Cardinal planes/points in paraxial optics

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Cardinal planes/points in paraxial optics Powered By Docstoc
					Optical systems:
Cameras and the eye

Hecht 5.7
Friday October 4, 2002


                         1
Optical devices: Camera
                    Multi-element lens




                             Film: edges
    AS=Iris Diaphragm        constitute field stop
                                                 2
Camera
   Most common camera is the so-called 35 mm
   camera ( refers to the film size)




                     27 mm


           34 mm
Multi element lens usually has a focal length of f =50 mm


                                                       3
Camera
 Object s = 1 m Image s’ ≈ 5.25 cm
 Object s = ∞    Image s’ = 5.0 cm
 Thus to focus object between s = 1 m and infinity,
 we only have to move the lens about 0.25 cm =
 2.5mm
 For most cameras, this is about the limit and it is
 difficult to focus on objects with s < 1 m




                                                       4
Camera




         AS=EnP=ExP Why?

                           5
Camera: Light Gathering Power
D = diameter of entrance pupil
L = object distance (L>> d)


                                 D




                l                    6
Camera: Brightness of image
Brightness of image is determined by the amount of
light falling on the film.
Each point on the film subtends a solid angle

                                D
                           D’
     dA D 2 D 2
 d  2      2
                
     r    4s'     4f 2



 Irradiance at any point
 on film is proportional
 to (D/f)2                          s’ ≈ f           7
 f-number of a lens

          Define f-number,          f
                                 A
                                    D
This is a measure of the speed of the lens
                                                 1
Small f# (big aperture) I large , t short    I  2
Large f# (small aperture) I small, t long       A



                                                     8
         Standard settings on camera lenses
        f# = f/D                               (f#)2
          1.2                                   1.5
          1.8                                   3.2
          2.8                                   7.8
          4.0                                   16
          5.6                                  31.5
            8                                   64
           11                                  121
           16                                  256
           22                                  484
Good lenses, f# =   1.2 or 1.8 (very fast) Difficult to get f/1   9
Total exposure on Film


                watts 
         E  I  2   t (exp osuretime )
                m 
           J
          2
           m

 Exposure time is varied by the shutter which has settings,
 1/1000, 1/500, 1/250, 1/100, 1/50
 Again in steps of factor of 2



                                                              10
Photo imaging with a camera lens
In ordinary 35 mm camera, the image is very small
(i.e. reduced many times compared with the object

 An airplane 1000 m in the air will be imaged with a magnification,




      M 
          5 10         2
                               5 10      5

            10 3
 Thus a 30 m airplane will be a 2 mm speck on film
 (same as a 2 m woman, 50 m)


Also, the lens is limited in the distance it can move relative to the film


                                                                             11
Telephoto lens
          L1                     L2




                    d               50 mm
A larger image can be achieved with a telephoto lens

Choose back focal length (bfl ≈ 50 mm)
Then lenses can be interchanged (easier to design)
The idea is to increase the effective focal length (and
hence image distance) of the camera lens.                 12
Telephoto Lens, Example

Suppose d = 9.0 cm, f2=-1.25 cm f1 = 10 cm

 Then for this telephoto lens

           P  P  P2  dP P2
                1         1                    Choose f = |h’| + bfl
           f  50 cm
Now the principal planes are located at
                                f'
           h'  H 2 ' H '   d       45cm
                                f1 '
                        f
           h  H1 H  d     360cm
                        f2                                         13
Telephoto Lens, Example
H’          h’ = - 45 cm




                                9 cm   5 cm

            f’= s’TP = 50 cm
                               Airplane now 1 cm long
     s'TP                      instead of 1 mm !!!!
             50
mTP       5  5  10
    
M     sc      5
         5      5
                                                    14
Depth of Field
            s2                            s2’

                        s1                       s1’




                                                                             d

                                                           x         x




                  so                            so’
  If d is small enough (e.g. less than grain size of film emulsion ~ 1 µm)       15
  then the image of these points will be acceptable
 Depth of Field (DOF)
       dso '
    x
        D
     so f  f  Ad 
s1 
       f 2  Adso                α   α   d
     so f  f  Ad    D
s2 
       f 2  Adso                x   x




                           so’
                                             16
Depth of field

                      2 Adso ( so  f ) f    2
      DOF  s2  s1 
                         f 4  A2 d 2 so
                                         2




    E.g. d = 1 µm, f# = A = 4, f = 5 cm, so = 6 m

                  DOF = 0.114 m

                i.e. so = 6 ± 0. 06 m

                                                    17
 Depth of field
                      Strongly dependent on the f# of the lens
                       Suppose, so = 4m, f = 5 cm, d = 40 µm

       so f  f  Ad 
                                            1200
                          10 ,000
s1                    
         f 2  Adso      25  1.6 A         1000


                               s1,s2 (cm)   800                s2
         DOF = s2 – s1                      600
                                                                          Depth of field (focus)

                                            400


    s f  f  Ad     10 ,000                                   s1
s2  o 2                                   200
      f  Adso       25  1.6 A
                                              0
                                                   0   2   4    6    8    10     12      14        16
                                                                                              18
                                                                     f#
19
Human Eye, Relaxed
                        20 mm



       15 mm




                        n’ = 1.33

 F             H H’                      F’




     3.6 mm

               7.2 mm               P = 66.7 D
                                                 20
Accommodation
 Refers to changes undergone by lens to
  enable imaging of closer objects
 Power of lens must increase
 There is a limit to such accommodation
  however and objects inside one’s “near
  point” cannot be imaged clearly
 Near point of normal eye = 25 cm
 Fully accommodated eye P = 70.7 for s =
  25 cm, s’ = 2 cm

                                            21
Myopia: Near Sightedness
      Eyeball too large ( or power of lens too large)




                                                        22
   Myopia – Near Sightedness
         Far point of the eye is much less than ∞, e.g. lf
 Must move object closer to eye to obtain a clear image




       Normal N.P.




Myopic       Myopic
F.P.         N.P.                                            23
Myopia
                    e.g. lf = 2m

                                       How will the
                 1 n' 1                near point be
                                     affected?
                 lf  s' f
                0.5 + 66.7 = 67.2 D

        is relaxed power of eye – too large!
To move far point to ∞, must decrease power to 66.7
         Use negative lens with P = -0.5 D             24
 Laser Eye surgery
   Radial Keratotomy – Introduce radial cuts to the
   cornea of the elongated, myopic eyeball

    Usually use the 10.6 µm line of a CO2 laser for
    almost 100% absorption by the corneal tissue      Blurred
                                                      vision




Front view                                              25
 Laser Eye surgery
   Radial Keratotomy – Introduce radial cuts to the
   cornea of the elongated, myopic eyeball
   Usually use the 10.6 µm line of a CO2 laser for
   almost 100% absorption by the corneal tissue
                                                     Distinct
                                                     vision




Front view                                              26
                                    Flattening
   Hyperopia – Far Sightedness
Eyeball too small – or lens of eye can’t fully accommodate

      Image of close objects formed behind retina




                                                         27
Hyperopia – Far Sightedness
             Suppose near point = 1m


       1 n'
          1  66.7  67.7 D
       1 s'

 Recall that for a near point of 25 cm, we need 70.7D

  Use a positive lens with 3 D power to correct this
  person’s vision (e.g. to enable them to read)
 Usually means they can no longer see distant
 objects - Need bifocals                               28
Correction lenses for myopia and hyperopia




  http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/V/Vision.html
                                                                         29


				
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