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The Nature of Light
Chapter 1
Physics 158, Introduction to Optics
Peter Beyersdorf
Document info
1. 1
Class Outline
Introductions/Announcements
Properties of light
Mathematical description of transverse waves
1. 2
Introductions
Instructor: Dr. Peter Beyersdorf
Office: Science 235! ! ! !
email: [email protected]
Phone: 924-5236
Office Hours
Tuesday and Thursday from Noon-1 pm
Webpage: TBA
Textbook: Pedrotti, “Introduction to Optics, 3rd edition”,
Modern Optics provides a framework for understanding and analyzing optical
wave propagation, interference, polarization and diffraction effects. This
framework is essential for those who will work with imaging systems, lasers,
optical communication systems, and optical measurements.
1.
Textbook
Jay Leno
commedian and
optics expert
Jay Leno
commedian and
optics expert
Katia Pedrotti
Star of “Big Brother”
and optics expert
Katia Pedrotti
Star of “Big Brother”
Anne Frank
Holocaust victim and
optics expert
Anne Frank
Holocaust victim and
optics expert
Leno M. Pedrotti
Professor of
Physics at
University of
Dayton
Leno M. Pedrotti
Professor of
3
1. 4
Course Grading
Your grade will come from a combination of homework (25%) and midterm
exams (25% each). Your final score will be the higher of the average of
these scores or your final exam score.
Homework (your lowest score will be dropped)!
Exam 1!
Exam 2
Exam 3 !
Final Exam
You will receive a normalized score (based on a curve) for each
component listed above.
It is your responsibility to convince me you have a conceptual
understanding of the subject matter. When you solve problems, presenting
your work in a clear, concise manner that shows the logical steps leading
to the final answer will go along way towards this objective. Work will
be graded on the quality of your solution, not just the correctness of
your answer.
1. 5
Policies and Rules
College and Departmental Policies
!
Students who wish to add or change lab sections must first obtain
written permission from the lab instructor indicating that a lab space is
available in the session for which the instructor is responsible.
!
You are responsible for understanding the policies and procedures about
add/drops, academic renewal, withdrawals, incompletes, classroom behavior, and
other policies described in the catalog. Please read your catalog thoroughly.
Class Rules
!
Place your personal electronics in quiet mode, and refrain from using
them in the classroom for non-class related work.
!
I encourage you to work on homework assignments in groups. You are
able to learn much more from each other than you can from me, and you will
find that if you take the time to help your classmates you will develop a
better understanding of the material yourself. Of course I am also available,
and am happy to meet with you during my scheduled office hours, and am
available on-and-off outside of these hours – just stop by my office.
!
It’s my job to help you learn. Help me help you – attend class,
participate in discussions and problem solving sessions, discuss with me problems
you are having and give me lots of feedback so I can teach more effectively!
1. 6
Questions to Ponder
Is a light ray a stream of particles or a wave in
some medium?
If light is a wave why does our eye see a
constant brightness for an object instead of a
brightness that cycles dark and light at the
frequency of the optical wave?
1. 7
What is Light?
It is a wave
propagates as a disturbance in the electric
and magnetic fields
can be described by a wavelength,
frequency, and other wave-like properties
It is made of particles
It interacts in a discreet manner
Its energy and momentum are quantized
1. 8
What is light?
Maxwell’s equations give rise to the “wave
equation” that describes the propagation of
electromagnetic waves.
1 ∂2E
∇ E= 2 2
vp ∂t
2!
In one-dimension and in free-space this can be
expressed as:
Which has solutions that are traveling waves
1. 9
What is light?
Like all electromagnetic waves, light
is a transverse electromagnetic wave
is a solution to the wave equation
propagates at a velocity of c (in free space)
is made up of quantum-mechanical particles called photons
What differentiates light from lower frequency electromagnetic
waves (microwaves, radio waves, etc),
The frequency of oscillation of light is too high for the
changing electric & magnetic fields to be directly observed
Only the average intensity of the electromagnetic wave
can be observed
What differentiates light from higher frequency
electromagnetic waves (x-rays, gamma rays)
Energy of optical photons is too low to ionize most
materials
1.10
The Electromagnetic Spectrum
Source: Louis Keiner, Coastal Carolina University
1.1. 11
Photoelectric Effect
Heinrich Hertz discovered, and Einstein explained,
that light above a given threshold frequency
would would produce a current when absorbed in
in metals. The explanation for this required the
idea of quantization of light
Image credit: hyperphsics.phy-astr.gsu.edu
1. 12
Quantum Aspects of Light
Light contains energy and momentum that is
carrier in discrete quantities by “photons”.
