n A j k + F = n H (6 k

28 eq nt η nt J nˆ A + F =. (7 nk and j ( n k Thee coupled ntegro-dfferental equaton are the MFIE for the unknown equvalent urface current dente, eq J and eq M. One hould note that Eq. (6 and (7 repreent a ytem of lnear equaton wth two equaton and two unknown. Moreover, the tangental component of Eq. ( create the followng MFIE for a PEC: J eq n ˆ A ext = n ˆ H nc. (8 Th MFIE an ntegro-dfferental equaton for the unknown equvalent urface current denty eq J...3 Method of moment The method of moment (MoM a robut numercal approach that olve EFIE or MFIE for the unknown equvalent urface current dente, eq J and eq M [4, 9, ]. In ung the MoM, a ere of fnte term or ba functon wth unknown ampltude coeffcent effectvely replace eq J and eq M. Th create a number of algebrac expreon whch matrx algebra technque readly olve. A a reult, the MoM ha the potental to formulate hgh-fdelty numercal oluton for the unknown equvalent urface current dente, eq J and eq M..3 Phycal optc approxmaton It mportant to remember that when ung urface equvalence n an electromagnetc catterng problem, the equvalent urface current dente, eq M, radate the unknown cattered feld, ct E and eq J and ct H, whch propagate n free pace 3

29 wth ndex of refracton n and mpedance η. Th facltate the ue of a free-pace Green functon G ( ; analytcal oluton for rr [cf. Eq. (6]; however, t tll dffcult to formulate eq J and eq M nce they are, by defnton, dependent on ct E and ct H, whch are unknown. The phycal-optc (PO approxmaton help to allevate thee contrant [4, ]. In eence, the PO approxmaton make ue of the geometrcal-optc (GO approxmaton to formulate the current dente nvolved n an electromagnetc catterng problem. Fgure 5 help to further explan th pont [8]. A hown, the current dente formulated wth the PO approxmaton equate to zero n the hadow regon of a catterng object an aumpton whch analogou to ung Krchhoff boundary condton n phycal or wave optc [3, 4]. When ung the PO approxmaton wth urface equvalence, one replace the unknown cattered feld wth reflected feld. pecfcally, ct ref E E and ct ref H H, o that the equvalent urface current dente, eq J and eq M, become ( ( eq ext nc ct nc ref J ˆ ˆ ˆ = n H = n H + H n H + H (9 and eq ˆ ext ˆ ( nc ct ˆ ( nc ref = = + + M n E n E E n E E. (3 Thee approxmaton aume that the catterng object and t aocated curvature are large compared to the wavelength of the ncdent feld, nc E and nc H 4. uch approxmaton are exact f the catterng object homogeneou, nfnte, and planar [4, ]. Wth that ad, the ncdent llumnaton follow the law of reflecton a drect 4. Th content wth geometrcal or ray optc, whch emerge a the lmt of phycal or wave optc when the wavelength approache zero, λ [, ]. 4

30 reult from the GO approxmaton [] o that the tangental reflected feld, nˆ E ref nc nc and nˆ H, relate to the tangental ncdent feld, nˆ E and nˆ H, n unque way for dfferent materal ubtrate,.e., delectrc, conductor, and a PEC. Appendx A explore thee relatonhp n more detal. ref Pont ource J M PO PO ¹ ¹ J M PO PO = = hadow Regon Fgure 5. A macro-cale decrpton of the phycal optc (PO approxmaton. Here, a pont ource llumnate a catterng object o that no current dente ext n the hadow regon predcted by the geometrcal optc approxmaton..3. General problem for delectrc A hown n Appendx A, the followng relatonhp hold true for delectrc accordng to the GO approxmaton [8]: ref nc nˆ E = nˆ E (3 ref nc and nˆ H = nˆ H, (3 where r the known Frenel reflecton coeffcent at the cloed urface of a catterng object. Accordngly, the equvalent urface current dente, Eq. (9 and (3, mplfy o that r r eq J and eq M, a gven n eq nc J ( r ˆ n H (33 5

31 eq nc and M ( + r ˆ n E. (34 Th dctate that when ung the PO approxmaton for a delectrc materal, the known nc nc tangental ncdent feld, nˆ E and nˆ H, n addton to the known Frenel reflecton coeffcent, r, are all that needed n determnng the analytcal form of and eq M..3. pecalzaton to conductor For very good conductor, the conductvty approache nfnty, σ [4]. A uch, the tangental ncdent electrc feld approxmate to zero all along the urface of a nc conductng materal,.e., nˆ E, and Eq. (33 and (34 mplfy to the followng relatonhp: eq J eq nc J ( r ˆ n H (35 and M eq. (36 Thu, the analy mplfe from that of delectrc. Only the electrc equvalent urface current denty eq J radate when ung the PO approxmaton for a conductng materal..3.3 pecalzaton to a perfect electrcal conductor A hown n Appendx A, the followng relatonhp hold true for a PEC accordng to the GO approxmaton [8]: ref nc nˆ E = nˆ E (37 ref nc and nˆ H = nˆ H. (38 Conequently, the equvalent urface current dente, (9 and (3, mply o that eq J and eq M, a gven n Eq. 6

32 eq nc J ˆ n H (39 and M eq. (4 = Th dctate that when ung the PO approxmaton for a perfectly conductng materal, nc the known tangental ncdent magnetc feld nˆ H all that needed n determnng the analytcal form of eq J..4 Mathematcal technque Mathematcal technque ext whch further mplfy the analy beyond the PO approxmaton. To help make thee mathematcal technque unambguou, frt conder the 3D electromagnetc catterng etup decrbed n Fgure 6. The analy alo refer to th etup a the macro-cale catterng geometry. Here, the vector, ρ = xxˆ + uuˆ, pont from the ource plane orgn to a tranvere locaton n the ource, nce v = n the ource plane; the vector, r = xˆ y yˆ + z zˆ, pont from the ource plane orgn to the urface plane orgn; the vector, r = x xˆ + y yˆ + z z ˆ, pont from the urface plane orgn to a pont on the cloed urface ; and the vector, r = xxˆ + yyˆ + zz ˆ, pont from the urface plane orgn to an obervaton pont. Th etup play a pvotal role n employng the far-feld approxmaton, plane-wave pectrum repreentaton, and method of tatonary phae, all of whch are mathematcal technque whch greatly mplfy the analy when ued under the rght aumpton. 7

33 (, -yz, z ( x, yz, r x u v ( n, h r r-r r r y ( x, y, z Fgure 6. A generc etup for 3D electromagnetc catterng problem. Here, the x axe algn n both the ource plane and the urface plane. Th aume otropy n the materal ubtrate..4. Far-feld approxmaton In the far feld, r r approxmately parallel to r, a hown n Fgure 7 [8]. Addtonally, k r r, and the followng approxmaton reult [4]: where r = r, r = r, and ˆ r rr for phae varaton r r = r + r r r, (4 r for ampltude varaton ( θ ( φ ( θ ( φ ( θ rr ˆ = x n co + y n n + z co. (4 Provded Eq. (4 and (4, the free-pace Green functon G ( rr (6, mplfe o that G x ( rr ( jk r r exp( jk r ; ; exp, a gven n Eq. exp = ( jk rr ˆ. (43 4π r r 4πr 8

34 From Eq. (43, the unknown cattered feld, ct E and ( and (3, atfy the followng approxmaton [4]: ( k ct ext ext ct ct N L k ct H, a formulated above n Eq. η E = F j + A E + E (44 η H = A + F H + H, (45 and j ( k ct ext ext ct ct N L k where E ct N ( jk r ( ˆˆ ˆˆ exp ext = jkη θθ + φφ N, (46 4π r ct ˆ ct HN = r E N, (47 η H ct L k exp η 4πr ( jk r ( ˆˆ ˆˆ ext = j θθ + φφ L, (48 and ct ct EL = ηˆ r H L. (49 Thee expreon depend on the far-feld-exteror magnetc and electrc vector potental, ext N and ext L, uch that ( ( exp ( jk ext ext eq N = N r = J r ˆ r r d (5 ext ext eq and = ( = ( exp ( jk L L r M r ˆ r r d. (5 It mportant to remember that the PO approxmaton mplfe the analy o that the known ncdent feld, nc E and equvalent urface current dente, nc H, help n determnng the analytcal form of the eq J and eq M, n Eq. (5 and (5. Movng forward, one can ue the plane-wave pectrum repreentaton to account for nc H. nc E and 9

35 z r-r r q q q r rr ˆ r p-f f y f dente, and Fgure 7. The mcro-cale geometry decrbng far-feld electromagnetc catterng. Here, the obervaton pont extend well pat what depcted and only a mall porton of the cloed urface dplayed. Th correpond to a zoomed-n decrpton of Fgure Plane-wave pectrum repreentaton In order to determne the analytcal form of the equvalent urface current eq J and eq M, the analy mut frt account for the known ncdent feld, nc H, whch propagate from the ource and llumnate the cloed urface [cf. Fgure 6]. Wth th n mnd, one can wrte the ncdent electrc feld pectrum x nc E nc E n term of t nc T e ung the plane wave pectrum repreentaton [5]. The followng expreon reult ung the macro-cale catterng geometry decrbed n Fgure 6: nc nc ( nc nc nc nc nc nc e ( x u ( x u E = E r = T k, k exp j k r + r dk dk, v (5 ( π and nc nc ( nc nc nc ( ( nc nc e e x u x u T = T k, k = E ρ exp j k x+ k u dxdu, v. (53

36 x u v Here, k nc = k nc xˆ + k nc uˆ + k nc v ˆ the ncdent propagaton vector and v the ourcefree half pace wth ndex of refracton n = and mpedance η = μ ε. nce the dvergence of the ncdent electrc feld equal zero n a ource-free half pace,.e., nc E =, t follow that n the patal-frequency doman, nc nc nc nc nc nc nc nc k T = k T + k T + k T =. (54 e x ex u eu v ev Th phycally tate that the pectrum of the ncdent electrc feld nc T e perpendcular to the ncdent propagaton vector nc nc k. Thu, for v k, k k T = T T, (55 nc nc nc x nc u nc ev nc ex nc eu kv kv o that the x and u component of nc T e unquely provde the v component. In a mlar fahon, the followng relatonhp provde the plane-wave pectrum repreentaton for the ncdent magnetc feld ( π nc H and t pectrum nc T h [5]: nc nc ( nc nc nc nc nc nc h ( x u ( x u H = H r = T k, k exp j k r + r dk dk, v (56 and nc nc ( nc nc nc ( ( nc nc h h x u x u T = T k, k = H ρ exp j k x+ k u dxdu, v. (57 nc From Maxwell equaton, E = jωμh doman, nc, o that n the patal-frequency k T =ωμt. (58 nc nc nc e h Here, ω = πν the free-pace angular frequency and ν the free-pace frequency. In an equvalent form, k nc = k ˆ nc ˆ nc k = ω μεk, where k ˆ nc the ncdent unt-

37 propagaton vector. Conequently, the pectrum of the ncdent magnetc feld relate to the pectrum of the ncdent electrc feld nc T e n the followng way: nc T h ˆ T = k T = ε k T = kˆ T. (59 nc nc nc nc nc nc nc h e e e ωμ μ η It follow that T h alo perpendcular to the ncdent propagaton vector nc o that for k, v nc k ; namely, nc nc nc nc nc nc nc nc k T = k T + k T + k T =, (6 h x hx u hu v hv k k T = T T. (6 nc nc nc x nc u nc hv nc hx nc hu kv kv Th ay that both the x and u component of nc T h unquely provde the v component. Before movng on n the analy, t mportant to note that the expreon gven n Eq. (53 and (57 for the ncdent pectrum, nc T e and nc T h, are mathematcally equvalent to takng the two-dmenonal Fourer tranform of the ncdent feld n the nc nc ource plane,.e., E ( ρ and H ( ρ, and gong to the patal-frequency doman. Wth that ad, the expreon gven n Eq. (5 and (57 allow one to then determne nc nc ncdent feld at any obervaton pont,.e., E ( r and H ( r. Phycally, th analogou to ummng up the contrbuton of a bunch of forward propagatng plane wave whch orgnate from the ource plane []. It alo mportant to note that the plane-wave pectrum repreentaton often reult n rather complex ntegral expreon. In practce, one mut employ addtonal mathematcal technque, uch a the method of tatonary phae, to olve thee complex ntegral expreon.

