Chapter Ⅳ Plane Surface and Prism
第四章 平面镜和棱镜
§4.1 Parallel beam——平行光束
§4.2 Critical angle and total refection ——临界角和全反射
§4.3 Roof prism——屋脊棱镜
§4.4 Plane-parallel plate——平行平板
§4.5 Optical tunnel of prism(unfolding prism)——棱镜的展开
§4.6 Image rotation——图像旋转
§4.7Application of prism in telescope——棱镜在望远镜中的应用
§4.8Refraction by a prism——折射棱镜
§4.9 Thin prism——薄棱镜
§4.10Direct-Vision prism——直视棱镜
§4.11Reflection and refraction of diverging beam——发散光束的反
射和折射
§4.1* Reflecting Mirror
Imaging characters: perfect, specular, virtual, erect, unit image.
Perfect: point →point; line →line; according reflecting law
Specular: (reversed) left-hand →right-hand
Virtual: diverging beam →converging beam
Figure 1:
Erect: upper → upper;
Unit:
n n n  n
 
l l
r
 r  , n   n 
 l   l ,  
nl 
1
n l
Odd times: specular
Even times: similar
Artificial plane surface are used in optical:
△
△
△
△
Changing image parity (Similarity)
Deviation of rays, (or beam fold)
Lateral displacements of rays,
Break light into its colors.(dispersion)
concentric beam
§4.2 Prism(internal total reflex prism)
1. parallel beam
△ The parallel beam remains parallel after reflection or refraction at a plane surface.
△ Refraction causes a change in width of a beam. Refraction remains the same width of beam.
Figure 2: rare medium to dense medium
Figure 3: dense medium to rare medium
When light beam is refracted, in particular(from glass to air)case, the refracted beam is narrow
up to zero.
2. The critical angle and total reflection
△ (1)from glass to air
To increase incident angle from to θto ic , the refractive angle increases from 0 to 90°, this
particular angle ic for which i=90°, is called critical angle.
In Snell’s law n sin 90  n sin ic ; ic  arcsin

Beyond the ic ,total reflection occurs.
△ (2) from air to glass(reversibility of light)
n
n
Figure 4

n sin 90  n sin ic ；
ic  arcsin
n
；
n
(3)Crown glass of index 1.520 surrounded by air. ic  41 8

(4) Total reflection is really total, no energy is lost upon reflection there will be small losses due
to absorption.
3. The commonest devices of this Kind are called total reflection prism(and optical fiber)
Figure 5 rectangular prism
Figure 6 The porro prism
Figure 7 The Dove prism
Figure 8 roof prism
△ If Dove prism: is rotated around the direction of the light, light beams rotate around each
other with twice the angle velocity of the prism.
△ Triple mirror: is made by cutting off the corner of a cube, three face intersect at the corner.
Any ray striking it will, after being internally reflected at each if three faces, be sent back
parallel to its original direction.
Figure 9 triple mirror
4.multiple mirror system
(1)Two-mirror system
A pair of mirrors mounted at fixed angle θ to each other causes a ray to be reflected through an
angle 2θ.
Note: these deflection 2θof the light are independent of the slope of the particular ray.
Figure 10The Penta prism
Figure 11 The Porro prism
Figure 12 The Rhomboid prism
Mirror at 45°
90°
0°
Ray deviation 90°
180°
0°
Prove according to the law of reflection.
*Two-mirror system is a constant-deviation system for any arbitrary tilting of the system.
*Two-mirror system is forms a similar image (right-hand coordinate→right in image space).
Single mirror system forms a specular image.
Figure 13 right angle prism
Figure 14 penta-prism
Figure 15
roof right-angle prism
Figure 16
Actual prism
§4.3 Roof-prism
Fuction:
Roof divided the beam into two halves. One half entering the left side and cross over to the right
side. The other half behaving conversely.
Note:
*the roof angle must be equal to 90°, otherwise double image will appear! Usual tolerance is a few
seconds of arc.
*A parallel beam entering the prism perpendicular to the end of face strikes the roof face ant the
angle of incidence equal to 60°, so coating of the roof face is not needed.
*The fuction of roof face is changing only the direction of image in sagittal plane.
Figure 17roof -penta prism
Figure 18 Schmidt roof prism
Figure 19 Lehman proof prism
Figure 20
Schmidt roof prism
lehman roof prism
half roof penta
§4.4
plane-parallel plate
1. Plane-surfaces are parallel to each other. Ray emerges parallel to its original direction.
2. With lateral displacement △y
Figure 21




y  AB sin  1  1   AB(sin 1  1 )  AB(sin 1 cos 1  sin 1 cos 1 )


AB 
d

cos 1
sin 1 ' 
n
sin 1 .
n
Nonlinear: y  d sin 1 (1 
n cos 1
n 1
)  d1
n cos  
n
1
 y increases with the angle Φ1
Linear: y  d
n 1
;
n

