﻿round 
  the 
  Focus 
  of 
  a 
  Lens, 
  at 
  Various 
  Apertures. 
  47 
  

  

  to 
  the 
  centre 
  or 
  the 
  focus, 
  and 
  to 
  the 
  usual 
  Oornu 
  spiral 
  ; 
  

   that 
  curve 
  is 
  well-known 
  from 
  the 
  distribution 
  of 
  light 
  

   outside 
  the 
  shadow 
  of 
  a 
  semiplane 
  ; 
  here 
  its 
  physical 
  

   meaning 
  is 
  different 
  : 
  it 
  represents 
  the 
  central 
  intensity 
  

   as 
  function 
  of 
  (the 
  area 
  of) 
  the 
  aperture. 
  It 
  has 
  its 
  first, 
  

   and 
  highest, 
  maximum 
  at 
  about 
  = 
  1*23, 
  as 
  already 
  men- 
  

   tioned. 
  A 
  number 
  of 
  isolated 
  points 
  below 
  and 
  near 
  this 
  

  

  curve 
  represent 
  the 
  intensity 
  at 
  the 
  circle 
  a! 
  = 
  . 
  — 
  =. 
  (i.e. 
  

  

  p=\, 
  for 
  the 
  said 
  example). 
  These 
  points 
  are 
  too 
  near 
  the 
  

   first 
  curve 
  to 
  be 
  joined 
  into 
  a 
  continuous 
  line. 
  The 
  next 
  

  

  curve, 
  T 
  10], 
  corresponds 
  to 
  «'= 
  — 
  ' 
  (i.e. 
  p 
  = 
  10\ 
  for 
  the 
  

  

  above 
  example) 
  ; 
  its 
  first, 
  as 
  well 
  as 
  its 
  second, 
  maximum 
  is 
  

   very 
  flat. 
  Its 
  interesting 
  feature 
  is 
  that 
  it 
  nearly 
  reaches 
  

   the 
  focal 
  curve 
  at 
  about 
  v 
  = 
  l*9. 
  The 
  moral 
  is 
  obvious 
  : 
  at 
  

   = 
  1*23 
  the 
  definition 
  as 
  well 
  as 
  the 
  light 
  intensity 
  are 
  

   excellent, 
  at 
  = 
  1*9 
  very 
  bad 
  ; 
  at 
  about 
  v 
  = 
  '2 
  3 
  much 
  better 
  

  

  40tt 
  

   again. 
  The 
  next 
  curve, 
  [20], 
  corresponds 
  to 
  a! 
  = 
  ./tttt^, 
  

  

  and 
  the 
  lowest, 
  T301, 
  to 
  «'= 
  , 
  — 
  . 
  

   L 
  ' 
  1/1000 
  

  

  To 
  form 
  an 
  opinion 
  about 
  the 
  intensity 
  as 
  function 
  of 
  

   distance 
  from 
  the 
  focus, 
  at 
  any 
  fixed 
  aperture 
  v<3*0, 
  it 
  is 
  

   enough 
  to 
  read 
  the 
  above 
  numerical 
  table 
  in 
  horizontal 
  rows 
  

   (instead 
  of 
  columns) 
  . 
  The 
  corresponding 
  curves 
  would 
  again 
  

   corroborate 
  the 
  above 
  remark, 
  viz. 
  that 
  the 
  greatest 
  central 
  

   intensity 
  and 
  the 
  best 
  definition 
  of 
  the 
  image 
  produced 
  by 
  a 
  

   plano-convex 
  lens, 
  of 
  refractive 
  index 
  n 
  and 
  curvature 
  

   radius 
  r, 
  are 
  obtained 
  at 
  an 
  aperture 
  corresponding 
  to 
  about 
  

   oa 
  1*23, 
  and 
  to 
  be 
  determined 
  for 
  any 
  concrete 
  case 
  by 
  (25). 
  

   If, 
  for 
  instance, 
  w=l*5 
  and 
  \ 
  = 
  J 
  micron, 
  then 
  the 
  best 
  

   relative 
  aperture 
  is, 
  in 
  round 
  figures, 
  

  

  for 
  r 
  = 
  5*56 
  cm. 
  2*78 
  cm. 
  6*9 
  mm. 
  1 
  mm. 
  

   h/r= 
  0-035 
  -041 
  -057 
  -090 
  

  

  The 
  last 
  of 
  these 
  would 
  be 
  a 
  very 
  minute 
  lens 
  indeed. 
  

   Practically, 
  not 
  going 
  much 
  below 
  r 
  = 
  3 
  cm., 
  the 
  best 
  

   relative 
  aperture 
  will 
  not 
  exceed 
  ^. 
  Opening 
  the 
  lens 
  up 
  

   to 
  0=1*9 
  would 
  spoil 
  the 
  central 
  intensity 
  and 
  the 
  definition 
  

   considerably. 
  The 
  next 
  best 
  aperture, 
  after 
  0=1*23, 
  would 
  

   correspond 
  to 
  about 
  0=2*3 
  : 
  the 
  next 
  favourable 
  opportunity 
  

   will 
  lie 
  a 
  little 
  beyond 
  0=3. 
  The 
  marginal 
  phase 
  retardation 
  

  

  is, 
  for 
  the 
  best 
  aperture 
  0=1*23, 
  i? 
  = 
  l'51 
  9 
  or 
  a 
  little 
  over 
  

   3/8 
  of 
  a 
  period. 
  

  

  