38 .4.3 Method of tatonary phae In ung the plane-wave pectrum repreentaton (along wth the far-feld and PO approxmaton, ntegral of the followng form often reult [8]: b ( ( ( F k = f x exp jkg x dx, a k. (6 a Here, f ( x lowly varyng n the nterval [ ab, ] and ( kg x rapdly ocllatng except near pecal pont where the rate of change of g( x tatonary wthn the nterval,.e., where d g ( x = g ( x =. (63 dx Thee pecal pont are called crtcal pont of the frt knd [6]. Away from thee pont, kg ( x rapdly ocllatng and the potve and negatve contrbuton of the ntegrand effectvely cancel out. In th cae, an aymptotc mathematcal technque known a the method of tatonary phae help n olvng the ntegral formulated n Eq. (6. The ntal analy aume that there only one crtcal pont of the frt knd; namely, at x x = and that f ( x and ( g x are both contnuou and well behaved n the nterval [ ab, ]. ubequently, the followng condton mut hold true: ( g ( x =, and ( g x, o that upon expandng n a Taylor ere, g x, ( ( ( ( ( ( x x ( ( ( x x = g x g x g x x x g x g x g x (64 3

39 x x and ( ( ( ( ( ( f x = f x + f x x x + f x + f ( x, (65 nce f ( x lowly varyng. ubttutng Eq. (64 and (65 nto Eq. (6, the method of tatonary phae dctate that [6] ( ( ( ( ( F k F k F k, a k +. (66 Here, ( x x ( F ( k = f ( x exp jkg( x exp jk g ( x dx (67 the contrbuton from the crtcal pont of the frt knd at x = x, and F ( ( k the contrbuton from the end pont. Thee end pont are called crtcal pont of the econd knd [6]; however, the preent analy neglect to formulate ther contrbuton 5. Wth ome mathematcal prowe, Eq. (66 evaluate to the followng expreon [6]: ( x ( ( π f π F( k F ( k = exp jkg ( x exp j gn g ( x k g x 4, (68 where gn g ( x ( x ( x f g > =. (69 f g < In general, f there are multple crtcal pont of the frt knd preent n the analy, then ther ndvdual contrbuton um together. The analy leadng up to Eq. (68 and (69 aumed one-dmenonal ntegraton; however, the method of tatonary phae extend to n-dmenonal ntegraton 5. Crtcal pont of the econd knd do not come nto play becaue of the nature of the feld aumed n th reearch effort. 4

40 [7, 8]. A uch, the contrbuton F ( ( k from the crtcal pont of the frt knd at ( x x x x= x =,,, n become ( x g ( x n π f π = exp exp gn { x } { x } ( F ( k jkg( j g( k x x, (7 Det 4 where ( x ( x ( x ( x g g g x x x x x g g g xg = x x x x x g g g xn x xn x xn ( x ( x ( x ( x ( x ( x n n x = x Det{ } denote the determnant operaton, and gn{ A} λ { A} λ { A}, (7 = denote the + gnature of a real ymmetrc non-degenerate matrx A. Here, λ ± { A} are the number of potve and negatve egenvalue of A..5 Coherence The feld of nteret n electromagnetc catterng problem are often random n nature. Goodman refer to uch feld a optcal dturbance [9]. Of prmary concern n the tattcal analy of optcal dturbance coherence. In eence, coherence decrbe the degree to whch one pont n a gven optcal dturbance relate to any other pont wthn the optcal dturbance n tme or pace. An optcal dturbance coherent when there a fxed relaton between one pont and all other pont wthn the optcal dturbance. On the other hand, an optcal dturbance then ncoherent when there no fxed relaton between one pont and any other pont. tattcal properte that fall 5

41 omewhere between the precedng decrpton provde for a partally coherent optcal dturbance. Mathematcally, one realze coherence through correlaton functon (, ; t, t Γ r r. Thee correlaton functon are, n general, dependent on two pont n pace, r and r, or two ntance of tme, t and t..5. elf-coherence functon When analyzng temporal coherence, an ndvdual ue what Goodman refer to a the elf-coherence functon Γ( r,τ [9]. pecfcally, T * * Γ ( r, τ = lm u(, t+ τ u (, t dt = u(, t+ τ u (, t T T r r r r, (7 T whch mply the tme autocorrelaton of an analytc functon u(, t n pace r. Throughout the analy, u(, t * and u (, t r at a ngle pont r repreent the optcal dturbance of nteret r repreent the complex conjugate of that optcal dturbance 6. Note that n wrtng Eq. (7, one aume that the optcal dturbance emanatng from a pont ource, o that only temporal coherence effect play a role. A uch, the temporal quantty, τ = t t, the tme nterval of nteret n quantfyng temporal coherence. Put mply, the elf-coherence functon Γ( r,τ gve a dtnct gauge for temporal coherence provded τ τc π Δω, where τ c the coherence tme and Δ ω the fnte angular bandwdth of the optcal dturbance [9]. The proce ued to meaure temporal coherence help n explanng th pont further. 6. Note that the calar feld analy preented here hold for vector feld,.e., each component of the vector feld. 6

42 In practce, coherence meaurement requre the nterference of lght ung optcal devce called nterferometer. The type of nterferometer ued depend hghly on the type of coherence to be meaured for a gven optcal ource. For example, when temporal coherence of concern, lght from a pont ource nterfered wth a delayed veron of telf. Th type of nterference requre ampltude plttng of the lght. A Mchelon nterferometer acheve th type of nterference and readly decrbed throughout the optc lterature Goodman treatment partcularly nghtful [9]. In the detecton plane of a Mchelon nterferometer, the rradance I ( τ cale wth the elf-coherence functon Γ( r,τ. Th ay that one can meaure temporal coherence through the nterference of lght..5. Mutual-coherence functon When analyzng patal coherence, an ndvdual ue what Wolf, Goodman, and many other refer to a the mutual coherence functon (MCF (,,τ partcular, * (,, τ u(, t τ u (, t Γ r r [3, 9]. In Γ r r = r + r, (73 whch a tme cro correlaton of an analytc functon u(, t r at two pont n pace, r and r. When dealng wth a ngle pont n pace r, Eq. (73 reduce to a elfcoherence functon Γ( r,τ, a gven n Eq. (7, o that n general, the MCF Γ( r, r,τ more robut n quantfyng coherence. In wrtng Eq. (73 and mlarly Eq. (7, one aume that the optcal dturbance tattcally tatonary, at leat n the wde ene. Th mean that the average optcal dturbance ha no explct tme dependence; ntead, 7

43 Γ r r depend only on tme dfference, τ = t t, not the actual value of the MCF (,,τ t and t. Phycally, th analogou to teady-tate/contnuou-wave operaton of the optcal ource [3]. In wrtng Eq. (73, one alo aume that the optcal dturbance emanatng from an extended ource, o that both temporal and patal coherence effect play a role. Temporal coherence effect play a role n the defnton of the MCF (,,τ Γ r r becaue there the potental for optcal path-length dfference between the extended ource and the two pont n pace, r and r. Thee optcal path-length dfference are neglgble when there ymmetry between the extended ource and the two pont, r and r, and when the lght quamonochromatc or narrowband, uch that Δω ω, where ω the mean angular frequency of the optcal dturbance [3, 9]. When thee condton are met, the analy treat the temporal properte wthn the MCF (,,τ vz., Γ (,, τ J(, exp( jωτ Γ r r eparately, r r r r (74 * and J(, =Γ (,, τ = = u(, t u (, t A uch, the mutual ntenty (, r r r r r r. (75 J r r gve a dtnct gauge for patal coherence provded the two pont n pace, r and r, tuate themelve wthn the patal A coherence area, ( c λ Ω, where λ the mean wavelength of the optcal dturbance and Ω the old angle ubtended from the extended ource to the two 8

44 pont [3, 9]. The proce ued to meaure patal coherence help n explanng th pont further. When patal coherence of concern, one would want to nterfere the lght from an extended ource wth a patally hfted, but not delayed veron of telf [9]. Th type of nterference requre wavefront plttng at two eparate pont. The Young double lt experment acheve th type of nterference and readly decrbed throughout the optc lterature the treatment of Goodman and Wolf are partcularly nghtful [3, 9]. In the detecton plane of Young double lt experment, the rradance I ( r, τ cale wth the MCF Γ( r, r,τ rradance I ( r cale wth the mutual ntenty J ( r, r, and f the etup allow for t, the can meaure patal coherence through the nterference of lght..5.3 Complex degree of coherence Normalzng the MCF Γ( r, r,τ. Th ay that an ndvdual, a gven n Eq. (73, an ndvdual obtan a quantty referred to a the complex degree of coherence (CDoC γ ( r, r, τ, where γ ( r, r, τ Γ( r, r, τ ( r, r, τ ( r, r, τ = Γ = Γ =. (76 Note that the complex degree of (elf coherence γ (, τ r follow from Eq. (7 when dealng wth a ngle pont n pace r [9], and mlarly, the (equal-tme complex degree of coherence (, j r r follow from Eq. (75 when dealng wth ymmetry n the optcal etup and narrowband lght [3]. Furthermore, one can relate the vblty 9

45 V ( r, τ of the rradance I ( r, τ (,, τ γ r r ung the followng relatonhp [3, 9]:, detected n ther repectve nterferometer, to the CDoC V ( r ( r τ I( r τ ( r τ + I( r τ max I, mn,, τ = = γ,, τ max I, mn, ( r r. (77 Together, Eq. (76 and (77 ay that the magntude of the CDoC γ (,, τ r r provde a normalzed unt of meaure for the amount of coherence (temporal or patal n an optcal dturbance at two pont n pace, r and r, and ome tme dfference τ = t t. For example, f ( r r γ,, τ =, two dfferent pont n pace are correlated and the optcal dturbance fully coherent; however, f ( r r γ,, τ =, two dfferent pont n pace are uncorrelated and the optcal dturbance ncoherent. A partally coherent optcal dturbance then atfe γ ( τ.5.4 Cro-pectral denty < r, r, <. The cro-pectral denty (CD W (,, ω patal coherence [3]. Explctly, π r r an alternatve way of analyzng Γ ( r, r, τ = W( r, r, ω exp( jωτ dτ (78 and (,, = Γ(,, exp( W r r ω r r τ jωτ dτ, (79 uch that the MCF Γ( r, r,τ and the CD W (,, ω Th ay that the CD W (,, ω r r form a Fourer tranform par. r r a way to analyze patal coherence n the pace- 3

46 frequency doman a oppoed to the pace-tme doman wth the MCF (,,τ Moreover, Wolf derve the followng reult [3]: whch ay that the CD W (,, ω * (,, ω = (, ω (, ω Γ r r. W r r U r U r, (8 r r the cro correlaton functon of an enemble { U ( r, ω } of ample functon U ( r, ω. Thee ample functon are the pace- and angular-frequency-dependent part of a monochromatc optcal dturbance,.e., (, = (, ω exp( ω u r t U r j t. It mport to remember that, n general, the Fourer tranform of an optcal dturbance doe not ext becaue t not abolutely ntegrable. However, the Wener- Khntchne theorem tate that for a random proce that zero mean and at leat wdeene tatonary, the autocorrelaton and the pectral denty form a Fourer tranform par [3, 9]. Th an mportant pont n the analy becaue when dealng wth a ngle pont n pace r, Eq. (8 reduce to an expreon for the pectral denty ( r, ω, where Thu, the pectral denty (, ω * (, ω = (,, ω = (, ω (, ω r W r r U r U r. (8 r a way to analyze elf coherence n the pacefrequency doman a oppoed to the pace-tme doman wth the elf-coherence functon Γ( r,τ. The proce ued to meaure patal coherence n the pace-frequency doman help n explanng th pont further. When meaurng patal coherence n the pace-frequency doman, one agan ue Young double lt experment. Narrow-band flter placed behnd the lt enure that 3

47 the optcal dturbance emanatng are pace- and angular-frequency-dependent enemble, { U ( r, ω } and { U ( r, ω }. Th allow an ndvdual to conder the pectrum of the lght n the detecton plane ntead of rradance. pecfcally, one can meaure the pectral denty ( r, ω. Wolf how that n the detecton plane of th modfed Young double lt experment [3], the pectral denty (, ω the CD W ( r, r, ω r cale wth. Th ay that an ndvdual can meaure patal coherence through the nterference of lght n the pace-frequency doman..5.5 pectral degree of coherence Normalzng the CD W ( r, r, ω, a gven n Eq. (8, an ndvdual obtan a quantty referred to a the pectral degree of coherence (DoC μ ( r, r, ω, where μ ( r, r, ω = W W ( r, r, ω ( r, r, ω W( r, r, ω. (8 A uch, one can then relate the vblty V ( r, ω of the pectral denty (, ω r, detected n the modfed Young double lt experment, to the DoC μ (,, ω the followng relatonhp [3]: V ( r ( r ω ( r ω ( r ω + ( r ω r r ung max, mn,, ω = = μ ( r, r, ω. (83 max, mn, Provded Eq. (8 and (83, the magntude of the DoC μ (,, ω r r provde a normalzed unt of meaure for the amount of patal coherence n an optcal dturbance at two pont n pace, r and r, and angular frequency ω. For ntance, f 3

