Download Optics I - Department of Applied Physics

Hong Kong Polytechnic University
Heat & Light (AP306)
Part II: Optics
Dr.Haitao Huang (黄海涛)
Department of Applied Physics, Hong Kong PolyU
Tel: 27665694; Office: CD613
PowerPoint can be downloaded from http://ap.polyu.edu.hk/apahthua
Textbook:
Introduction to Optics, 2nd Edition, F.L.Pedrotti and L.S.Pedrotti (Prentice Hall
1993)
Reference books:
Optics, 4th Edition, E.Hecht (Addison Wesley
Optics, 3rd Edition, A.Ghatah (McGraw Hill 2005)
Waves—Berkeley Physics Course V.3, F.S.Crawford, Jr., (McGraw Hill 1968)
Heat & Light----by Dr.H.Huang, Department of Applied Physics
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Nature of Light
Duality in Nature:
Rays of light can be regarded as streams of (very small) particles emitted
from a source of light and traveling in straight lines.
Light also takes a wave motion, spreading out from a light source in all
directions and propagating through an all-pervasive medium. Light can be
viewed as the electromagnetic radiation in a particular region of spectrum.
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Electromagnetic radiation may vary in wavelength (or frequency) and in intensity.
Variation in wavelength is called electromagnetic spectrum.
Variation in intensity is described by radiometry and photometry.
Radiometry involves only physical measurement.
Photometry involves the response of human eye.
Radiant flux emitted per unit of solid angle by a point source in a specific direction is
called the radiant intensity Ie (watts per sr=steradian),
dΦe
Ie 
d
Irradiance Ee (watts per square meter) is the radiant flux received on a unit surface
area,
Φe 4I e I e
Ee 
A

4r 2

r2
Radiance Le (watts per square meter per sr) is the radiant intensity per unit of
projected area, perpendicular to the specific direction,
dI
d 2I
Le 
e
dA cos

e
d dA cos 
Radiant intensity of a black body (or a perfectly diffuse scattering surface) follows the
Lambert’s law,
I e    I e 0 cos 
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Visual Sensitivity:
When we use photometric quantities, we are measuring the properties of visible
radiation as they appear to the normal eye, rather than as they appear to an
“unbiased” detector.
The visual
wavelength.
sensitivity
varies
with
The sensitivity curve, sometimes
referred to as the luminosity curve of an
equal energy spectrum, give the
relative brightness, as assessed by the
average eye, of the different colors of
the spectrum when the incident
energies at each wavelength have
been reduced to the same mechanical
value. For the average light adapted
eye at moderate intensities (photopic
vision) the maximum visual effect is
obtained with light of wavelength
555nm (yellow-green).
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Hong Kong Polytechnic University Production and Measurement of Light
Standard Source and Candela
A standard must give a constant luminous output for an indefinite period of time.
This restriction excludes all gas lamps, electric filament lamps, and discharge tubes
from being used for standardization.
The present primary standard source of light is based on the concept of a black
body radiator. A black body is a device which absorbs entire radiation incident on it
at all temperatures. When at a constant temperature a black body will be radiating
the same energy that it receives, the total quantity depending on its temperature.
Thus a black body is also a perfect emitter of radiation, and will emit more energy
than any other body at the same temperature. The radiation from a black body is
independent of the nature and material of the body.
A hollow vessel with a blackened interior and single small
hole provides the basis for a standard source.
The unit of measurement of the luminous intensity of a
source is candela. One candela is equal to one sixtieth
(1/60) of the luminous intensity per square centimeter of a
black body at the temperature of solidification of platinum.
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baffle
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Hong Kong Polytechnic University Production and Measurement of Light
Sources
•
•
•
•
•
Sunlight, skylight
Incandescent sources: blackbody; Nernst glower and globar; tungsten filament
Discharge lamps: monochromatic and spectral sources; high-intensity sources
(carbon arc, compact short arc, flash and concentrated zirconium arc);
fluorescent lamps
Semiconductor light-emitting diodes (LEDs)
Laser
Detectors
•
•
Thermal detectors: thermocouples and thermopiles; bolometers and thermistors;
pyroelectric detector; pneumatic or golay
Quantum detectors: phototubes and photomultipliers; photoconductive;
photovoltaic; photographic
reflectance or
reflection factor

