INSIGHT  INTO  RADIATION  FORCES  ON  SPHERES  FROM  PARTIAL  WAVE  PHASE  SHIFTS   Philip  L.  MARSTON1,*  and  Likun  ZHANG  2   1

 Physics  and  Astronomy  Dept.,  Washington  State  University,  Pullman,  Washington,  99164-­‐‑2814,  USA     2  Physics  and  Astronomy  Dept.,  University  of  Mississippi,  University,  Mississippi  38677,  USA   *Corresponding  author:  [email protected]    

  Abstract   Though   to   some   extent   partial   wave   phase   shift   methods  have  been  neglected  in  classical  scattering  theory   for   a   few   decades,   recent   publications   reveal   their   utility   for   understanding   acoustic   radiation   forces   on   spheres   in   standing  waves  and  in  beams.  This  approach  incorporates   the  spherical  symmetry  and  it  efficiently  parameterizes  the   scattering   and   associated   radiation   forces.   “Designer   spheres”  become  parameterized  to  reach  a  given  objective.  

1 Historical  Introduction  

By   the   1930s   it   was   common   to   formulate   quantum   mechanical   scattering   theory   using   partial-­‐‑wave   phase   shifts   [1].     By   1946   Van   de   Hulst   [2]   realized   that   the   analogous   formulation   of   electric   and   magnetic   partial   waves   was   helpful   for   understanding   electromagnetic   scattering   by   spheres   [2].     See   also   [3].     By   the   late   1950s   some  relevant  expressions  had  been  published  concerning   acoustic   radiation   forces   on   spheres   in   plane   traveling   waves   [4,5],   though   it   has   recently   been   found   [6]   that   an   aspect  of  those  results  was  incorrect.    The  acoustic  case  is   especially   straightforward   to   illustrate   and   it   has   been   recently   found   that   related   radiation   force   properties   for   spheres   in   traveling   waves,   beams,   and   standing   waves   can   be   expressed   using   partial   wave   phase   shifts.     This   approach   is   especially   valuable   for   gaining   insight   into   situations   where   only   a   few   partial   waves   are   involved.     The   traveling   wave   case   is   partially   analogous   to   the   expression   of   quantum   mechanical   momentum   transfer   cross  sections  [6].   1.1 Phase  Shifts  and  Scattering  Amplitudes   For  several  decades  it  has  been  common  to  express  the   far-­‐‑field  complex  scattering  by  spheres  using  the  exp(-­‐‑iωt)   time  convention  and  the  dimensionless  form  function  [6,7]       ! f(θ) = (–i /ka) !!!   (2n + 1) (sn – 1) Pn(cosθ) ,   (1)     where   Pn   is   a   Legendre   polynomial,   θ   is   the   scattering   angle,   and   the   complex   functions   sn(ka)   depend   on   material  properties,  k  =  ω/c,  a  is  the  sphere  radius,  and  c  is   the  speed  of  sound  in  the  outer  media.    For  the  usual  case   of   passive   spheres   it   may   be   shown   that   |sn|   <   1   [8].     Though   the   general   case   of   momentum   transfer   allowing   for  energy  absorption  has  been  formulated  [6],  for  brevity   in   the   present   summary   absorption   will   be   neglected   so   that  |sn|  =  1.    In  that  situation  each  sn  is  characterized  by  a  