48 ( r r μ,, ω =, two dfferent pont n pace are correlated and the optcal dturbance patally coherent; however, f ( r r μ,, ω =, two dfferent pont n pace are uncorrelated and the optcal dturbance patally ncoherent. A patally partally < r, r, <. coherent optcal dturbance then atfe μ( ω.5.6 Gauan chell-model ource Referencng Fgure 7, n the ource plane at followng form: where, = x, x+ u, * (,, ω = (, ω (, ω v =, the CD W (,, ω r r take the W ρ ρ U ρ U ρ, (84 r ˆ uˆ. Conequently, the CD W ( ρ, ρ, ω of a Gauan chellmodel (GM ource take the followng form [3]: (,, ω = (, ω (, ω μ (, ω W ρ ρ ρ ρ ρ ρ, (85 uch that ( ρ ω r = w, A exp (86 and μ( ρ ω Note that the parameter r, = exp. (87 A, w, and are pace ndependent but are, n general, dependent on angular frequency ω. Th dependence omtted for brevty n the notaton. Alo note that upon ubttutng Eq. (85 nto Eq. (8, the magntude of the DoC μ ( ω ρ,ρ become, 33

49 ρ ρ μ( ρ,ρ, ω = μ( ρ ρ, ω = exp, (88 whch depend only on the dtance between two pont and not on the pont themelve. Th the clac charactertc of a chell-model ource [3]. mlar to the Gauan laer beam ource [], the three parameter A, w, and phycally decrbe the GM ource. For ntance, the ource beam wdth w the radal dtance ρ where the ource magntude A fall to e t ntal on-ax value. Th gve a nce gauge for the phycal ze of the emanatng beam. Lkewe, the ource coherence length the dtance between two pont ρ ρ where the magntude of the DoC μ ( ω ρ,ρ fall to e t ntal on-ax value. Th a, drect reult of the relatonhp found n Eq. (88. In practce, f ρ ρ, then the two pont are correlated and the GM ource patally coherent; converely, f ρ ρ, then the two pont are uncorrelated and the GM ource patally ncoherent. Partal patal coherence then atfe < ρ ρ <. In ung the GM ource formulated n Eq. (85-(87, the analy tractable for a varety of feld of practcal nteret. For example, the GM ource reduce to a pont ource when the ource beam wdth approache zero, w, or a plane wave when the ource beam wdth approache nfnty, w. One can alo ue the GM ource to model patally coherent Gauan laer beam. Here, an ndvdual allow the ource coherence radu to approach nfnty,. On the other hand, when the ource coherence radu approache zero,, one obtan a patally ncoherent Gauan 34

50 beam ource. Th mplcty and veratlty make the GM ource deal for nvetgaton concerned wth patal coherence..5.7 Cro-pectral denty matrx When analyzng patal coherence n the pace-frequency doman wth electromagnetc vector feld, one ue the cro-pectral denty matrx (CDM W r r (,,ω [3]. In general, the cro-pectral denty matrx CDM W( r, r,ω the dyadc (outer product created from electrc feld vector of the followng form: uch that ( ( ˆ ( ˆ l, ω = Ex l, ω + Eu l, ω ( l =, Ex( rl, ω = ( l =, E ( r, ω Er r x r u W r r E r E r (,, ω (, ω (, ω u l ( r, ω ( r, ω Ex = * * Ex(, ω Eu(, ω E r r u * * ( r, ω ( r, ω ( r, ω ( r, ω * ( r, ω ( r, ω ( m, ; n, ( r, r, ω ( m, ; n, m n mn, (89 Ex Ex Ex E u =, (9 * * Eu(, ω Ex(, ω Eu(, ω Eu(, ω r r r r = E E = x u = x u = W = x u = x u where denote Hermtan conjugate. In Eq. (89, E ( r, ω and E (, ω x l u r are member of tattcal enemble whch are at leat wde-ene tatonary, and n referencng Fgure 7, are analytc functon n two mutually orthogonal drecton perpendcular to the drecton of propagaton,.e., the v drecton. Th ay that the vector-feld reult preented n Eq. (9 analogou to the calar-feld reult gven above n Eq. (8. l 35

51 Accordngly, the D ( r, ω and the DoC μ (,, ω r r are determned from the CDM W r r (,,ω ung the followng relatonhp [3]: and μ( r, r, ω where { } = ω ω { } ( r, = Tr W( r, r, Tr { W( r, r, ω } ( ω { W r r } { W( r r ω } (9, (9 Tr,, Tr,, Tr denote the trace operaton and r = r,. Th ay that the vector-feld reult preented n Eq. (9 and (9 drectly relate to the calar-feld reult gven above n Eq. (8-(83. The magntude of the DoC μ (,, ω r r reultng from electromagnetc vector feld alo provde a normalzed unt of meaure for the amount of patal coherence,.e., μ( ω r, r,. Referencng Fgure 7, the CDM W( r, r,ω of a Gauan chell-model (GM ource take the followng element-baed form [3]: ( r, r, ω = ( ρ, ω ( ρ, ω μ ( ρ ρ, ω ( m =, ; n =, W x u x u mn m n mn uch that (, ω = A exp ( m = x, u m m wm, (93 ρ ρ (94 ρ ρ ρ ρ. (95 and μ (, ω = B exp ( m = xu, ; n = xu, mn mn mn Note that the element-baed parameter A m, w m, B mn, and mn are pace ndependent but are, n general, dependent on angular frequency ω. Th dependence omtted for brevty n the notaton. Alo note that the CDM W( r, r,ω gven n Eq. (93-(95 36

52 analogou to the calar-feld reult gven above n Eq. (85-(87; however, there are addtonal contrant, vz., B mn = when m = n, (96 Bmn when m n, (97 B mn = B, (98 * nm and mn = nm. (99 Nonethele, the GM ource preented n Eq. (93-(99 deal for nvetgaton concerned wth patal coherence..6 Polarzaton Gven electromagnetc vector feld and the CDM W( r, r,ω, a defned above n Eq. (9, polarzaton relatonhp reult. The frt polarzaton relatonhp of nteret the pace- and angular-frequency-dependent degree of polarzaton (DoP P (, ω r [3]. Partcularly, where agan, { } DoP P (, ω P 4Det,,, =, ( ( r ω { W( r r ω } { W( r r ω } ( Tr,, Det denote the determnant operaton and r = r,. In general, the r provde a normalzed unt of meaure for the amount of polarzaton n an optcal dturbance [3, 3]. When ( r wherea when ( r then atfe P ( ω P, ω =, the optcal dturbance polarzed, P, ω =, the optcal dturbance unpolarzed. Partal polarzaton < r, <. 37

53 The econd polarzaton relatonhp of nteret the pace- and angularfrequency-dependent angle of polarzaton (AOP ψ ( r, ω { Wxu ( rrω } ( rrω W ( rrω. pecfcally, Re,, ψ ( r, ω = tan, ( π < ψ π. ( Wxx,, uu,, Th angle depcted n Fgure 8 n term of a polarzaton ellpe. The em-major and em-mnor axe of th ellpe atfy the followng relatonhp [3]: and 8 { ( r, ω = 4 ( r, r, ω + ( r, r, ω ( r, r, ω a Wxu Wxx Wuu { Wxu ( rrω } Wxx ( rrω Wuu ( rrω 4 Re,, +,,,, 8 { ( r, ω = 4 ( r, r, ω + ( r, r, ω ( r, r, ω b Wxu Wxx Wuu { Wxu ( rrω } Wxx ( rrω Wuu ( rrω 4 Re,, +,,,, (, (3 repectvely. A uch, the pace- and angular-frequency-dependent ellptcty ε( r, ω follow a where χ (, ω ellptcty ε(, ω ( r, ω ( r, ω b ε( r, ω = = tan χ, ω, π 4< χ π 4 a ( r (, (4 r the ellptcty angle, whch alo depcted n Fgure 8. In general, the r provde a normalzed unt of meaure for the polarzaton tate of the optcal dturbance [3, 3]. When ( r ε, ω =, the em-major and em-mnor axe of the polarzaton ellpe equal each other,.e., a(, ω = b(, ω r r. Th correpond to a crcularly polarzed optcal dturbance. On the other hand, when ( r ε, ω =, the em- 38

54 mnor ax of the polarzaton ellpe equal zero,.e., ( r b, ω =. Th correpond to a lnearly polarzed optcal dturbance. An ellptcally polarzed optcal dturbance then atfe ε( ω < r, <. u χ ψ x Fgure 8. A decrpton of the polarzaton geometry ued n the analy. The thrd polarzaton relatonhp of nteret the angular-frequency-dependent two-pont toke vector (,,ω followng relatonhp: rr [3]. Per e, the component of th vector atfy the ( r, r, ω xx ( r, r, ω uu ( r, r, ω ( r, r, ω xx ( r, r, ω uu ( r, r, ω ( r, r, ω xu ( r, r, ω ux ( r, r, ω ( r, r, ω = ( r, r, ω ( r, r, ω = W + W = W W = W + W 3 j Wux W xu (5 or * * ( r, r, ω = ( r, ω ( r, ω + ( r, ω ( r, ω Ex Ex Eu E u * * ( r, r, ω = Ex( r, ω Ex( r, ω Eu( r, ω Eu( r, ω * *. (6 ( r, r, ω = Ex( r, ω Eu ( r, ω + Eu( r, ω Ex( r, ω * * 3( r, r, ω = j Eu(, ω Ex(, ω Ex(, ω Eu(, ω r r r r 39

55 Htory how that the ngle pont toke vector (,,ω rr, where r = r,, a very veratle tool n term of analyzng polarzaton [3, 3]. For example, P ( r, ω = ( rr,, ω + ( rr,, ω + ( rr,, ω ( rr,, ω 3, (7 whch ay that one can obtan the DoP P(, ω r two eparate way wthn the analy. 4

56 3 Background rough urface catterng lterature revew A mentoned n Chapter, the purpoe of th dertaton to extend the rough urface catterng lterature to nclude the effect of partally coherent electromagnetc beam llumnaton. In upport, recent publcaton derved a D calar-equvalent oluton for the catterng of partally coherent beam from tattcally rough urface ung the phycal-optc (PO approxmaton [, ]. Thee publcaton erve a the ba for th dertaton; however, modern-day reearch n rough urface catterng date back to the work of Lord Raylegh around the turn of the th century [3-33]. Wth th ad, one can dtnguh the publhed lterature n rough urface catterng nto two man categore. The frt category deal wth the reearch predomnately concerned wth the catterng of fully coherent and fully ncoherent llumnaton from rough urface, wherea the econd category deal wth the reearch predomnately concerned wth the catterng of partally coherent llumnaton from rough urface. 3. Fully coherent and fully ncoherent llumnaton everal dfferent reearch communte come to mnd when revewng the rough urface catterng lterature pertanng to fully coherent and fully ncoherent llumnaton. The frt couple dentfy themelve wth the rough urface catterng reearch performed by the optc and photonc communte for metrology and manufacturng applcaton. The text wrtten by tover hghlght th pont [34]. Converely, the econd couple dentfy themelve wth the rough urface catterng reearch performed by the radofrequency/mcrowave and vble/near-nfrared communte for ynthetc aperture radar and remote enng applcaton. The three-volume text by Ulaby et al. hghlght th pont [35]. Wth ome excepton, the common approache employed by thee reearch 4

57 communte are the geometrcal-optc (GO, lnear-ytem, perturbaton, PO, and fullwave method. One may refer to work of Beckmann and pzzchno [36], Ihmaru [37], Oglvy [38], Voronovch [39], Warnck and Chew [4], Elfouhaly and Guérn [4], Neto-Veperna [4], Maradudn [43], and Fung and Chen [44], for excellent ummare on rough urface catterng technque ung fully coherent and fully ncoherent llumnaton. 3.. Geometrcal-optc method When employng GO method, one typcally ue a bdrectonal reflectance dtrbuton functon (BRDF or t polarmetrc counterpart, a polarmetrc BRDF (pbrdf, to model rough urface catterng. Ncodemu wa the frt to ntroduce the BRDF n 965 [45]. Defned n radometrc term, the BRDF the reflected radance dvded by the ncdent rradance [34]. A uch, the BRDF typcally characterze how lght reflect from urface n term of a pecular and dffue component or a polarzed and unpolarzed component [46]. Both emprcal and analytcal BRDF ext n practce. Meaurement help n formulatng emprcal BRDF. For example, n preparaton for the NAA Apollo mon, the analy of lght cattered from the lunar urface led reearcher to conclude that the moon urface compoed of a partculate materal [47]. On the other hand, both PO and GO approxmaton help n formulatng analytcal BRDF []. Ung the GO approxmaton, the emnal BRDF paper that of Torrance and parrow [48], wherea the oft-referenced pbrdf paper that of Pret and Meer [49]. Many other model ext baed on ther work. Thee nclude BRDF for applcaton n pave vble/near-nfrared remote enng [5] and computer graphc [5] un lterature revew partcularly nghtful [5]. 4