luminous flux per unit area reflected by a surface
luminous flux per unit area incident on the surface
transmittance

luminous flux per unit area transmitted by the body
luminous flux per unit area incident on the surface
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Geometrical Optics
Law of Reflection
When a ray of light is reflected at an
interface dividing two uniform media, the
reflected ray remains within the plane of
incidence includes the incident ray and
the normal to the point of incidence.
Law of Refraction (Snell’s Law)
When a ray of light is refracted at an
interface dividing two uniform media, the
transmitted ray remains within the plane
of incidence and the sine of the angle of
refraction is directly proportional to the
sine of the angle of incidence.
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Geometrical Optics
Huygens’ Principle
Wavefront: the locus of the points which are in the same phase. For example, if we
drop a small stone in a calm pool of water, circular ripples spread out from the point
of impact, each point on the circumference of the circle (whose center is at the point
of impact) oscillates with the same amplitude and same phase and thus we have a
circular wavefront.
Each point of a wavefront is a source of secondary disturbance and the
wavelets emanating from these points spread out in all directions with the
speed of the wave. The envelope of these wavelets gives the shape of the new
wavefront.
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Geometrical Optics
Heat & Light----by Dr.H.Huang, Department of Applied Physics
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Geometrical Optics
Fermat’s Principle
The Fermat’s Principle is also called the “principle of least time”. The field of
geometrical optics can be studied by using Fermat’s principle which determines the
path of the rays.
The ray will correspond to that path for which the time taken is an extreme in
comparison to nearby paths, i.e., it is either a minimum or a maximum or
stationary.
The mathematical form of the Fermat’s principle is,   nds  0
where n is the refractive index and the integration is done along the path.
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Geometrical Optics
Principle of Reversibility:
Any actual ray of light in an optical system, if reversed in direction, will retrace the
same path backward. This principle is directly derived from the Fermat’s principle.
Reflection in Plane Mirrors
Specular reflection: the reflection from a perfectly smooth surface which obeys the
law of reflection.
Specular reflection from the xy-plane: the
reflected ray PQ remains within the plane
of incidence, making equal angles with the
normal at P. If the direction of the incident
ray is described by its unit vector, r1=(x, y,
z), then the reflected ray is r2=(x, y, z)
A virtual image of a point object S is
formed at point S since the rays of
light appear to come from that point.
SA=SA
The image is the same size as the
object. The image is reversed, laterally
inverted and upright.
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Geometrical Optics
Refraction through Plane Surfaces:
n1 sin 1  n2 sin 2
For rays making a small angle with the normal to the
surface, a reasonably good image can be located. In this
approximation, we only allow paraxial rays to form the
image. Since n1>n2, the virtual image S of the point object
appears to be nearer than it actually is.
A critical angle of incidence C is reached when the angle of
refraction reaches 90°. For angles of incidence larger than
this critical angle, the incident ray experiences total internal
reflection, as shown.
 n2 