If  is larger: y  d sin 1 1 




2
n  sin 1 
cos 1
§4.5 The optical tunnel of a prism
1.
A prism is essentially a combination of a block of glass and one or more mirrors, so
we can unfold a prism by reflecting it successively in each of the mirror surfaces.
Figure 22 Dove prism
Figure 23 penta-prism
a) The figure show that optical tunnel is a sloping block of glass, entering and emerging
rays are parallel.
b) The entering and emerging faces are paralleled and axis of the beam goes straight
through the middle.
2. Image displacement due to thick glass plate as shows above; reflecting prism can be
unfolded into a plane paralleled plate.
a) Inserting a thick glass plate into a converging beam, causes the image displacement along
the axis.
The image will move away from the lens.
Figure 24
By an amount

cos I 1 
y
 d 1 
cos I 1


tgI1
n
cos
I
1 

n 1
Paraxial: s  d
n
s
① The image displacement will increase as I increases, it leading to a considerable amount
of overcorrected spherical aberration, and also leads to chromatic aberration.
② For paraxial rays I I′ are small, cosines cos I  1, cos I   1, image shift becomes
s
n 1
d
n
③ since the image magnification by a plane surface is unity, the size of he shifted image
will be unchanged.
④ Equivalent air layer
The normal of second surface crosses the incident ray at C, and the first surface at B,
BC  d called equivalent air layer
d d sd 
n 1
d
d
n
n
Equivalent air layer can be used in calculating the size of prism.
§4.6 Image Rotation
1. Rotating mirrors
If a mirror is rotated through an angle θ about an axis perpendicular to the plane of incidence,
the reflected ray turns through twice as great an angle 2θ.
Figure 25 reflecting ray turns
Normal is in position N the angle AOA′=2I
Normal is in New position N′ the angle AOA″=2(I+θ)
Reflecting ray turns 2(I+θ)-2I=2θ
2. Image Rotating prism (Dove prism)
Figure 26
 The essential requirements are that the axes of the entering and emerging beams must lie in
the same straight line, which constitutes the axis of rotation; there must be an odd number of
reflecting surface.
 For Dove prism, the figure shows:
① The image rotates in the same direction as the prism is turned, but twice prism speed.
② Because of odd number of reflectors, the image is left-hand.
§4.7 The Application of prism in telescope
1. The porro prism
Figure 27
The first prism inverts the image and reverse the direction of the light.
The second prism merely reverses the direction of light, so the saggittal direction of the image is
also reversed.
2. The Panoramic telescope
Figure 28
⑴ the head prism (right-angle prism) can be rotated about the vertical axis to scan the horizonal
image rotates at prism speed.
⑵ These rotation is compensated by use of a Dove prism, Dove prism must be rotated at half the
speed of the head prism.
⑶ As there are now two mirrors in system, it is necessary to add two more, this is done by use of
amici roof prism (roof right-angle prism).
*Dove prism is mounted ahead of the object because it must be used in paralleled light.
3. The trajectory camera
Figure 29 The trajectory camera
t
§4.8 Refraction by a prism
Figure 30



Deviation:    1   2   i1  i1    i2  i2   i1  i2  

The
angle
of
minimum
 
deviation

occurs
at
that
particular
angle
of
incidence

 
i1  i2 , i1  i2 
2
To prove these angle equal, refer to optical measurement: refractometer
Application:
▲ If the angle α and δ are measured, the most accurater measurement of refractive index n can
be get.
▲ When prism are used in spectroscopes and spectrographs, they are always set as nearly as
possible at minimum deviation. Because slight of incident light would cause astigmatism.
§4.9 Thin prism (optical wedge)
Ⅰ When refracting angle α becomes small enough to ensure that its sine may be equal to the
angle themselves. (a few degree)picture
Figure 31
n

n0
sin
1
      
2

sin
  n  1


2
(in air)
Ⅱ Combination of two thin prism of equal power two thin prism can be rotated in opposite
directions in their own plane.
Figure 32-a 
 0,   2n  1
Figure32-b
  180  ,   0
▲ When the two thin prism are parallel, power is twice that of either one.
▲ When the two thin prisms are opposed, power is zero.
How the power and direction of deviation depend on the angle between the two prisms?
Deviation:    1   2  2 1 2 cos  
2
2
β: the angle between the two thin prisms
(using vector, because
1 ,  2 , very small)
2 12 1  cos    2 1 cos

2
Figure 33
tan  
 
 2 sin 

 tan
 1   1 cos 
2

2
 : between the resultant deviation and due do prism alone.
§4.10 Direct-Vision Prism
Primary fuction of direct-vision prism is to produce a visible spectrum, central color of which
emerges from prism parallel to the incident light, and is in line with the incident light’
Figure 34
Combination usual glass prism and flint-glass prism, placed back to back.
§4.11Reflection and Refraction of Divergent Rays
ⅠReflection
Figure 35
Divergent pencil of light is reflected at a plane surface remains divergent.
According to the law of reflection object distance l=image distance l′ point A′ is said to be a
virtual image of A. They appear to come from a source at A′, actually, do not pass through A′.
ⅡRefraction
Figure 36
△An object is in glass plastic or water the image appears closer to the surface.
△Refractive angle is larger than incident angle . extending these emergent rays backward, we
locate their intersection in pairs, they are image points, virtual image.
How far away these images is (from the object)?
ⅢImage formed by paraxial rays
Figure 37
Consider the right triangles
h  ltg  l tg 
l  l 
sin  cos  
n cos  
l
cos  sin  
n cos 
If Φ and Ф′ are very small
l 
n
l  n
l→ 
n
l
n
The ration of the image distance for paraxial rays is just equal to the ration of the index of
refraction.