58 3.. Lnear-ytem method In the late 97, Harvey and hack developed a lnear-ytem formulaton of rough urface catterng baed on calar dffracton theory [53-55]. In th approach, a urface tranfer functon characterze the catterng proce much lke the optcal tranfer functon doe for aberraton found wthn an magng ytem. The Fourer tranform of th urface tranfer functon then yeld a cattered radance dtrbuton functon cloely related to the BRDF. In the late 98, Harvey et al. modfed th theory to nclude the effect of grazng ncdence at X-ray wavelength [56]. Th helped n the degn of X-ray telecope. Mot recently, Krywono et al. modfed the theory once agan to a non-paraxal regme [57, 58]. Th calar non-paraxal lnear-ytem formulaton of rough urface catterng clam to produce accurate reult for rougher urface than the theore baed on perturbaton method and for larger ncdent and cattered angle than the theore baed on PO method [59-6] Perturbaton method The perturbaton approach to rough urface catterng model the urface roughne a a mall perturbaton relatve to the cae of a perfectly mooth urface. A uch, th approach requre that the urface roughne be mall compared to the wavelength of the ncdent radaton [37]. The lterature credt Rce wth the groundbreakng paper on th ubject [6]; however, t mportant to note that Lord Raylegh ntated the ue of many of the mathematcal technque [3-33]. Thu, the lterature often refer to the perturbaton formulaton of rough urface catterng a Raylegh-Rce theory. It alo mportant to note that dfferent approache found wthn the lterature tend to yeld mlar reult up to a ffth-order perturbaton expanon [63]; 43

59 nonethele, perturbaton method are the oldet and mot wdely ued n the rough urface catterng lterature Phycal-optc method The PO approach to rough urface catterng ue the PO approxmaton [4, ], whch analogou to ung Krchhoff boundary condton n phycal or wave optc [3, 4]. Th done o that ntead of atfyng the exact boundary condton, a done wth perturbaton method, the feld and t normal dervatve mplfy on the catterng urface. Accordngly, th approach doe not requre that the urface roughne be mall compared to the wavelength of the ncdent radaton [35]. The lterature typcally credt Beckmann wth the tralblazng work on th ubject [36], and one often ee the ttle of Beckmann-Krchhoff theory or the Krchhoff approxmaton ued n practce. It mportant to note that the PO approxmaton typcally allow an ndvdual to calculate cloed-form expreon where other approxmaton/theore would not. uch the cae when conderng the catterng of fully coherent laer beam llumnaton from rough urface [64-68] Full-wave method When employng full-wave method, one typcally ue the method of moment [4, 9, ], the fnte dfference tme doman [9, 69, 7], or the fnte element method [9, 7, 7] to atfy Maxwell equaton and model rough urface catterng. Th problem ha a rch htory that date back to the late 97. ome of the early notable work n th feld that of Bahar [73-76], Axlne and Fung [77], Thoro [78], and Colln [79, 8]. The topcal revew wrtten by Warnck and Chew outlne many uch full-wave technque [4]. 44

60 3. Partally coherent llumnaton The rough urface catterng lterature pertanng to partally coherent llumnaton the ole reult of the prolferaton of laer-baed ytem, uch a thoe found n actve-llumnaton ytem for drected-energy and remote-enng applcaton [8-85]. In recent tme, the tattcal behavor of the laer-target nteracton; n partcular, the reultng peckle pattern, ganed conderable nteret. nce the preence of peckle typcally detrmental n applcaton nvolvng coherent lght, technque for uppreng peckle naturally followed. ome of the early notable lterature n uch feld a metrology and remote enng nclude the reearch effort of Danty [86], Fuj and Aakura [87, 88], Pederen [89], Goodman [9], Parry [9], and Yohmura et al. [9]. A recent text wrtten by Goodman revew many uch technque [93]. One way to uppre peckle n actve-llumnaton ytem to ue partally coherent lght ntead of fully coherent laer lght. A a reult, th reearch topc becomng more and more popular due predomnately to the work of Wolf n creatng h unfed theory of coherence and polarzaton [, 3]. Th unfed theory help n explanng correlaton-nduced change n coherence, polarzaton, and pectrum of partally coherent lght. In partcular, much of the publhed lterature ue the properte of a partally coherent electromagnetc beam whoe cro-pectral denty matrx poee a Gauan chell-model (GM form [3]. A the name mple, chell wa the frt to conjecture uch an electromagnetc ource n 96 [94, 95]. nce then, much effort ha gone nto undertandng the phyc behnd GM ource/beam. ome of the publhed lterature nclude reearch n realzablty condton [96-98], expermental generaton [98-], numercal mulaton [-4], free-pace propagaton [5-7], 45

61 turbulent propagaton [8-], or ource/beam of mlar form [-3]. The topcal revew wrtten by Gbur and Ver [4] and a recent text wrtten by Korotkova [5] are mot thorough n revew. In regard to the catterng of partally coherent llumnaton, mot of the current lterature deal wth the catterng from low-contrat urface,.e., where the ndex of refracton dffer only lghtly from unty [6-5]. Thee are catterng urface n whch the Born approxmaton vald [3, 3, 4, 5] the topcal revew wrtten by Zhao and Wang thoroughly revew th problem [6]. In vew of th, there are far fewer publcaton on the catterng of partally coherent llumnaton from rough urface. Of the publhed work to date, the followng approache are common: the phae-creen model, the ABCD matrx formulaton, and the coherent-mode repreentaton. 3.. Phae-creen method In 975, Goodman developed a phae-creen formulaton of rough urface catterng baed on calar dffracton theory [9, 93]. In th approach, a phae-creen tranmttance functon characterze the catterng proce much lke an aperture tranmttance functon doe n phycal or wave optc [4]. Hoover and Gamz mot recently employed th approach [7]. In o dong, Hoover and Gamz aumed dealzed quamonochromatc plane-wave llumnaton. Th allowed for the applcaton of the generalzed Van Cttert-Zernke (VCZ theorem to the mutual ntenty functon on the phae-creen urface. The VCZ theorem relate the rradance to the mutual ntenty through a Fourer tranform [3, 9]. Hoover and Gamz work ultmately lead to the formulaton of a generalzed BRDF oluton whch wa the um of a coherent and 46

62 ncoherent component; however, ther work dd not drectly account for partally coherent beam llumnaton. 3.. ABCD-matrx method The text wrtten by Andrew and Phllp bet decrbe the prncple behnd the ABCD-matrx approach to rough urface catterng [8]. In general, the ABCD-matrx approach decrbe paraxal wave propagaton through any complex optcal ytem. When modelng the rough urface catterng ung the ABCD-matrx approach, a phaecreen tranmttance functon agan characterze the catterng proce. However, the ncluon of a oft-gauan aperture n the model account for the ze of the catterng urface and accompanyng dffracton effect. Korotkova dcue th pont n her text [5, 9]. The reearch of Hanen et al. [3] and Yura and Hanon [3] ued th approach to look at rough urface catterng from a target whch produced partally developed peckle. Wu and Ca alo decrbed an approach to enng the catter from rough urface ung ABCD-matrx method and partally coherent beam llumnaton va the GM formulaton [3]; however, th work only applcable to mall-angle catterng geometre wth very rough urface [33-35] Coherent-mode method The text wrtten by Otrovky bet decrbe the prncple behnd the coherentmode approach to rough urface catterng [36]. Huttunen et al. ued th approach along wth the PO approxmaton to look at the catterng from two-dmenonal mcrotructured meda [37],.e., an olated groove or lt n a perfectly conductng materal ubtrate. Th unque approach to rough urface catterng may prove ueful for future reearch effort. 47

63 4 Methodology for the 3D vector oluton Fgure decrbe the geometry ued to obtan a 3D vector oluton for the problem propoed above n Chapter. A hown, a zero mean D ample functon (, h= h x y decrbe the urface heght at the rough nterface wth tandard devaton σ h and correlaton length h. Th gve re to a tattcally rough urface. patally partally coherent electromagnetc beam llumnaton (parameter gven below emanate from the ource plane pecfed by the coordnate ( x, uv,, whch are dfferent from the urface-plane coordnate (,, x yz. A uch, the vector, ρ = xxˆ + uuˆ, pont from the ource plane orgn to a tranvere beam locaton nce v = n the ource plane; the vector, r = xˆ y yˆ + z z ˆ, pont from the ource-plane orgn to the urface-plane orgn; and the vector, r = xxˆ + yyˆ + zz ˆ, pont from the urface plane orgn to an obervaton pont. Note that n the ource and the urface plane, the x axe algn, whch aume that the urface of the homogeneou medum tattcally otropc [38, 43] wth mpedance η. Above, the medum free pace wth mpedance η. 4. Incdent feld cro-pectral denty matrx A mentoned above, patally partally coherent electromagnetc beam llumnaton emanate from the ource plane. Wth th n mnd, the analy ue a Gauan chell-model (GM form for the ncdent feld cro-pectral denty matrx (CDM (, W ρ ρ [3, 94], uch that 48

64 W ( ρ, ρ E ( ρ E ( ρ * * Ex( ρ Ex ( ρ Ex( ρ Eu ( ρ = * * Eu( Ex ( Eu( Eu ( ρ ρ ρ ρ, (8 = Wmn ( ρ, ρ ( m = x, u; n = x, u ρ ρ ρ ρ = Amexp A n exp B mn exp 4w 4w mn where denote correlaton, denote Hermtan conjugate, and denote complex conjugate. In Eq. (8, the element-baed parameter ampltude n the x and u drecton, repectvely, A m and A n are the beam w the ource wdth, and the element-baed parameter B mn and mn = nm are the correlaton ampltude and correlaton length, repectvely. Note that namely, B mn follow addtonal contrant [3, 97]; B B mn mn = when m= n when m n. (9 B mn = B * nm Alo note that, n general, the ncdent feld E and the parameter A m, w, mn, and B mn are radan frequency ω dependent [3]; however, the analy omt th dependence for brevty n the notaton The analy preented n th chapter ue the MK ytem of unt n addton to the engneerng gn conventon for the tme-harmonc varaton (cf. Footnote, p. 5. In addton, ome of the notaton mplfed from that preented n Chapter, e.g., E nc = E. Th done for brevty n the notaton. 49

65 (, -y, z r ( h z r ( x, yz, x u v q q ( x, yh, f = 3p r f y L ( h x L (a ŝ ˆp kˆ ( h ( h q ˆn r q ŝ kˆ r ˆ -n t q pˆ r kˆ t (b Fgure 9. The macro-cale (a and mcro-cale (b catterng geometry of a D tattcally rough urface of length L and wdth L. 5

66 4. cattered feld The preent analy ue the PO approxmaton to develop a far-feld expreon for the cattered feld E ( r. For th purpoe, one can wrte the ncdent feld E ( term of t pectrum ( k, x ku T ung the plane-wave pectrum repreentaton [5]. r n Ung the macro-cale catterng geometry gven n Fgure 9a, the followng expreon reult: jk r jk r ( = ( kx, ku e e dkxdku E r T ( ( π and ( x u = ( jkxx jkuu T k, k E ρ e e dxdu, ( whch are vald n the ource-free half pace where v. In Eq. ( and (, k = k ˆ ˆ ˆ ˆ k = kxx+ kuu+ kvv the ncdent propagaton vector, k = π λ the freepace wavenumber, and λ the free-pace wavelength. For mot drected-energy and remote-enng engagement cenaro, all of the obervaton pont of nteret are n the far feld. A uch, the cattered electrc feld E ( r depend on the far-feld vector potental, L( r and ( relatonhp 8 : N r, ung the followng jkr e E ( r jk ( φθ ˆˆ θφ ˆˆ L( r η( θθ ˆˆ+ φφ ˆˆ N( r, ( 4π r 8. In the far feld, r D λ >, where D L and r L; conequently, the analy neglect all contrbuton to the cattered feld becaue ther contrbuton cale a E that are n the radal ˆr drecton r n, where n =,3,, and are neglgble [4]. 5