 n1 
 C  sin 1 
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Geometrical Optics
Imaging by an Optical System
An optical system includes any number of reflecting and/or refracting surfaces, of
any curvature, that may alter the direction of rays leaving an object.
The region may include any number of intervening media, but we assume each
individual medium is homogenous and isotropic, and so characterized by its own
refractive index. The rays from the object point O spread out radially in all directions
in real object space. In real object space the rays are diverging and spherical
wavefronts are expanding. Suppose that the optical system redirects these rays in
to the real image space, the wavefronts are contracting and the rays are converging
to a common image point I. From Fermat’s principle, the rays are said to be
isochronous. Further, by the principle of reversibility, if I is the object point, the image
point will be O. The points O and I are said to be conjugate points for the optical
system. Both points O and I can be imaginary.
Reflecting or refracting surfaces that form perfect images are called Cartesian
surfaces.
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Geometrical Optics
Spherical Surface:
Spherical mirrors may be either concave or convex relative to an object point O,
depending on weather the center of curvature C is on the same or opposite side of
the surface.
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Geometrical Optics
Reflection at a Spherical Surface:
One ray originating at O is normal to the
spherical surface at its vertex V and the other is
incident arbitrary at P. The intersection of the
two rays (extended backward) determines the
image point I conjugate to O. The image is
virtual, located behind the mirror surface.
  
2     
     2
sin   tan   
h h
h
  2
s s
R
1 1
2
 
s s
R
Similarly, for a concave mirror, we can have
1 1 2
 
s s R
1 1
2
 
s s
R
1. The object distance s is positive
when O is a real object. When O
is a virtual object, s is negative.
2. The image distance s is positive
when I is a real image, and
negative when I is a virtual image.
3. The radius of curvature R is
positive when C is to the right of V,
corresponding to a convex mirror,
and negative when C is to the left
of V, corresponding to a concave
mirror.
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Geometrical Optics
Focal Length:
For an object at infinity, the image distance is the focal
length,
f 
R
2
The focal length is positive for a concave mirror and
negative for a convex mirror. The mirror equation can
be written as,
1 1 1
 
s s f
Lateral magnification: m 
hi si

h0 s0
We assign a (+) magnification to the case where the
image has the same orientation as the object and a ()
magnification when the image is inverted relative to the
object.
m
si
s0
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Geometrical Optics
Graphic diagram for spherical mirrors:
(a) parallel to the principal axis are reflected though F;
(b) passing through F are reflected parallel to the principal axis;
(c) passing through C is reflected back along the same path.
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Geometrical Optics
Example: An object 3 cm high is placed 20 cm from (a) a convex and (b) a concave
spherical mirror, each of 10 cm focal length. Determine the position and nature of the image
in each case.
Solution: (a) Convex mirror: f=-10 cm and s=+20 cm.
From mirror equation,
m
s 1

s 3
1 1 1
we get s=-6.67 cm
 
s s f
The image is virtual, 6.67 cm to the right of the mirror vertex, and is erect and 1 cm high.
(b): Concave mirror: f=+10 cm and s=+20 cm.
From mirror equation we obtain s=20 cm and, therefore, m=1
The image is real, 20 cm to the left of the mirror vertex, and is inverted and the same
size as the object.
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Geometrical Optics
Refraction at a Spherical Surface:
n1 sin 1  n2 sin 2
n11  n2 2
n1      n2     
h h
h h
n1     n2   
 s R
 s R 
n1 n2 n1  n2
 
s s
R
Employing the same sign convention for mirrors,
n1 n2 n2  n1


s s
R
Lateral magnification:
h 
h 
n1  0   n2  i 
 s 
 s 
m
hi
n s
 1
hO
n2 s
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Geometrical Optics
Example: A real object is positioned in air, 30 cm from a convex spherical surface of
radius 5 cm. To the right of the interface, the refractive index is 1.33. What is the image
distance and lateral magnification of the image?
Solution: s=30 cm and R=5 cm.
1 1.33 1.33  1



30
s
5
This gives cm which means the image is real.
n1 s 
140  1

indicating an inverted image.
1.3330
n2 s
As shown in the figure, suppose now the second medium is only 10 cm thick, forming a thick
lens, with a second, concave spherical surface, also of radius 5 cm. The refraction by the first
surface will be unaffected by this change. Inside the lens, before a real image is formed, rays
are intercepted and again refracted by the second surface to produce a different image.
We have s2=30 cm and R2=5 cm.
m
1.33 1 1  1.33
 