real  phase  shift  δn  such  that  sn  =  exp(i2δn)  where  the  δn  are   functions  of  ka  and  of  material  properties.   1.2 Radiation  Forces  and  Wave-­‐‑Field  Parameters   When  expressing  axial  radiation  forces  Fz  on  spheres  in   progressive   waves,   it   is   conventional   to   introduce   a   dimensionless   force   function   YP(ka)   such   that   Fz   =   πa2(I0/c)YP  [8,9].    For  standing  waves  the  proportionality  Fz   ∼  πa2YS  sin(2kzh)  is  used  where  kz  is  the  axial  wave  number   (which   may   differ   from   k   in   a   standing   wave   of   superposed  beams)  and  the  parameter  h  is  such  that  z  =  -­‐‑h   corresponds   to   a   pressure   antinode   of   the   incident   standing  wave  with  z  =  0  being  the  centre  of  the  sphere  [9].     Various  relations  between  YP  and  YS  and  the  δn  have  been   recently  derived  which  also  involve  wave-­‐‑field  parameters   [6,  9].   1.3 Spheres  in  Bessel  Beams  and  Bessel  Standing  Waves   For   these   cases   it   is   convenient   to   introduce   the   wave   field  parameter  β  such  that  kz  =  k  cosβ.    The  wave-­‐‑field  is   axisymmetric  and  the  sphere  is  on  the  symmetry  axis.  For   a  given  type  of  sphere  it  is  appropriate  to  evaluate  YP  and   YS   in   the   (ka,   β)   plane   in   three-­‐‑dimensional   plots   [9].     For   example,  negative  radiation  force  “islands”  in  YP  plotted  in   the  (ka,  β)  plane  were  originally  discovered  in  that  way  [7].     By   expressing   YP   and   YS   using   the   phase   shifts   δn,   novel   conditions   on   the   δn   have   been   found   for   which   the   extrapolated  contours  having  YP  =  0  or  YS  =  0  vanish  on  the   β  axis  [9].  For  finite  ka  the  direction  of  the  radiation  force   changes  when  β  crosses  such  a  contour.  In  certain  cases  it   is   possible   to   express   the   intercept   for   such   contours   directly   in   terms   of   the   ratio   of   the   monopole   and   dipole   phase   shifts   δ0/δ1   when   ka   is   small   [9].   Related   contours   have   been   investigated   for   average   forces   associated   with   quantum  mechanical  and  optical  scattering  [10].    

2 Forces  on  Spheres  in  Plane  Progressive  Waves  

         The   phase   shift   approach   may   also   be   used   to   advantage   even   in   the   simplest   case   of   plane   progressive   waves  in  which  case  from  Eq.  (16)  of  [6],  YP  reduces  to     !                  YP  =  (2/ka)2 !!!  (n+1)  sin2(δn  -­‐‑  δn+1)  ,     (2)     which   is   manifestly   non-­‐‑negative.   When   dissipation   is   included  using  the  most  elementary  methods  of  modifying   the   needed   expressions,   there   are   additional   positive   contributions  to  YP  [6,8].    

 

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3 Forces  on  Spheres  in  Plane  Standing  Waves  

The   phase   shift   approach   may   also   be   used   to   advantage  in  the  case  of  plane  standing  waves.  In  that  case   from   Eq.   (28b)   of   [6]   and   Eq.   (18)   of   [9],   in   the   plane   standing  wave  limit,  YS  reduces  to:     ! YS  =  -­‐‑(2/ka)2 !!!  (n+1)(-­‐‑1)n  sin(2δn  -­‐‑  2δn+1) ,     (3)     a   result   differing   in   several   ways   from   the   better-­‐‑known   progressive   wave   case,   Eq.   (2).   When   ka   is   small,   the   δn   may  be  expanded  in  a  series  involving  powers  of  ka  such   that   the   coefficients   anj   directly   involve   the   material   properties  of  the  sphere  and  the  surrounding  fluid  [9].  By   truncating  that  expansion  while  retaining  terms  involving   δ0,  δ1,  and  a  single  term  for  δ2,  a  simple  approximation  for   the   lowest   ka   root   of   YS(ka)   =   0   has   been   derived;   see   Eq.   (A2)   of   [9].   In   the   discussion   which   follows   it   will   be   convenient   denote   the   corresponding   approximation   for   the  ka  value  of  that  root  as  x0.  From  that  result  it  has  been   possible  to  derive  improved  low  ka  approximations  for  Ys   for   fluid   and   solid   spheres   [11,12]   having   the   following   form:         YSLFc(ka) = YSLF(ka) [1 − (ka/x0)2] , (4)     where  YSLF(ka),  published  in  1955  [13], is  the  standard  low   frequency  approximation  which  is  linear  ka.  (YSLF  is  based   only   on   Rayleigh   scattering   and   as   reviewed   in   [9]   has   been  widely  used  in  the  design  of  acoustical  manipulation   devices.)   Various   comparisons   with   calculations   of   YS(ka) based  on  well-­‐‑known  exact  formulations  show  that   for   situations   where   the   term   (ka/x0)2 is   significant,   the   YSLFc   approximation   in   Eq.   (4)   improves   on   the   standard   approximation   YSLF   [11,12].   This   is   done   in   a   way   that   directly   reveals   the   significant   of   the   material   parameters   through  the  dependence  of  x0  on  the  anj  [9,11,12].            For  both  the  plane  progressive  wave  and  plane  standing   wave   situations,   Eq.   (2)   and   (3),   the   phase-­‐‑shift   approach   also   recovers   various   specialized   approximations   at   small   ka.  For  example,  the  density  dependences  of  King’s  rigid-­‐‑ movable-­‐‑sphere  force  expressions  [14]  are  recovered  [12].  