67 ˆ ( ( e jk rr Lr Mr d, (3 = and ( ( ˆ = e jk rr N r J r d. (4 In Eq. (-(4, ˆθ and ˆφ are unt vector n the polar (vertcal polarzaton and azmuth (horzontal polarzaton drecton, repectvely, M( r and ( J r are the equvalent urface current dente, repectvely, and the vector, r = x xˆ + y yˆ + h z ˆ, pont from the urface-plane orgn to a pont on the tattcally rough urface. Ung the mcro-cale catterng geometry gven n Fgure 9b and the PO approxmaton [4, ], M r M E r ( ( (5 and J( r J E ( r where J and M are dyadc, uch that η J = nˆ ˆ pˆ pˆ ˆ ( r ( r and = ( + r + ( + r In Eq. (7 and (8, ˆ = ˆ ( x, y, (6 (7 M nˆ ˆ ˆ ˆ ˆ p p. (8 n n the D unt outward normal vector gven by n h ˆ ˆ ˆ x x hy y + z nˆ = =, (9 n + h + h x y where h x y (, h h( x, y h h x y = = h = = x x y y. ( 5

68 Furthermore, ˆ and ˆ p are the unt perpendcular and parallel vector, wherea r and r are the correpondng Frenel reflecton coeffcent, repectvely. Referencng the mcro-cale catterng geometry n Fgure 9b, the followng relatonhp reult: ˆ k nˆ ˆ = pˆ = ˆ kˆ pˆ = ˆ kˆ ˆ k nˆ r r. ( Thu, n arrvng at the relatonhp found n Eq. (7 and (8, one mut ue the GO approxmaton []; pecfcally the law of reflecton, uch that ˆ ˆ ˆ ˆ r nk = nk 9. Baed on Eq. (9, t mportant to note that the ntegraton n Eq. (3 and (4 over the parameterzed rough urface,.e., d = n dx dy. Conequently, ung Eq. ( and ubttutng Eq. (5-(9 nto Eq. (3 and (4, one obtan the followng expreon: ( Lr L L jk r jqr = L T k, k e e dk dk dx dy ( π L L ( x u x u ( and N( r where ( ˆ L L jk r jqr = N T ( kx, ku e e dkxdkudx dy, (3 η ( π L L q= k rˆ k = q xˆ + q yˆ + q z ˆ, L = n M, and N = n J. Wthout further x y z mplfcaton, no analytcal expreon ext for the far-feld vector potental, L( r and N( r, gven n Eq. ( and (3. Th becaue the ntegrand n Eq. ( and (3 are complcated functon of urface heght and urface lope; namely, h, h x, and h y wth repect to the ntegral over the parameterzed rough urface. One 9. ee Appendx B for more detal. 53

69 typcally mplfe thee ntegral ung the tatonary-phae (P approxmaton [, 35],.e., ( qr ( qr. (4 x y A a reult, the relatonhp found n Eq. ( mplfy, uch that and n turn, h q q x y x hy, (5 qz qz L L and N N n Eq. ( and (3. mlar to the PO approxmaton, the P approxmaton phycally dctate that reflecton from the rough urface locally pecular and exclude all local dffracton effect [, 35]. 4.3 cattered feld cro-pectral denty matrx The analy preented here develop cloed-form expreon for the cattered W r, r W r, r feld CDM (. In general, ( uch that n the far feld W r, r E r E r ( ( ( depend on the cattered feld E ( r, * * ( r ( r ( r ( r E E E E θ θ θ φ =. (6 * * Eφ ( Eθ ( Eφ ( Eφ ( r r r r = W = = ( r, r ( p θφ, ; q θφ, pq Ung Eq. (, one determne the matrx element found n Eq. (6 a * ( r, r =Ω ( φˆ ˆ( φˆ ˆj ( r ( r Wθθ L Lj = xyz,, j = xyz,, * ( φˆ ˆ( θˆ ˆj L ( r N ( r * + η( θˆ ˆ( φˆ ˆ j N( r Lj( r + η ˆ * ( θ ( θ j ( r ( r j + η j ˆ ˆ ˆ N N, (7 54

70 * ( r, r =Ω ( φˆ ˆ( θˆ ˆj ( r ( r Wθφ L Lj = xyz,, j = xyz,, * ( φˆ ˆ( φˆ ˆj L ( r N ( r * η( θˆ ˆ ( θˆ ˆ j N( r Lj( r + η ( ( ( * θˆ ˆ φˆ ˆ j N r Nj ( r + η j * ( r, r =Ω ( θˆ ˆ( φˆ ˆj ( r ( r Wφθ L Lj = xyz,, j = xyz,, * ( θˆ ˆ( θˆ ˆj L ( r N ( r * η( φˆ ˆ ( φˆ ˆ j N( r Lj( r + η ( ( ( * φˆ ˆ θˆ ˆ j N r Nj ( r η j +, (8, (9 and where * ( r, r =Ω ( θˆ ˆ( θˆ ˆj ( r ( r Wφφ L Lj = xyz,, j = xyz,, * ( θˆ ˆ( φˆ ˆj L ( r N ( r * η( φˆ ˆ( θˆ ˆ j N( r Lj( r + η ˆ * ( φ ( φ j ( r ( r j η j ( 4π ˆ ˆ ˆ N N, (3 jk r jk r e e Ω = k. (3 rr In addton, ung Eq. (3-(5, one determne the element-baed correlaton found n Eq. (3 from the followng relatonhp:. In ung Eq. (3-(5, one mut aume that all obervaton pont are n the far feld (cf. Footnote 8, p. 5, the phycal-optc approxmaton hold (cf. Appendx B, and the effect of hadowng/makng and multple catterng are neglgble [38, 4]. 55

71 ( ( r Lr L = ( π 4 ( ( r Lr N L L L L L L L L = L T T L ( kx, ku ( kx, ku jkvr jkvr jqxx jqxx jqyy jqyy jqzh jqzh e e e e e e e e ( π 4 L L L L η L L L L dk dk x x dk dk dx dx dy dy u u L T T N ( kx, ku ( kx, ku jkvr jkvr jqxx jqxx jqyy jqyy jqzh jqzh e e e e e e e e dk dk x dk dk dx dx dy dy x u u, (3, (33 ( ( r N r L = ( π 4 L L L L η L L L L N T T L ( kx, ku ( kx, ku jkvr jkvr jqxx jqxx jqyy jqyy jqzh jqzh e e e e e e e e dk dk dk dk dxdx dydy x x u u, (34 and ( ( r N r N = ( π 4 η L L L L L L L L N T T N ( kx, ku ( kx, ku jkvr jkvr jqxx jqxx jqyy jqyy jqzh jqzh e e e e e e e e dk dk x dk dk dx dx dy dy x u u, (35 where r = r. Inherent n Eq. (3-(35 the aumpton that the ncdent feld plane-wave pectrum tattcally ndependent of the rough urface. Th aumpton phycally ntutve; thu, Eq. (3-(35 contan two eparate correlaton. 56

72 The frt correlaton wth repect to the ncdent feld plane-wave pectrum. Th correlaton equvalent to a dyadc [cf. Eq. (]; namely, T where Φ=Φ ( k,,, x kx ku ku k k k k = Φ, (36 ( x, u T ( x, u. The econd correlaton wth repect to the parameterzed rough urface. Th correlaton a jont charactertc functon χ of the random varable h = h( x, y and h h( x, y =, uch that e e jqzh jqzh = χ. (37 In practce, one mut chooe a form for th jont charactertc functon. A very common choce for the tattcal dtrbuton of the rough urface to aume that the urface heght are Gauan dtrbuted and Gauan correlated. In o dong, the jont probablty denty functon p p( h, h and h take the followng form [36]: where ( x x, y y = of the random varable h + h Γ hh p = exp, (38 πσ σ h ( Γ h Γ Γ=Γ the urface autocorrelaton functon, uch that ( x x ( y y Γ= exp exp. (39 h h Htory how that one typcally chooe Gauan-Gauan (G-G model for analytcal convenence [93]; however, other model ext n practce. For example, the tretched exponental-tretched exponental (E-E model better characterze urface roughened by random ndutral procee [38]. Bau et al. hghlghted th pont wth proflometer h 57

73 meaurement of andblated metallc urface [39, 4]. Unfortunately no general analytcal form ext for the E jont charactertc functon; neverthele, the analy of Bau et al. alo howed that G-G model were tll farly good approxmaton for andblated metallc urface [39, 4]. Thu, Fourer tranformng the jont probablty denty functon p n Eq. (38 yeld the dered jont charactertc functon for the preent analy [36],.e., χ jqzh jqzh = p e e dhdh where χ χ( k,,, x kx ku ku; x x, y y, (4 σ h = exp ( qz+ qz exp( σ hqzqzγ =. Note that throughout the lterature, numerou other urface model ext n addton to G-G and E model. Ung the relatonhp found n Eq. (36-(4, the ntegrand n Eq. (3- (35 tll contan complcated functon wth repect to the ource and urface parameter. To mplfy the analy, one can eparate thee complcated functon nto ampltude and phae term, vz., ( ( r Lr L = ( π 4 L L L L L L L L f e dk dk dk dk, (4 dx dx dy dy jkg x x u u. For example, a recent publcaton explored the ue of non-gauan urface autocorrelaton functon [4]. 58

74 and ( ( r Lr N = ( π 4 ( ( r N r L L L L L η L L L L = ( π 4 ( ( r N r N L L L L η L L L L = ( π 4 η L L L L L L L L f e f e f e Here, f = χ ( L Φ L f = χ L N dk dk dk dk, (4 dx dx dy dy jkg x x u u dk dk dk dk, (43 dx dx dy dy jkg x x u u dk dk dk dk. (44 dx dx dy dy jkg x x u u ( Φ f = χ N L ( Φ, (45, (46, (47 and f = χ ( N Φ N (48 are ampltude dyadc that contan all of the ampltude term, and ( yˆ vˆ ( yˆ vˆ g = r + y k r + y k v v ( yu ˆ ˆ ( + ( x k xk + y k yk k x x u u k ( xr ˆ ˆ x xr ˆ ˆ x ( yr ˆ ˆ y yˆ rˆ y + + (49 a common phae functon that contan all of the phae term. Wthout further mplfcaton, no cloed-form expreon ext for the ntegral relatonhp gven n Eq. (4-(44. 59

75 To mplfy the ntegral found n parenthe n Eq. (4-(44, the analy ue an aymptotc mathematcal technque known a the method of tatonary phae (MoP [6, 7, 37]. In o dong, one aume that the ampltude term are lowly varyng n the nterval (,. One mut alo aume that the phae term are rapdly ocllatng n the nterval (, except near pecal pont where the rate of change zero or tatonary. Thee pecal pont are called crtcal pont of the frt knd [6]. Away from thee pont, the phae term are rapdly ocllatng and the potve and negatve contrbuton of the ntegrand n Eq. (44 effectvely cancel out. Ung the MoP to mplfy Eq. (4-(44 ha two mplcaton wth regard to the macro-cale catterng geometry gven n Fgure 9a. The frt mplcaton wth repect to the v component of the ncdent propagaton vector ( ( v x u k ; namely, k = k k k. (5 In partcular, the analy aume that k v k and x k v k ; a a reult, u k n f, f, f, and f v, ( x, ( u, k n g k k k k k. (5 Th phycally mple that the ncdent electromagnetc feld are hghly drectonal beng predomnately drected along the v drecton n Fgure 9a. The econd mplcaton that the dtance from the ource-plane orgn to the urface-plane orgn mut be much, much greater than half the urface length,.e., r L, whch typcally the cae for mot drected-energy and remote-enng engagement cenaro. To provde ome dea 6

76 of how much greater, lettng r = 5L, r = 5L, and r = L reult n percentage error of 8%, 4%, and %, repectvely. A uch, ung the MoP to mplfy Eq. (4-(44 reult n the followng relatonhp: ( π r ( ( r Lr L ( π ηr ( ( r Lr N ( π ηr ( ( r N r L and N( r N ( r where,,, and ( π η r k k k k, (5, (53, (54, (55 are dyadc that contan all of the ampltude and phae term evaluated at the crtcal pont of the frt knd whch one determne a The analy explctly defne,, ( yu ˆ ˆ k k k x k y x,, u,, r r, and. (56 n Appendx D for dfferent materal ubtrate,.e., delectrc, conductor, and a perfect electrcal conductor (PEC. Provded Eq. (5-(56 and Appendx D, one tll left wth ntegral wth repect to the parameterzed rough urface. Thee ntegral take the followng elementbaed form:. ee Appendx C for more detal. 6