 30 s 2
5
So, s2=9 cm. It is a real image.
m
1.339  2
1 30 5
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Geometrical Optics
Thin Lenses:
Two spherical surfaces are involved in a thin lens.
n1 n2 n2  n1
 
1st refracting surface of radius R1,
s1 s1
R1
n2 n1 n1  n2
 
2nd surface of radius R2,
s2 s 2
R2
1 1 n2  n1  1 1 
  
 
s s
n1  R1 R2 
For thin lens, s2  s1
lensmaker’s equation: 1  n2  n1  1  1 


f
Lateral magnification:
n1
s
m
s
 R1
R2 
1 1 1
 
Thin lens equation:
s s f
A (convex) lens is thicker in the middle: positive focal
length.
A (concave) lens thinner in the middle: negative focal
length.
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Geometrical Optics
Ray Diagram for Thin
Lenses:
Thin lenses are best
represented,
for
purpose
of
ray
construction, by a
vertical line with edges
suggesting the general
shape of the lens.
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Geometrical Optics
Example: Find and describe the intermediate and final images produced by a
two-lens system such as the one sketched in Fig.3.16(a). Let f1=15 cm, f2=15 cm,
and their separation be 60 cm. Let the object be 25 cm from the first lens.
Solution: For the first lens, s1=25 cm.
1 1 1
 
s1 s1 f1
s1  37.5 cm
m1  
s1
 1.5
s1
Thus the first image is real and inverted.
For the second lens we have s2=6037.5=22.5 cm. Then,
1 1
1
 
s 2 s 2
f2
s2  9 cm
m2  
s 2
 0.4
s2
Thus the final image is virtual and inverted.
The overall magnification is m=m1m2=0.6.
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Geometrical Optics
Vergence and Refractive Power:
The reciprocal of distance is defined as vergence, which represents the curvature of
wavefront incident or emerge from a lens. They are,
V 1 s
V   1 s
The change in curvature from object space to image space is due to the refracting
power P of the lens, given by 1/f.
With these definitions, the thin lens equation can be written as,
V V   P
When the lengths are measured in meters, the powers have units of diopters (D).
The refractive power of a surface is (n2n1)/R.
When several thin-lenses are placed together, the overall refractive power is the sum
of the individual refractive powers,
P  P1  P2  P3  ...
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Geometrical Optics
Newtonian Equation for the Thin Lens
hi
f

hO x
hi
x

hO
f
xx  f 2
f
x
m 
x
f
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Matrix Methods in Paraxial Optics
Thick Lens:
Thick lens is a lens whose thickness along its optical axis cannot be ignored without
leading to serious errors in analysis.
Six Cardinal points:
• first and second system focal points (F1 and F2);
• first and second principal points (H1 and H2);
• first and second nodal points (N1 and N2).
Planes normal to the axis at these points are called cardinal planes.
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Matrix Methods in Paraxial Optics
Basic Equations:
1 nL  n nL  n nL  n nL  n t



f1
nR2
nR1
nnL
R1 R2
n
f 2   f1
n
n L  n
r
f1t
nL R2
 n n  n 
v  1   L
t  f1
n
nL R2 


f1 f 2

1
s0 si
s
nL  n
f 2t
n L R1
 n n n 
w  1   L
t  f 2
 n nL R1 
Distances are directed,
positive or negative by a
sign convention that makes
distances directed to the
left negative and distances
to the right, positive.
nsi
m
nso
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Matrix Methods in Paraxial Optics
Example: Determine the focal lengths and the principal points for a 4-cm thick, biconvex
lens with refractive index of 1.52 and radii of curvature of 25 cm, when the lens caps the
end of a long cylinder filled with water (n=1.33).
Solution: nL=1.52, n=1, n=1.33, t=4 cm,
The first refraction surface is a convex plane, so R1=25cm.
The second one is a concave plane, so R2=25cm. According to equation
1 nL  n nL  n nL  n nL  n t



