﻿round 
  the 
  Focus 
  of 
  a 
  Lens, 
  at 
  Various 
  Apertures. 
  35 
  

  

  cr 
  = 
  0, 
  or 
  (3 
  = 
  0, 
  for 
  cophasal 
  vibrations 
  (77 
  = 
  0), 
  and 
  for 
  

   small 
  6 
  1, 
  the 
  integral 
  (6) 
  reduces 
  to 
  the 
  familiar 
  form 
  

  

  w 
  = 
  (j 
  (*6)6dO 
  = 
  ^J 
  1 
  {ad 
  1 
  ), 
  

   giving 
  for 
  the 
  intensity 
  the 
  well-known 
  expression 
  

  

  Formula? 
  (5) 
  and 
  (6) 
  are 
  valid 
  for 
  any 
  angular 
  aperture 
  

  

  2 
  < 
  », 
  and 
  for 
  any 
  given 
  axially 
  symmetrical 
  phase 
  dis- 
  

  

  tribution 
  77 
  = 
  77 
  (6). 
  The 
  axial 
  displacement 
  (of 
  P) 
  from 
  the 
  

   centre 
  enters 
  only 
  through 
  the 
  factor 
  e^ 
  eose 
  ; 
  the 
  transversal 
  

   displacement 
  a 
  enters 
  through 
  the 
  zeroth 
  Bessel 
  function, 
  

   and 
  the 
  phase 
  heterogeneity 
  through 
  e 
  ir 
  >. 
  Notice 
  in 
  passing 
  

   that, 
  by 
  (6), 
  a 
  phase 
  distribution 
  of 
  the 
  type 
  77 
  =# 
  . 
  cos#* 
  

   is 
  equivalent 
  to 
  a 
  rigid 
  shift 
  of 
  the 
  whole 
  image 
  (luminous 
  

  

  region) 
  along 
  the 
  axis 
  by 
  — 
  ~ 
  and 
  is, 
  therefore, 
  unessential. 
  

  

  In 
  the 
  case 
  of 
  a 
  wave 
  issuing 
  from 
  a 
  lens, 
  with 
  the 
  centre 
  

   placed 
  in 
  its 
  focus, 
  and 
  NO 
  along 
  its 
  optical 
  axis, 
  the 
  series 
  

   development 
  of 
  77 
  does 
  not 
  at 
  all 
  contain 
  such 
  a 
  term, 
  i. 
  e., 
  

   practically, 
  no 
  term 
  in 
  6 
  2 
  . 
  In 
  fact, 
  it 
  will 
  be 
  seen 
  that, 
  with 
  

   the 
  above 
  choice 
  of 
  the 
  reference 
  sphere, 
  the 
  series 
  for 
  77 
  

   starts, 
  for 
  any 
  "uncorrected"" 
  lens 
  (such 
  as 
  the 
  simple 
  plano- 
  

   convex 
  lens), 
  with 
  # 
  4 
  , 
  the 
  next 
  term 
  being 
  in 
  6*. 
  

  

  In 
  all 
  practical 
  cases, 
  connected 
  with 
  lenses, 
  the 
  angular 
  

   semiaperture 
  X 
  hardly 
  exceeds 
  4° 
  or 
  5°. 
  Under 
  these 
  

   circumstances 
  we 
  can 
  write 
  in 
  (6), 
  both 
  in 
  the 
  factor 
  

   of 
  J 
  and 
  in 
  J 
  itself, 
  sin 
  0=6, 
  and 
  in 
  the 
  exponential, 
  

   £cos0==/3 
  — 
  i/36 
  2 
  . 
  The 
  first 
  term, 
  /3, 
  giving 
  only 
  the 
  

   factor 
  e 
  ip 
  outside 
  the 
  integral, 
  does 
  not 
  influence 
  the 
  value 
  

   of 
  \w\ 
  2 
  and 
  can, 
  therefore, 
  be 
  rejected. 
  

  

  Thus, 
  introducing 
  the 
  new 
  variable 
  

  

  u 
  = 
  6\ 
  

   the 
  formula 
  for 
  small 
  Q 
  x 
  will 
  be 
  

  

  2w 
  = 
  \ 
  ««»-*«> 
  J 
  (* 
  y/u).du,- 
  .... 
  (7) 
  

  

  where 
  77 
  is 
  a 
  given 
  function 
  of 
  u. 
  The 
  corresponding 
  

   intensity 
  at 
  P 
  will 
  be 
  determined 
  by 
  (5), 
  a 
  and 
  ft 
  being 
  

  

  * 
  Which 
  in 
  the 
  case 
  of 
  a 
  small 
  9 
  X 
  becomes 
  rj 
  = 
  —lg$ 
  2 
  , 
  the 
  additive 
  

   constant 
  term 
  g 
  being 
  irrelevant. 
  

  

  D2 
  

  

  