77 where ( m x, u; n x, u ( yu ˆ ˆ k ( yu ˆ ˆ ( yu ˆ ˆ k ( yu ˆ ˆ L L L L k k k Ψ mn = Φmn x, x, y, y r L L L L r r r k k k χ x, x, y, y ; x x, y y r r r r ( xr ˆ ˆ xr ˆ ˆ exp ( yr ˆ ˆ yr ˆ ˆ exp jk x x jk y y ( ( k k yu ˆ ˆ exp j ( x x exp j y y r r ( yˆ vˆ( exp jk y y dx dx dy dy, (57 = =. In Eq. (57, Φ mn equvalent to the Fourer tranform of the ncdent feld CDM element found n Eq. (8,.e., jkxx jkxx jkuu jkuu mn Wmn ( r, r e e e e dxdx dudu Φ = π A A B m n mn = exp amn kx amn kx + b mnkxkx ( amn bmn where ( m x, u; n x, u = = and ( ( ( ( exp a k a k + b k k mn u mn u mn u u a a = + b b = a = b = mn mn mn mn mn mn 4w mn 4 amn bmn 4 amn bmn mn ( ( b, (58. (59 One can reduce the ntegral found n Eq. (57 nto cloed-form expreon. For th purpoe, the analy frt perform the followng varable tranformaton: x = x x x = x + x y = y y y = y + y, (6 d a d a o that Eq. (57 mplfe nto the followng expreon: 6

78 π AmAnB mn kσ h Ψ mn = exp ( ϑz+ ϑz 4 ( amn bmn L L ya L L xa xd yd exp kσϑϑ h z zexp exp L ya L L xa L h h k k exp ( amn + b mn xdexp ( amn b mn xa r r ( yˆ uˆ k ( ( yˆ uˆ mn mn d exp ( mn mn k exp a + b y a b ya r r k k exp j ( ϑx + ϑx xd exp j ( x x xa ϑ ϑ k k exp j ( ϑy+ ϑy yd exp j ( ϑy ϑy ya exp where ( m x, u; n x, u k = = and j xaxd j r ( yˆ uˆ k exp yayddxddxadyddya r, (6 ϑ = xr ˆ ˆ ϑ = yˆ r ˆ yˆ vˆ ϑ = zˆ rˆ zˆ v. ˆ (6 x,, y,, z,, From here, one mut handle the exponental term contanng the urface autocorrelaton functon (.e., the frt exponental term nde the ntegral above. Htory how that there are two eparate way to go about th. The frt to expand the ad exponental term n a Taylor ere and proceed wth the evaluaton of the ntegral [36]. Mathematcally, th approach applcable to all urface; however, becaue the ere lowly convergent, the analy lmt th approach to mooth-to-moderately rough urface the next ub-ecton develop a cloed-form expreon for th cae. The other approach nvolve expandng the urface autocorrelaton functon [cf. Eq. (39] n a Taylor ere and retanng only the frt and econd order term [36]. Th treatment 63

79 applcable to very rough urface ub-ecton 4.3. develop a cloed-form expreon for th cae mooth-to-moderately rough urface When conderng mooth-to-moderately rough urface, one mut expand the exponental term contanng the urface autocorrelaton functon found n Eq. (6 n a Taylor ere. pecfcally, l ( kσϑϑ h z z x d y d exp kσϑϑ h z zexp exp = h h l= l!. (63 l l exp x exp d y d h h ubttuton of Eq. (63 nto Eq. (6 allow one to then eparate the ntegral over the parameterzed rough urface,.e., ( amn bmn l ( kσϑϑ h z z π AmAnB mn kσ h Ψ mn = exp ( ϑz + ϑz 4 l= l! L exp a b x a j x x xa r L L xa l k exp x exp d ( a mn + b mn xd xa L h r exp L L k k ( mn mn exp ( ϑ ϑ k ( ϑ + ϑ x j xa r x r x d d a ( yu ˆ ˆ ( mn mn k exp ( ϑ ϑ k exp a b y a j y y ya r ( yu ˆ ˆ ( ϑ ϑ ( yu ˆ ˆ ( mn mn L ya l k exp y exp d a + b y d ya L h r k exp + dx dx j ya r y y yd dyddya r. (64 64

80 Th a very mportant tep n the analy. It allow for the development of a cloed form expreon for mooth-to-moderately rough urface condton wthout havng to convert to polar coordnate. It a relatvely traght forward proce to evaluate the ntegral over x d and y d n Eq. (64; however, complex error functon reult due to the parameterzed rough urface [, ]. Thee complex error functon model dffracton caued by the ncdent radaton over-llumnatng the rough urface and are neglgble under certan condton. A uch, the preent goal n the analy to determne condton n whch one can extend the x d and d y lmt of ntegraton to (, condton for thee approxmaton occur when n Eq. (64. The neceary l k exp ( mn + mn > l k ( yu ˆ ˆ exp ( mn + mn > x exp d a b x d δ x h r y exp d a b y d δ y h r, (65 where δ x and δ y are uer-defned parameter and denote the pont at whch the exponental functon wth repect to x d and y d no longer mantan gnfcant value n Eq. (64. Furthermore, f l = and x = y = L, the argument of the exponental are d at a mnmum and the x d and y d lmt of ntegraton are at a maxmum. One then derve the followng condton from Eq. (65: d ( δ r ln ( δ y r ln x L> L> kw kw yˆ uˆ. (66 65

81 Thee condton phycally mean that the projected fully coherent ncdent beam ze mut ft on the rough urface. In atfyng thee condton the x d and y d lmt of ntegraton extend to (, n Eq. (64. Note that f δx = δy = δ, the econd yu ˆ ˆ = co [cf. Fgure condton n Eq. (66 become more trngent nce ( θ 9a]. One then determne how well the ncdent beam ft by δ the maller the δ, the more accurate the approxmaton. Alo note that f the projected fully coherent ncdent beam ze doe not ft on the rough urface, complex error functon reult [, ], and one ha to evaluate the follow-on ntegral expreon numercally. Aumng that the condton n Eq. (66 hold, ubequent evaluaton of the ntegral over x d and y d mplfe Eq. (64, o that 3 3. One mut complete the quare n the exponental term and ue of the followng ntegral relatonhp [5, p. 66]: exp( at exp( jbt dt = π a exp b ( 4a where a >., 66

82 ( kσϑϑ h z z l 3 π r hamanb mn kσ h mn exp ( ϑ z ϑ z ( mn mn x y a b l= l! lmnl mn kr h kr h exp ( ϑx + ϑx exp x y ( ϑy+ ϑy 8l mn 8lmn L k k h exp ( amn b mn xaexp x x a r 8 L lmn kr h k exp ( ϑx + ϑx x exp x a ( 4 j ϑx ϑx xa mn dxa l L 4 k ( yu ˆ ˆ ( ( yu ˆ ˆ k h exp a mn b mn ya y y a r 8 L l mn Ψ = + where ( m x, u; n x, u ( yu ˆ ˆ ( k ϑ + ϑ exp ( ϑ ϑ kr exp = = and h y y ya j y y y y a 4l mn ( = k a + b + lr x lmn h mn mn ( ( yu y ˆ ˆ lmn = k h amn + bmn + lr dy a, (67. (68 mlar to the analy preented above, one can extend the x a and y a lmt of ntegraton to (, n Eq. (64 and ubequently Eq. (67. Here, the neceary condton for thee approxmaton occur when ( δ r ln ( δ y r ln x 4 4 L> + L> + kw ˆ ˆ αmn kw yu α, (69 mn where mn α mn = (7 w the element-baed ource rato, and δ x and δ y are agan uer-defned parameter, repectvely. They denote the pont at whch the exponental functon wth repect to 67

83 x a and y a no longer mantan gnfcant value n Eq. (67. The condton gven n Eq. (69 phycally mean that the projected partally coherent ncdent beam ze mut ft on the rough urface. In atfyng thee condton the x a and y a lmt of ntegraton extend to (, n Eq. (67. Addtonally, f δx = δy = δ, the econd condton n Eq. (69 become the mot trngent wthn the mooth-to-moderately rough urface analy. Aumng that the condton n Eq. (69 hold, one can then evaluate the remanng ntegral n Eq. (67 4. In o dong, the followng cloed-form expreon reult: x ( lmn r h( ϑx ϑx y ( ( ˆ ˆ ( lmn r h y u ϑy ϑy ( kσϑϑ h z z l 4 4 4π r hamanb mn kσ h mn exp ( ϑ z ϑ z ˆ ˆ ( mn mn x y k y u a b l= l! l mnlmn Ψ = + kr h exp + x x 4l mnlmn kr h exp y y 4 + l mnlmn r exp + x l mn ( k hb mn lr ( ϑx ϑx r exp + y ( yˆ uˆ l mn kr exp j 3 h x y lmnlmn ( k ( ˆ ˆ ( hb y u lr mn ϑy ϑ y x ( ϑ ϑ + ( ϑ ϑ y lmn x x lmn y y (7 where ( m x, u; n x, u = = and 4. ee Footnote 3, p

84 x 4( ( yu ˆ ˆ 4( = r + a b x lmn h mn mn lmn. (7 = r + a b y y lmn h mn mn lmn At frt glance, the ummaton omewhat obcure the phycal nterpretaton of Eq. (7. However, further examnaton of th cloed-form expreon how that the exponental term on the thrd and fourth lne generally drve the angular extent of the pectral denty (D, wherea the exponental term on the fourth and ffth lne generally drve the angular extent of the pectral degree of coherence (DoC. Before explorng thee pont further n the next chapter, the analy conder very rough urface Very rough urface When conderng very rough urface n the analy, one mut expand the urface autocorrelaton functon found nde the frt exponental term n Eq. (6. Here, one retan only the frt and econd order term n a Taylor ere. To make th concept manfet, the analy frt wrte the jont charactertc functon found n Eq. (4 n an alternatve form, where ( yˆ uˆ k ( yˆ uˆ k k k χ x, x, y, y ; xd, yd = r r r r. (73 ϑz ϑ z x d y d exp kσϑϑ h z z + exp exp ϑz ϑ z h h In the cae of very rough urface condton,.e., 5 The followng crteron: σh.5λ, help n dcernng the tranton pont from the mooth-to-moderately rough urface regme to the very rough urface regme and an emprcally determned relatonhp wthn the analy. 69

85 k σϑϑ, (74 h z z the alternatve form found n Eq. (73 mantan gnfcant value when ϑz ϑ z x d y d + exp exp. (75 ϑz ϑz h h nce all of the obervaton pont of nteret are n the far feld, f one then conder that ϑz ϑz, (76 ϑ ϑ z z the relatonhp found n Eq. (75 only poble for mall x d and y d. Wth Eq. (73-(76 n mnd, t make nce to expand the exponental functon found n Eq. (75 and retan only the frt and econd order term, o that x d y d xd yd exp exp h h h. (77 h ubttutng Eq. (77 nto Eq. (6 allow one to agan eparate the ntegral over the parameterzed rough urface. A mentoned before, th allow for the development of a cloed-form expreon for very rough urface condton wthout havng to convert to polar coordnate. Carryng out the ubequent ntegraton 6, the followng cloed-form expreon reult: 6. ee Footnote 3, p

86 4π r A A B k σ Ψ = ϑ ϑ where ( m x, u; n x, u 4 4 h m n mn h mn exp ( z z ˆ ˆ ( mn mn x y k yu a b mnmn kr x h exp ( mn r h( ϑx + ϑ x x x 4mnmn kr h y exp ( ( ˆ ˆ ( y y mn r h yu ϑy + ϑy 4mnmn kr exp ( x hb mn + rσ hϑϑ z z ( ϑx ϑx mn kr exp + y ( yu ˆ ˆ mn 3 kr h y x exp j x y mn ( ϑx ϑx + mn ( ϑy ϑy mnmn = = and ( ( ˆ ˆ hb mn yu rσϑϑ h z z ( ϑy ϑy ( = k a + b + k r σϑϑ x mn h mn mn h z z ( ( y u y mn = k ˆ ˆ h amn + bmn + k rσϑϑ h z z x 4( ( yˆ uˆ 4( = r + a b x mn h mn mn mn = r + a b y y mn h mn mn mn, (78. (79 In ung the cloed-form expreon gven n Eq. (78, the analy mut atfy the condton found n Eq. (69. The cloed-form expreon obtaned n Eq. (78 remarkably phycal. For ntance, the exponental term on the econd and thrd lne of Eq. (78 are predomnately reponble for the angular extent of the cattered D. Thee exponental term are functon of the um of the quare of the obervaton projecton,.e., ϑ x, and ϑ y,. On the other hand, the exponental term on the fourth and ffth lne of Eq. (78 determne the angular extent of the cattered DoC. Note that thee term are functon 7

87 of the dfference of the obervaton projecton,.e., ϑx ϑx and ϑy ϑy. Thu, one can tate that the cloed-form expreon obtaned n Eq. (78 allow the cattered feld CDM to mantan t GM form wth repect to ϑ x, and ϑ y, thee pont more cloely n the next chapter.. The analy examne 7