f1
nR2
nR1
nnL
R1 R2
we can get f1=35.74cm to the left of the first principal plane. From
f2  
n
f1
n
we have f2=47.53cm to the right of the second principal plane. Equations
r
n L  n
f1t
nL R2
s
nL  n
f 2t
n L R1
give us r=0.715cm and s=2.60cm.
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Matrix Methods in Paraxial Optics
Matrix Method:
In the paraxial approximation, changes in height and direction of a ray can be
expressed by a linear matrix equation that links the image and object rays. The
matrix is characteristic of an optical system.
Translation Matrix:
 y1  1 L  y0 
   0 1   
 0 
 1 
The 22 ray-transfer matrix represents the effect of translation on the ray.
1 L 
0 1 


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Matrix Methods in Paraxial Optics
Refraction Matrix:
The refraction of a ray is illustrated in the figure.
We need to add a sign convention for the angles: angles are considered positive for
all rays pointing upward, either before or after a reflection; angles for rays pointing
downward are considered negative.
(y,) and (y,) can be related by the followings,
y  y
n  n 
y n
y
 
R n
R
n
y n
y y
           
n
R n 
R R
        

1n 
n
  1 y  
R  n 
n
1
 y 
    1  n  1
   R  n 
0 y
 
n  
n   
1

1  n 
 R  n  1

 
0
n
n 
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Matrix Methods in Paraxial Optics
Reflection Matrix:
Consider the reflection at a spherical surface as shown.
We have, y=y, and
           
 
2y 2
 y 
R R
 y  1
    2
   R
1
2
 R
0  y 

1  

0

1

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Matrix Methods in Paraxial Optics
Thick-Lens and Thin-Lens Matrices:
In traversing the thick lens, the ray undergoes two refractions and one translation.
For the first refraction  y1 
 y0 
   M 1  
 1
 0
 y2 
 y1 
   M 2  
 2
 1
 y3 
 y2 
For the second refraction    M 3  
 2 
 3 
 y3 
 y0 
Finally    M 3 M 2 M 1  
 3 
 0 
For the translation
 1
The entire lens M   n L  n
 nR
2

For thin lens
0 1 t  1
0


nL 
n  nL n 
0 1 
  nL R1 nL 
n  
1
0  1 0

M   nL  n  1  1  1   1 1


R R    f
n


2
1




Heat & Light----by Dr.H.Huang, Department of Applied Physics
32
Hong Kong Polytechnic University
Matrix Methods in Paraxial Optics
System Ray-Transfer Matrix:
For an optical system which includes any number N of translations, reflections, and
refractions, the following general relation exists.
 yN 
 y0 

M
M



M
M
N
N 1
2
1
 

 N
 0 
The ray transfer matrix (or system matrix) is
M  M N M N 1    M 2 M 1
The determinant of the system matrix has a very useful property:
Det M 
n0
nf
where n0 and nf are the refractive indices of the initial and final media of the optical
system.
Heat & Light----by Dr.H.Huang, Department of Applied Physics
33
Hong Kong Polytechnic University
Matrix Methods in Paraxial Optics
Example: Find the system matrix for the thick lens. R1=45cm, R2=30cm, t=5cm,
nL=1.60, and n=n=1.
Solution: Using system matrix,
We get
 1
M   n L  n
 nR
2

 23
1
0
1
0

 1 5 
 
1    24

 1
M  1

 50 1.6 0 1  120 1.6   7
1200
0 1 t  1
 nn
nL  
L
0 1 
  nL R1
n  
25 
8
17 

16 
0
n
nL 
Heat & Light----by Dr.H.Huang, Department of Applied Physics
34
Hong Kong Polytechnic University
Matrix Methods in Paraxial Optics
Significance of System Matrix Elements:
 y f   A B   y0 
   
  
C
D
 0 
 f 
D=0
A=0
B=0
C=0
Heat & Light----by Dr.H.Huang, Department of Applied Physics
35
Hong Kong Polytechnic University
Heat & Light----by Dr.H.Huang, Department of Applied Physics
36