﻿48 
  Light 
  Distribution 
  round 
  the 
  Focus 
  of 
  a 
  Lens. 
  

  

  Returning 
  once 
  more 
  to 
  the 
  table 
  on 
  p. 
  45, 
  notice 
  that 
  the 
  

   v-values 
  arrayed 
  in 
  the 
  first 
  column 
  have 
  themselves 
  a 
  simple 
  

   meaning, 
  their 
  squares 
  being 
  proportional 
  to 
  the 
  " 
  normal 
  " 
  

   central 
  intensity 
  N 
  which 
  would 
  correspond 
  to 
  a 
  perfect 
  

   wave. 
  Thus 
  the 
  relative 
  central 
  intensity, 
  which 
  is 
  the 
  

   squared 
  ratio 
  of 
  chord 
  and 
  arc 
  of 
  the 
  Cornu 
  spiral, 
  is 
  

   given 
  by 
  

  

  I 
  :N 
  =(V+B 
  1 
  ')^, 
  

  

  i. 
  e. 
  simply 
  by 
  the 
  figures 
  of 
  the 
  second 
  divided 
  by 
  the 
  squares 
  

  

  247 
  

   of 
  those 
  of 
  the 
  first 
  column. 
  Thus, 
  for 
  v 
  = 
  0'5, 
  I 
  : 
  N 
  =^^., 
  

  

  and 
  for 
  v=l, 
  or 
  rj 
  equal 
  to 
  a 
  quarter-period, 
  I 
  :N 
  ='8004, 
  

   agreeing 
  with 
  Lord 
  Rayleigh's 
  result 
  of 
  1879 
  *, 
  and 
  so 
  on. 
  

   The 
  relative 
  intensity 
  at 
  the 
  focus 
  decreases 
  steadily, 
  of 
  

   course 
  ; 
  the 
  absolute 
  intensity, 
  due 
  to 
  the 
  defective 
  wave, 
  

   increases 
  only 
  up 
  to 
  a 
  certain 
  maximum 
  and 
  oscillates 
  in 
  

   decreasing 
  amplitudes 
  round 
  a 
  limit 
  corresponding 
  to 
  the 
  

   asymptotic 
  value 
  A 
  2 
  + 
  B 
  2 
  = 
  i. 
  The 
  central 
  intensity 
  by 
  

   itself 
  cannot, 
  of 
  course, 
  inform 
  us 
  about 
  the 
  " 
  definition." 
  

   It 
  so 
  happens 
  however 
  that, 
  in 
  the 
  case 
  of 
  the 
  plano-convex 
  

   lens 
  at 
  least, 
  the 
  best 
  aperture, 
  v==l*23, 
  for 
  the 
  focal 
  absolute 
  

   intensity 
  is 
  also 
  sensibly 
  the 
  best 
  for 
  the 
  definition 
  of 
  the 
  

   image. 
  

  

  So 
  much 
  about 
  the 
  distribution 
  of 
  intensity 
  in 
  the 
  focal 
  

   plane 
  of 
  the 
  lens. 
  The 
  intensity 
  at 
  points 
  outside 
  the 
  focal 
  

   plane 
  will 
  require 
  but 
  a 
  few 
  remarks, 
  owing 
  to 
  the 
  relative 
  

   smallness 
  of 
  the 
  axial 
  intensity 
  gradient 
  (for 
  small 
  6) 
  already 
  

   hinted 
  at. 
  Consider, 
  for 
  instance, 
  the 
  points 
  of 
  the 
  optical 
  

   ads 
  itself, 
  that 
  is, 
  a 
  / 
  = 
  a 
  = 
  0. 
  Then, 
  according 
  to 
  (22), 
  the 
  

  

  arc 
  element 
  is 
  ^ 
  = 
  (^l) 
  dv, 
  precisely 
  as 
  for 
  the 
  focal 
  or 
  

  

  Cornu-spiral 
  itself, 
  and, 
  the 
  sloping 
  angle 
  and 
  the 
  curvature 
  

   of 
  a 
  /3-curve, 
  

  

  2 
  / 
  \ 
  J 
  /2 
  

  

  £ 
  = 
  T" 
  +/9 
  © 
  "' 
  k 
  = 
  WM 
  +- 
  8 
  > 
  ■ 
  ■ 
  ( 
  26 
  > 
  

  

  where 
  /3 
  = 
  27ro-/\ 
  measures 
  the 
  axial 
  coordinate 
  of 
  the 
  point 
  

   in 
  question. 
  Take, 
  for 
  instance, 
  our 
  previous 
  example 
  

  

  71=1-5, 
  r/\= 
  ilO 
  6 
  . 
  Then, 
  by 
  (19), 
  

  

  * 
  Phil. 
  Mag. 
  viii. 
  p. 
  409. 
  Rayleigh's 
  figure, 
  ! 
  8003, 
  differs 
  hut 
  

   insignificantly 
  from 
  the 
  above 
  one. 
  

  

  