88 5 Exploraton of the 3D vector oluton The purpoe of th chapter to explore the 3D vector oluton obtaned above n Chapter 4. A tated n Chapter, by formulatng the 3D vector oluton n a manner content wth Wolf unfed theory of coherence and polarzaton [, 3], all phycal mplcaton nherent n Wolf work apply here. Accordngly, one can readly formulate the cattered pectral degree of coherence (DoC (, μ r r, the normalzed cattered pectral denty (D ( r, and the cattered degree of polarzaton (DoP P ( N r from the cloed-form expreon developed above for the cattered feld cro pectral denty matrx (CDM W ( r, r. The analy ue the followng relatonhp [3]: ( r, r μ = Tr { W ( r, r } (, (8 Tr, Tr, { W r r } { W ( r r } and ( r Tr { W ( r, r } N ( r =, (8 max Tr { W ( r, r } 4Det, P =, (8 { W ( r r } { W ( r r } ( Tr, where Tr{ } denote the trace operaton, Det{ } denote the determnant operaton, and r = r, correpond wth a ngle obervaton pont. Thee relatonhp contan meaurable quantte n practce and erve a metrc n whch to compare the 3D vector oluton to prevouly valdated oluton and emprcal meaurement. Much of the analy preented n th chapter ue a 5.8 cm 5.8 cm Labphere Infragold coupon [4]. It alo ue a nomnal far-feld etup, where 73

89 λ =.6 μm, r = r, = 85 cm, and w =.9 mm. A uch, the Labphere Infragold coupon mantan the followng complex ndex of refracton: n= 3.45 j63.6 [43]. Note that a KLA Tencor Alpha-tep IQ urface Profler [44] determned the urface tattc of the Labphere Infragold coupon a σ =.9 μm, 6.9 μm, and σ =.44 rad [cf. Eq. (9] ung four cm can (tep ze. μm. Thee urface h tattc relate to very rough urface condton [cf. Eq. (74]. 5. Comparon wth the D calar-equvalent oluton In order to compare the 3D vector oluton to the prevouly valdated D calarequvalent oluton [, ], the analy aume horzontally polarzed (-pol llumnaton and an n-plane catterng geometry,.e., A = B = B = and h h = u ux xu φ = φ =. Th provde the etup needed to make a far comparon between the, 9 two oluton. 5.. Angular pectral degree of coherence radu An ndvdual can formulate a cloed-form expreon that decrbe the angular extent over whch the catter feld correlated,.e., the angular DoC radu. In general, the angular DoC radu provde a gauge for the average peckle ze oberved n the far feld and a quantty of mportance when dealng wth drected-energy and remoteenng applcaton. Note that the analy preented here hghly analogou to that performed for the D calar-equvalent oluton [, ]. Becaue of the ummaton n Eq. (7, t not poble to derve a cloed-form expreon for the angular DoC radu for mooth-to-moderately rough urface. Thu, the preent analy lmted to very rough urface. Aumng that Eq. (69 hold, o 74

90 that the ncdent llumnaton ft on the rough urface, the exponental term on the fourth and ffth lne of Eq. (78, n general, determne the angular extent of the cattered DoC. Provded -pol llumnaton and an n-plane catterng geometry, only Ψ xx ext wthn the analy, and the dfference of the obervaton projecton mplfy, uch that ( ( ( ( ϑ ϑ = n θ co φ n θ co φ = (83 x x and ϑy ϑy n ( θ n ( φ n ( θ n ( φ n ( θ n ( θ = =. (84 Conequently, the followng correlaton exponental γ reult from Eq. (78 for kr γ = exp + σ ϑ ϑ n n y ( ˆ ˆ θ θ y u xx y r xx exp n ( θ n ( θ y ( ˆ ˆ y u xx ( ( ˆ ˆ hb xx y u r h z z ( ( Upon ettng γ equal to e, the followng expreon reult: Ψ xx :. (85 n ( θ n ( θ xx e y yu ˆ ˆ A y r xx r ( ˆ ˆ h yu ( yu + 8w xx 4 xx ˆ ˆ h σϑϑ h z z yu ˆ ˆ = + r + w k w r. (86 4 Becaue the magntude of the argument of γ large (pecfcally the krσϑϑ term, n ( θ n ( θ h z z for Eq. (85 to have a gnfcant value. Th mple that θ θ and that γ approxmately a functon of Δ θ = θ θ. Ung th nght, Eq. (86 mplfe becaue 75

91 n co ( θ = n ( θ +Δθ = n ( θ co( Δ θ + co( θ n ( Δθ n ( θ + co( θ Δθ ( θ = co( θ +Δθ = co( θ co( Δθ n ( θ n ( Δθ co( θ, (87, (88 and ϑϑ ( yˆ uˆ ( θ + ( θ ( θ + ( θ co ( θ ( θ ( θ co co co co = z z co + co. (89 After ome mple algebra, the expreon for the angular DoC radu become where xx xx w 8w Δθ + e ϖ r + ( α xx Ω kσh ( + ϖ + σh ( + ϖ α = a ource rato [cf. Eq. (7], ϖ co( θ co( θ, (9 = a projecton rato, Ω = w r the ource half angle (vewed from the rough urface, and σ = h σ h h (9 the urface lope tandard devaton [36] Baed on the aumpton ued wthn the analy [cf. Footnote, p.56], vald urface lope tandard devaton mut atfy the followng condton: σ h [35, 66]..5 rad 76

92 For all ntent and purpoe, one can neglect the term nvolvng Ω σ h n Eq. (9. Th rato reult n value on the order of 4 for mot drected-energy and remote-enng engagement cenaro. Wth th ad, one can alo clam that the ource term contaned n the radcal above much greater than the urface term. Thu, factorng out the ource term and ung the bnomal approxmaton yeld ( α xx Ω + Δθ e ϖ + + ( αxx 4 kw σh ( + ϖ. (9 Ω ϖ + ( α For mot cae of nteret, one can neglect the econd term contaned wthn the parenthe n Eq. (9. It only provde a mall correcton to the angular DoC radu due to the urface parameter. A a reult, the angular DoC radu become a xx functon of only the ource parameter. Th hghly analogou to the reult obtaned by the D calar-equvalent oluton [, ]. It alo content wth the clac, narrowband, fully coherent llumnaton reult derved by Goodman [9]. 5.. Angular pectral denty radu An ndvdual can alo formulate a cloed-form expreon for the angular D radu. In general, the angular D radu provde a gauge for the ze of the average power dtrbuton oberved n the far feld. Th a quantty of mportance when dealng wth drected-energy and remote-enng applcaton. The analy, yet agan, hghly analogou to that performed for the D calar-equvalent oluton [, ]. 77

93 A wa the cae for the angular DoC radu, the ummaton n Eq. (7 doe not allow for a cloed-form expreon for the angular D radu for mooth-to-moderately rough urface. Furthermore, one mut lmt the analy to near normal ncdence, o ˆ ˆ = co. Aumng that Eq. (69 hold, o that the ncdent llumnaton that yu ( θ ft on the rough urface, the exponental term on the econd and thrd lne of Eq. (78 predomnantly determne the angular extent of the cattered D for very rough urface. Provded -pol llumnaton, only Ψ xx ext wthn the analy. In addton, for an n-plane catterng geometry and a ngle obervaton pont,.e., r = r,, the obervaton projecton mplfy, uch that ( ( ϑ = ϑ, = n θ co φ = (93 x and ϑ, n ( n ( n ( y ϑy θ φ θ x = = =. (94 Conequently, the followng power-dtrbuton exponental β reult from Eq. (78 for Ψ xx : kr h β = exp y y mnmn y ( mn r h n ( θ Upon ettng β equal to e, the followng expreon reult: nce n ( θe co ( θe of co( θ e, where ( θ σ h ( θ. (95 ( α + kw xx e = Ω + + e + n co. (96 =, Eq. (96 manpulate nto a quadratc equaton n term 78

94 ( α + xx = ( + σh co ( θe + 4σh co( θe + ( Ω + σh +. (97 kw olvng th quadratc equaton (only the potve root make phycal ene, the angular D radu become θ ( ( + α = co + + σh Ω +. (98 xx e + σ h kw co + σ h For mot cae of nteret, the ource and urface term contaned wthn the radcal n Eq. (98 are neglgble. They only provde a mall correcton to the angular D radu due to the ource parameter. Thu, the angular D radu become a functon of only the urface parameter. Th hghly analogou to the reult obtaned by the D calarequvalent oluton [, ] Fully coherent llumnaton valdaton In order to valdate the angular rad developed above, the preent analy ue the Labphere Infragold coupon and the nomnal far-feld etup (decrbed above wth fully coherent llumnaton at normal ncdence, o that = w and θ =. Provded th etup, Fgure how a comparon between the 3D vector oluton, the D calar-equvalent oluton [, ], and a full-wave D method of moment (MoM oluton [39, 4]. The D MoM oluton obtaned the cattered feld from 4 ndependent rough urface realzaton mulated ung the method decrbed by Yura and Hanon [45] wth a Gauan-Gauan (G-G probablty dtrbuton functon xx 79

95 (PDF [cf. Eq. (38] 8. Note that the reult match up well and that the cloed-form expreon for the angular rad behave a predcted. For ntance, the dahed vertcal lne n Fgure a, whch repreent the angular DoC radu [cf. Eq. (9], dentfe the correct e locaton, wherea the dahed vertcal lne n Fgure b, whch repreent the angular D radu [cf. Eq. (98], come cloe to the e locaton. The analy performed n the next ub-ecton further explan th mall dagreement wth repect to the angular D radu; nonethele, the reult n Fgure help to valdate the 3D vector oluton aumng fully coherent llumnaton and very rough urface condton..8 3 = /.8 3 = / 7 ("3.6.4 N ( D MoM D ol. 3D ol "3 [mrad] (a [deg] (b Fgure. Comparon between a full-wave D method of moment (MoM oluton, the D calar-equvalent oluton, and the 3D vector oluton for fully coherent llumnaton at normal ncdence of a very rough conductng urface. (a how the magntude of the cattered pectral degree of coherence a a functon of the dfference between two polar angle, wherea (b how the normalzed cattered pectral denty a a functon of a ngle polar angle. 8. A full-wave 3D MoM oluton unrealzable at optcal wavelength wth the current fully coherent etup the computatonal amplng and memory requrement are far too great for modern dektop computer. 8

96 In order to valdate the 3D vector oluton for mooth-to-moderately rough urface condton, the analy aume the ame nomnal far-feld etup ued above but vare the urface heght tandard devaton of the Labphere Infragold coupon, o that σ =. λ,. λ, and.3λ. Fgure how the reult for th etup. Note that for h both the magntude of the cattered DoC (Fgure a and the normalzed cattered D (Fgure b, the reult devate for mall σ h between the 3D and D oluton. Th mot lkely due to fact that the D calar-equvalent oluton confned to a ngle plane, wherea the 3D vector oluton not. The reult of Hyde et al. howed excellent agreement between the D calar-equvalent oluton and a full-wave D MoM oluton ung mlar etup parameter []. Wth th ad, the D oluton do not capture all the phyc related to the 3D vector problem the next ub-ecton examne th pont further at normal and non-normal ncdence ung partally coherent llumnaton = / D 3D 7 (" = / N ( < h =.6 < h =.6 < h = "3 [mrad] (a [deg] (b Fgure. Comparon between the D calar-equvalent and 3D vector oluton for fully coherent llumnaton at normal ncdence of mooth-to-moderately rough conductng urface. (a how the magntude of the cattered pectral degree of coherence a a functon of the dfference between two polar angle. (b how the normalzed cattered pectral denty a a functon of a ngle polar angle. 8

97 5..4 Partally coherent llumnaton valdaton The preent analy vare the coherence of the ncdent llumnaton, o that α xx =,,.5, and.5 [cf. Eq. (7], where w =.9 mm and xx = α xxw. Thee value relate to a coherent ource and to a relatvely ncoherent ource, repectvely. In addton, the preent analy vare the urface roughne of the Labphere Infragold coupon, o that σ h =. rad,.5 rad,. rad, and.5 rad [cf. Eq. (9], where = 6.9 μm and σ = σ. Thee value relate wth mooth-to-very rough h h h h urface condton, repectvely. Fgure and Fgure 3 below how reult for the magntude of the cattered DoC and the normalzed cattered D for partally coherent llumnaton at normal ncdence,.e., θ =. Note the excellent agreement between the D and 3D oluton (mlar to that acheved for the fully coherent llumnaton analy. Fgure 4 how the reult for the normalzed cattered D for partally coherent llumnaton at non-normal ncdence,.e., θ = 4. Here, the reult devate between the D and 3D oluton a σ h ncreae. Th devaton determntc n practce. For - pol llumnaton and an n-plane catterng geometry, the D and 3D oluton have the ame functonal dependence n the exponental term whch predomnately drve the angular extent of the cattered D. The analy explored th functonal dependence above n the dervaton of the angular D radu. Wth that ad, there a mall dfference contaned n the ampltude term when comparng the D and 3D oluton. 8

