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XII.    On  the  Diffraction  Theory  of  Microscopic  Vision. 
By  Albert  B.  Porter  *. 
1.  A   LTHOUGH  thirty-two  years   have   passed    since 
JljL     Professor  Ernst  Abbe  f  proposed  his  diffraction 
theory  of  microscopic  vision,  it  is  still  to  some  extent  a  matter 
of  controversy  among  microscopists  J,  and  is  perhaps  less 
familiar  to  physicists  than  its  importance  warrants.  The  theory 
may  be  briefly  stated  in  the  following  form.  If  a  lens  is  to 
produce  a  truthful  image  of  an  illuminated  object,  it  must 
have  an  aperture  sufficient  to  transmit  the  whole  of  the 
diffraction  pattern  produced  by  the  object  ;  if  but  part  of  this 
diffraction  pattern  is  transmitted,  the  image  will  not  truthfully 
represent  the  object,  but  will  correspond  to  another  (virtual) 
object  whose  whole  diffraction  pattern  is  identical  with  that 
portion  which  passes  through  the  lens  ;  if  the  structure  of  the 
object  is  so  fine,  or  if  the  aperture  of  the  lens  is  so  narrow, 
that  no  part  of  the  diffraction  pattern  due  to  the  structure  is 
transmitted  by  the  lens,  then  the  structure  will  be  invisible 
no  matter  what  magnification  is  used.  Abbe  and  others  have 
devised  a  number  of  interesting  experiments  §  to  illustrate 
the  theory,  but  the  complete  mathematical  development  has 
never  been  published  ||. 
2.  The  particular  case  in  which  the  object  is  a  transmission 
grating  consisting  of  alternate  opaque  and  transparent  lines 
may,  however,  be  treated  by  means  of  a  simple  application 
of  Fourier's  theorem.  Let  a  and  b  be  respectively  the  widths 
of  the  transparent  and  opaque  lines  on  the  gratings^  and  let 
A  be  the  amplitude  of  the  (monochromatic)  light  which  will 
be  assumed  to  fall  upon  the  grating  at  perpendicular  incidence  ; 
then  the  distribution  of  amplitude  in  the  light  passing  through 
the  grating  will  be  as  shown  in  tig.  1.  The  function  repre- 
sented by  this  curve  may  be  developed  in  a  cosine  series  by 
means  of  the  formula 
n    s      t  7       7          irx      ,          2ttx                    7           rairx   , 
J\x)  —  i^o  +  <>i  cos  — ■  +  be,  cos -f  .  .  .  4  bm  cos  — ; !-.... 
C  C  (> 
in  which  bm=-  \     fix)  cos  — —  dx. 
C  Jo*  c 
*  Read  before  the  American  Physical  Society,  April  22,  1905.  Com- 
municated by  the  Author. 
f  Archivfilr  mikroskopische  Anatomie,  ix.  pp.  413-403  (1887)  :  Gescnn- 
melte  Abhandlungen,  i.  pp.  4o-100  (1904). 
%  Gage  on  The  Microscope,  9t-.li  edition,  p.  21  (1904). 
§  Mueller-Pouillet's  Lehrbuch  der  Physik,  9th  edition,  II,  i.  p.  712  \ 
Lewis  Wright's  '  Light,'  2nd  edition,  p.  198. 
|i  Mueller-Pouillet,  id.,  p.  703. 
