Download Lecture Note No. 2

Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
Geometrical Optics and Optical Elements
Lecture Note No. 2
L >> : Geometrical optics (Image quality is determined by rays. Refraction and reflection only.
No absorption, no optical aberration errors. Also called ray or Gaussian optics.)
L ~ : Diffraction-limited optics (Image clarity is determined by diffraction)
■ Thin Lens Formula
Under paraxial ray assumption (   0 ), l o ~ s o and li ~ si :
n1 n2 n2  n1


s o si
R
… for a single refraction surface.
Thus, as si   , so  f o 
And as s o   , si  f i 
n1
R
n2  n1
n2
R
n2  n1
yo , y i :
+ if above the optical axis
x o , so , f o :
+ if left of the vertex (V)
xi , si , f i :
+ if right of the vertex
R:
+ if the center (C) is right of the vertex (V)
Subscript 1:
Left or ray incident surface
Subscript 2:
Right or ray emitting surface
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Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
♦ Extending of the Single surface-equation to double-surfaced lens gives:
Combining (a) and (b), using so2   si1  d , gives:
 1
1 1  nl
1  1
 
  
 1 
s o si  nm
R
R
f
2 
 1
where f  lim si  s o  lim so  si
HW#1. EoC Problem 5.22
♦ Magnification:
Transverse magnification:
MT 
yi
s
x
f
 i  i 
yo
so
f
xo
Longitudinal magnification:
ML 
dxi d f 2 / xo
f2

  2  M T2
dxo
dxo
xo

♦ Combination of thin lenses:
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
Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
si 2 
f 2 d  f 2 so1 f1 / so1  f1 
d  f 2  so1 f1 / so1  f1 
M T  M T1  M T 2 
f 1 si 2
d so1  f1   so1 f1
f.f.l. = lim so1 
si 2 
As d  0 ,
f.f.l. = b.f.l. =
f1 d  f 2 
;
d   f1  f 2 
f1 f2
 f
f1  f2
b.f.l. = lim si 2 
or
HW#2. EoC Problem 5.37
■ Mirrors
♦ Planar Mirrors
*Mirror rotation:
Rate of image rotation = 2 x Rate of mirror rotation
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so 1 
1
1
1


f
f1 f 2
f 2 d  f1 
d   f1  f 2 
Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
♦ Spherical Mirror
SA   i  so  R
PA   r  si  R 
SA ~ so , PA ~ si ,  i ~ SC / SA,  r ~ CP / PA
1 1
2 1
  
so si
R f
*For paraxial region, a parabolic mirror can be approximated as a spherical mirror
with R  2 f or f = R/2.
f
*F-Number 
D: Aperture open diameter
D
When the open area is doubled, the diameter is increased
by 1.4 (= 2 ) and F-number decreases to 0.714 (= 1 / 2 ).
HW#3. EoC Problem 5.59
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Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
■ Prisms
For fixed n and  ,  depends only on  i1 , and from
 i1   t 2
d
 0 (for minimum deviation),
d i1
and  t1   i 2
This states that the ray for which the deviation is a minimum traverses the prism symmetrically,
that is, parallel to its base. This explains the difference in the spherical aberration (SA)
depending on the orientation of a plano-convex lens for collimation. (Fig. 6.16)
We get
n
sin min    / 2
sin  / 2
… The most accurate technique for determining n of a transparent material
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Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
■ Thick Lens Formula
After a great deal of algebraic manipulation, the focal length measured from the principal plane,
i.e., f  ffl  h1 or f  bfl  h2 is given by:
1
1
1 nl  1d l  1 1
 nl  1 



f
nl R1 R2  so si
 R1 R2
h1  
f nl  1d l
R2 nl
MT 
yi
s
x
f
 i  i 
yo
so
f
xo
h2  
f nl  1d l
R1nl
(see E-o-C Prob. 6.18)
Note: f , so , si are measured w.r.t. the Principal Plane
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Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
■ Ray Tracing
The ray vector is defined as:
n  
ri1   i1 i1 
 yi1 
The lens system matrix is given as:
 D2 d l
1  n
l
A (D1, D2, dl, nl)= 
d
l

 nl
D1D2 dl 
nl 
 , det A = 1
Dd

1 1 l

nl
 D1  D2 
The transfer matrix across “1-2” surfaces, separated by d21 in a medium n, is given as:
0
 1
T21  

d 21 / n 1
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Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
HW#4:
f1 = -30 cm
f2 = 20 cm
d32 = 10 cm
d 21  2 cm, d 43  2 cm, nl  1.5, nm  1.0
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Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
■ Mirrors
The matrix expression is given as:
 n r    1  2 n   n  i 
R 

 y 
 r   0
1   yi 
rf = M ri
rf (reflected ray vector) = M (mirror matrix) ri (incident ray vector)
For a flat mirror ( R   , n = 1.0),
  1 0
M= 
,
 0 1
 
rf =  r  ,
 yr 
Therefore,  r   i and yr  yi
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 
ri =  i 
 yi 
Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
■ Aberrations:
Departures from the idealized conditions of Gaussian or ray optics.
Becomes more substantial in the non-paraxial region.
♦ Spherical Aberration:
The non-paraxial lens formula is given as:
n
n1 n2 n2  n1


 h2  1
s o si
R
 2so
2
 1 1
n
    2
2si
 so R 
1 1
  
 R si 
2



where the height h is the distance of ray measured w.r.t. the optical axis.
The focal length f  lim so  si  decreases with increasing h: Spherical aberrations
HW#5:
Which of the following two configurations provides a better definition of the focal volume?
Explain the physical reason for your answer.
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Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
♦ Chromatic Aberration [n increases with  ]:
 1
1
1 
 , shows
 nl  1 
f
 R1 R2 
that the focal length decreases with decreasing  . Thus, the focal length of blue ray is smaller
than the focal length of red ray, i.e., f B  f R and nB  nR
Since nl   increases with decreasing  , the thin lens equation,
*Fraunhofer Lines (Table 6.1):
Reference spectral lines generated by different substances such as H, Na, He, …
C
D or d
b or c
F
F, g or K





Red
Yellow
Green
Blue
Violet
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Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
♦ Achromatic Doublet Lens to Reduce CA:
For both lens1 and lens2,
 1
1
1 

 n1d  1

f1d
 R11 R12 
and
Abbe Number V for d (Yellow) is defined as:
Vd 
nd  1
n F  nC
**The achromatic condition is given as:
V1d
1

f1d
f d V1d  V2 d 
V2 d
1

f 2d
f d V2 d  V1d 
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 1
1
1 

 n2 d  1

f 2d
 R21 R22 
Micro/Nano-Scale Optical Imaging Techniques, Lecture Note. 1
Prof. K. D. Kihm, Spring 2008
HW#6: Design an achromatic lens of f d = 0.5 m choosing BK1 material [ nC = 1.50763, n d =
1.51009, nF = 1.51566] for lens 1 (equi- or double-convex) and F2 material [ nC = 1.61503, n d
= 1.62004, nF = 1.63208] for lens 2 (concave).
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