98 .8, xx =, xx =, xx =.5, xx = = / < h =.5 rad 7 (" (" = / < h =. rad 3 4 "3 [mrad] (a. 3 4 "3 [mrad] (b.8 3 = / < h =. rad.8 3 = / < h =.5 rad 7 (" (" "3 [mrad] (c 3 4 "3 [mrad] (d Fgure. Comparon between the D oluton (crcle and 3D oluton (lne for partally coherent llumnaton at normal ncdence of mooth-to-very rough conductng urface. (a-(d how the magntude of the cattered pectral degree of coherence a a functon of the dfference between two polar angle for varyng ource parameter rato and urface lope tandard devaton. 83

99 3 = /, xx = 3 = /, xx =.8.8 N (3.6.4 N ( [deg] (a 3 = /, xx = [deg] (b < h =. rad < h =.5 rad < h =. rad < h =.5 rad 3 = /, xx = N (3.6.4 N ( [deg] (c [deg] (d Fgure 3. Comparon between the D oluton (crcle and 3D oluton (lne for partally coherent llumnaton at normal ncdence of mooth-to-very rough conductng urface. (a-(d how the normalzed cattered pectral denty a a functon of a ngle polar angle for varyng ource parameter rato and urface lope tandard devaton. 84

100 3 =4 /, xx = 3 =4 /, xx =.8.8 N (3.6.4 N ( [deg] (a 3 =4 /, xx =.5 < h =. rad < 3 [deg] h =.5 rad < (b h =. rad < h =.5 rad 3 =4 /, xx =.5.8 N (3.6.4 N ( [deg] (c [deg] (d Fgure 4. Comparon between the D oluton (crcle and 3D oluton (lne for partally coherent llumnaton at non-normal ncdence of mooth-to-very rough conductng urface. (a-(d how the normalzed cattered pectral denty a a functon of a ngle polar angle for varyng ource parameter rato and urface lope tandard devaton. 85

101 The 3D vector oluton contan a ampltude factor; wherea, the D x xx y xx calar-equvalent oluton contan only a y xx ampltude factor [cf. Eq. (7 and (79]. Thee ampltude factor (n addton to polarzaton term appear n front of the exponental term whch predomnantly drve the angular extent of the cattered D. Functonally, thee ampltude factor tend to puh the cattered D to the rght, wherea the polarzaton term tend to pull the cattered D to the left (for potve ncdent angle the oppote true for negatve ncdent angle. The addtonal ampltude factor contaned n the 3D oluton puhe the cattered D, o that the peak of the far-feld power dtrbuton alway algn wth the pecular drecton,.e., where θ = θ. Converely, Fgure 4 how that for the D oluton, a h σ ncreae the D peak hft more and more to the left and doe not algn wth the pecular drecton 9. Th due to the lack of the aforementoned ampltude factor. Before movng on n the analy, t mportant to note that thee ad ampltude factor are alo reponble for the dcrepancy een n the angular D radu (cf. the vertcal dahed lne n Fgure and Fgure 3. The angular D radu only come cloe to the e locaton becaue thee ampltude factor tend to puh out the wng of the far-feld power dtrbuton. Neverthele, th behavor determntc n nature and the angular D radu, a derved above, adequately characterze the behavor of the cattered D. 9. The full-wave D MoM oluton of Bau et al. how mlar behavor [39, 4]. 86

102 5. Comparon to a polarmetrc bdrectonal dtrbuton functon The analy preented here compare the 3D vector oluton to a polarmetrc bdrectonal dtrbuton functon (pbrdf developed by Pret and Meer [49]. In general, the pbrdf of Pret and Meer aume fully ncoherent llumnaton; thu, there no coherence nformaton contaned wthn the oluton. Intead, the pbrdf of Pret and Meer provde the Mueller matrx for tattcally rough urface that are characterzed by a G-G PDF [cf. Eq. (38]; a a reult, f the analy aume unpolarzed llumnaton, then the frt column of the Mueller matrx (gven by the pbrdf oluton become the cattered toke vector [3, 3]. Provded th cattered toke vector, the analy can then compare the normalzed cattered D and the cattered DoP between the pbrdf oluton and the 3D vector oluton. 5.. Normalzed pectral denty valdaton The preent analy ue the Labphere Infragold coupon and the nomnal farfeld etup (decrbed above. It alo aume unpolarzed llumnaton, o that Ax = Au and B = B =. Fgure 5 how reult for th etup wth partally coherent xu ux llumnaton at non-normal ncdence, where α = α =.5 and θ =. Note the exact agreement between the pbrdf and 3D oluton for the normalzed D wth varou catterng geometre. Alo note that one can obtan the normalzed cattered D from the pbrdf oluton by cone correctng the frt term of the cattered toke vector [34],.e., multplyng by co( θ xx uu, and dvdng by the max value. 87

103 .8 3 = /? =9 /.8 3 = /? =45 / N (3.6.4 N ( [deg] (a pbrdf 3D ol [deg] (b.8 3 = / 3 = /.8 3 = / 3 =! / N (?.6.4 N (? ? [deg] (c 3? [deg] (d Fgure 5. Comparon of the normalzed cattered pectral dente obtaned from a polarmetrc bdrectonal dtrbuton functon (pbrdf and the 3D vector oluton for unpolarzed llumnaton at non-normal ncdence and a very rough conductng urface. (a depct an n-plane catterng geometry, wherea (b depct an out-of-plane catterng geometry wth reult a a functon of a ngle polar angle. Converely, (c and (d depct b-tatc catterng geometre a a functon of a ngle azmuth angle. Note that the mnmum occur at the mono-tatc obervaton pont n both (c and (d. 88

104 5.. Degree of polarzaton valdaton The ntal analy preented here agan ue the Labphere Infragold coupon and the nomnal far-feld etup (decrbed above. It alo aume unpolarzed llumnaton throughout, o that Ax = Au and Bxu = Bux =. Fgure 6 how reult for partally coherent llumnaton at non-normal ncdence, where agan α = α =.5 and θ = Note the exact agreement between the pbrdf and 3D oluton for the cattered DoP wth varou catterng geometre. r Now the analy aume a tactcal engagement cenaro, o that λ =.64 μm, = r, = km, w =.54 cm, xx = uu =.5w, and θ = Th etup correpond wth partally coherent llumnaton of NKB7 gla, where n =.57 [43], at Brewter angle. The reult preented n Fgure 7a how exact agreement between the pbrdf and 3D oluton for cattered DoP aumng very rough urface condton, where h = λ and σh = λ. Fgure 7b then how reult for both very rough and mooth-to-moderately rough urface condton ung only the 3D vector oluton, where agan h = λ but σh = λ and.λ. It mportant to note that the cattered DoP only ext where lght ext n the analy. Th make ene conderng that, by defnton, the cattered DoP depend on the cattered D [cf. Eq. (8 and (8]. Wth that ad, Fgure 7b how that the cattered DoP doe not depend on urface roughne, at leat for the aumpton ued wthn the analy [cf. Footnote, p.55]. It alo doe not depend on coherence, at leat for unpolarzed ncdent llumnaton th content wth the example gven by Wolf for otropc beam parameter [3],.e., when xx = uu (Appendx E how an example where th not the cae. xx uu. 89

105 8 6 3 = /? =9 / = /? =45 / P (3 #!3 4 P (3 #! [deg] (a pbrdf 3D ol [deg] (b.8 3 = / 3 = /.8 3 = / 3 =! / P (? #!3.6.4 P (? #! ? [deg] (c 3? [deg] (d Fgure 6. Comparon of the cattered degree of polarzaton obtaned from a polarmetrc bdrectonal dtrbuton functon (pbrdf and the 3D vector oluton for unpolarzed llumnaton at non-normal ncdence and a very rough conductng urface. (a depct an n-plane catterng geometry, wherea (b depct an out of plane catterng geometry wth reult a a functon of a ngle polar angle. Converely, (c and (d depct b-tatc catterng geometre a a functon of a ngle azmuth angle. Note that the mnmum occur at the mono-tatc obervaton pont n both (c and (d. 9

106 .8 pbrdf 3D ol..8 P ( =56.43 / P ( =56.43 / [deg] (a. < h =6 < h = [deg] (b Fgure 7. Comparon of the cattered degree of polarzaton for unpolarzed llumnaton at Brewter angle of a delectrc urface wth varyng roughne condton. (a how the reult obtaned from a polarmetrc bdrectonal dtrbuton functon (pbrdf and the 3D vector oluton for non-normal ncdence at Brewter angle wth very rough urface condton. (b how the reult from the 3D vector oluton for both very rough and mooth-to-moderately rough urface condton. 5.3 Comparon to emprcal meaurement To compare the 3D vector oluton wth emprcal meaurement, the analy ue reult from the Complete Angle catter Intrument (CAI at the Ar Force Inttute of Technology [46]. Fgure 8 decrbe the catterng geometry aocated wth the CAI for both n-plane meaurement (Fgure 8a and out-of-plane meaurement (Fgure 8b. Provded Fgure 8, the analy ue the followng angle tranformaton, developed by Germer and Amal [48], to relate the CAI catterng geometry to that of the 3D vector oluton (cf. Fgure 9: ( ( θ = co co α co β, (99. Goldten readly decrbe the prncple behnd dual-rotatng-retarder polarmety [47, p. 357]. 9

107 ( ( ( ( ( ( θ = co co β n α n δ + co α co β co δ, ( 3π = +. ( and tan (, tan φ b a ( b, a In Eq. (, tan ( ba, return the nvere tangent of b a after takng nto account the quadrant of the pont ( ab,. Wth th n mnd, ( ( ( ( ( b = n δ co γ n α n β + co n γ + co co co n n n ( δ ( α ( γ ( β ( α ( γ ( ( ( ( ( ( ( δ ( α ( γ ( α ( β ( γ a = co δ co γ n α co α n β n γ + n co co n n n ( ( ( ( (, (, (3 b = co α co γ n β n α n γ, (4 and a co( γ n ( α co( α n ( β n ( γ =. (5 Provded Eq. (99-(5, the followng analy ue the Labphere Infragold coupon and the nomnal far-feld etup (decrbed above. Thee etup parameter bet match thoe ued by the CAI. It alo aume unpolarzed partally coherent llumnaton at non-normal ncdence, o that Ax = Au, B = B =, αxx = αuu =.5, and θ = 5.3. In-plane meaurement xu ux For n-plane meaurement, β = γ =. A a reult, the angle tranformaton. gven above mplfy, o that α = θ = and δ = θ. Fgure 9 how reult for the n-plane comparon tudy. Note that the reult for the CAI meaurement are hghly ocllatory for both the normalzed cattered D and the cattered DoP. Th mot lkely due to the fact the meaurement contan peckle. pnnng the ample would 9

108 average out th peckle; however, the current CAI etup doe not allow for contnuou pnnng of the ample. In addton, the CAI meaurement, wth repect to the normalzed cattered D, how that the Labphere Infragold coupon doe not mantan a G-G model for the underlyng urface tattc. Intead, a tretched exponentaltretched exponental (E-E model better characterze the Labphere Infragold coupon [38]. Th een by comparng the reult to a full-wave D MoM oluton [39, 4]. Here, the analy obtaned the cattered feld from 4 ndependent rough urface realzaton mulated ung the method decrbed by Yura and Hanon [45] wth both G-G and E-E PDF [cf. Eq. (38]. Before movng on n the analy, t mportant to note that the CAI meaurement do not contan enough fdelty, wth repect to the cattered DoP, to compare them to thoe obtaned by the 3D vector oluton Out-of-plane meaurement For the out-of-plane meaurement, the CAI ued the followng etup parameter: α = and β = 5. Accordngly, one can ue the angle tranformaton gven above n Eq. (99-(5 to relate the 3D vector oluton to the CAI meaurement. Fgure how reult for the out-of-plane comparon tudy. Note agan that the CAI meaurement, wth repect to the normalzed cattered D, how that the Labphere Infragold coupon doe not mantan a G-G model for the underlyng urface tattc. Alo note that the CAI meaurement do not contan enough fdelty, wth repect to the cattered DoP, to compare them to thoe obtaned by the 3D vector oluton. 93

109 Detector cattered Beam Coupon Incdent beam (a Incdent Beam β γ Detector α δ (b Fgure 8. Decrpton of the Complete Angle catter Intrument at the Ar Force Inttute of Technology. (a how the catterng geometry ued to collect n-plane meaurement [4], wherea (b how the catterng geometry ued to collect outof-plane meaurement [48]. 94