4 Applications  and  Relevance  to  Optical  Forces  

The   examples   reviewed   here   are   sufficient   to   illustrate   the   utility   of   phase   shift   based   approaches   when   considering   the   relationship   between   radiation   forces   and   scattering.    Since  this  approach  incorporates  the  symmetry   and  parameterizes  the  radiation  forces  (see  for  example  Eq.   (2))  it  gives  an  alternative  to  searching  material  parameter   space   for   achieving   some   desired   outcome.   See   Fig.   1.   In   the   phase   shift   approach,   the   phase-­‐‑shift   space   would   be   searched  (for  classes  of  wave-­‐‑fields  of  interest)  to  discover   if   a   particular   outcome   is   ever   achievable.   If   a   relevant   successful  region  in  phase  shift-­‐‑space  can  be  determined,  it   becomes  a  separate  issue  as  to  how  to  achieve  the  desired  

set  of  phase  shifts  for  a  given  ka.  A  sphere  selected  in  this   way  may  be  described  as  a  “designer  sphere.”   Since  it  has  been  known  for  over  25  years  that  acoustical   radiation   force   expressions   for   right   circular   cylinders   [15,16]   have   analogous   forms   for   the   dependence   on   the   partial  wave  coefficients  (sn – 1)  with  corresponding  sphere   cases,   similar   applications   to   cylinders   (with   minor   modifications)  are  anticipated.   Historically   there   has   been   much   to   be   gained   by   comparing   phenomena   and   analytical   results   of   optical   and  acoustical  scattering  [17,18,19,20].     In   light   scattering   it   appears   advantageous   to   combine   phase-­‐‑shift   based   approaches   along   with   the   utility   of   scattering   and   radiation-­‐‑force   formulations   based   on   Generalized  Lorenz-­‐‑Mie  Theories  (GLMT)  [21].  The  phase   shift   and   “designer   sphere”   approach   of   Fig.   1   may   be   used.   There   are   separate   phase   shifts   corresponding   to   electric  and  magnetic  multipole  scattering  contributions.  A   preliminary   investigation   by   one   of   us   (P.   L.   M.)   of   the   optical  plane  progressive  wave  case  suggests  however  that   in   addition   to   terms   partially   analogous   to   Eq.   (2),   proportional   to   sin2(δn   -­‐‑   δn+1)   with   electric   and   magnetic   multipole   phase   shifts   considered   separately,   there   will   also   be   contributions   to   the   radiation   force   from   terms   jointly  involving  electric  and  magnetic  phase  shifts.  

5 Acknowledgement  

P.  L.  M.  acknowledges  the  support  of  the  U.  S.  Office  of   Naval   Research   under   award   number   N000141512603.   L.   Z.  acknowledges  the  support  of  the  2013–2014  F.  V.  Hunt   Postdoctoral  Research  Fellowship  in  Acoustics.      

  Figure  1  Approach  based  on  partial-­‐‑wave  phase  shifts  for   achieving  a  given  scattering  or  radiation  force  objective  for   sufficiently  symmetric  objects.    

6 References   [1] N.   F.   Mott   and   H.   S.   W.   Massey,   The   Theory   of   Atomic   Collisions   (University  Press,  Oxford,  1965).   [2] Van   de   Hulst,   H.C.,   Optics   of   spherical   particles,   Recherches   Astronomiques  de  l'ʹObservatoire  d'ʹUtrecht,  11:  1-­‐‑87  (1946).   [3] H.  C.  Van  de  Hulst,  Light  Scattering  by  Small  Particles  (Wiley,  New   York,  1957).  

 

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