The energy of a single photon is
E = !ω = hf =
hc
λ
The momentum of a single photon is
h
p! = !!k = k̂
λ
1.13
1.14
Properties of light
What are some distinguishing properties of a
light wave?
Wavelength (frequency)
Direction of propagation (spatial mode)
Irradiance (amplitude)
Polarization
Phase
1.15
Describing EM waves
Amplitude
Expressions for a monochromatic wave
Polarization
Direction of
propagation
Wavelength
useful relations
Frequency
Initial phase
1. 16
Antenna Example
A car radio antenna has a length that is λ/4
which gives the optimal reception efficiency. If
the length of the antenna is 78cm what station
is it optimized for?
1.17
Complex representation of waves
Since manipulating trigonometric functions is difficult, it is common
practice to express a wave in complex form instead
The time (and/or space) dependance is often omitted as it is
understood the wave is oscillating at frequency ω, and so it can
always be accounted for by adding it in explicitly after calculating
how the wave evolves in space
Also, it is understood that when expressing a wave in complex form,
the actual wave is the real part of the expression.
This form is referred to as the phasor representation of the wave
1.18
Phasor Example
Two waves of equal frequency copropagate in
the +x direction. One has an amplitude of 2
units and a (initial) phase of -π/4, while the
other has an amplitude of 1 unit and a (initial)
phase of π/2
Express each wave as a trig function and as
a phasor
Find an expression for their sum
1.19
Phasor Example
In trigonometric form:
E1 (x, t) = 2 cos(kx − ωt − π/4)
E2 (x, t) = cos(kx − ωt + π/2)
In phasor form:
!1 (x, t) = 2ei(kx−ωt−π/4)
E
!2 (x, t) = ei(kx−ωt+π/2)
E
1.20
Phasor Example
Factor out common terms:
"
!1 (x, t) + E
!2 (x, t) = 2e
E
−iπ/4
+e
iπ/2
#
ei(kx−ωt)
! ! ! ! add as components of vectors in the complex plane
E!1 + !
E2
!1 (x, t) + E
!2 (x, t)
E
=
"
2 cos π/4 − 2i sin π/4
#
+ cos π/2 + i sin π/2 ei(kx−ωt)
(1.414 − 0.414i)ei(kx−ωt)
! 2!
! ! ! ! E!!1 ! !E
! ! ! ! ! ! ! ! Express in terms of an amplitude and phase
!1 (x, t) + E
!2 (x, t)
E
=
1.474 ei(kx−ωt−0.28 rad)
1.21
Irradiance of waves
The oscillations of the electric and magnetic fields are too
fast in an optical wave to be directly measured. We can
only observe the power delivered by the wave
Power per unit area is called irradiance and is proportional
to the square of the (electric or magnetic) field. We
typically ignore the constants of proportionality since we
are often only concerned with relative irradiance.
1
2
Irms = n!0 cErms
2
! T
1
Irms (t) ∝< E 2 (t) >=
E 2 (t" )dt"
T T −t
The irradiance is given by
or in the phasor picture, by
Irms (t) ∝ |E(t)|2 = EE ∗ = E ∗ E
1.22
Irradiance Example
A wave given by
E1 (x, t) = E0 cos(k1 x − ω1 t)
adds with a wave that is expressed by
E2 (x, t) = E0 cos(k2 x − ω2 t)
Which direction are these waves going?
Write an expression for the irradiance of the
sum at x=0 as a function of time
1.23
Irradiance Example
Using the phasor representation
E1 (x, t) = E0 ei(k1 x−ω1 t)
E1 (x = 0, t) + E2 (x = 0, t)
!
"
= E0 e−iω1 t + e−iω1 t
#
$
−i(ω1 +ω2 )t/2
+i(ω1 −ω2 )t/2
−i(ω1 −ω2 )t/2
= E0 e
e
+e
=
or
E2 (x, t) = E0 ei(k2 x−ω2 t)
2E0 e−i(ω1 +ω2 )t/2 cos ((ω1 − ω2 )t/2)
Irms (t) ∝ E ∗ E = 4|E0 |2 cos
!
! ! ! !Irms!(t)!= !4I0 !cos !
!
"
ω1 − ω2
! 2 ! t!
ω1 − ω2
t
2
with
"
I0 ≡
1
n!0 c|E0 |2
2
Does this violate conservation of Energy?
1.24
Radiometry
Many terms are used to describe the properties of
light. Some are subjective others are objective
Color
Wavelength/Bandwidth
Luminous intensity
Luminance (brightness)
Radiance
Irradiance
1.25
Color
A quality of light based on human perception of
the relative stimulation of the red, green and
blue photosensitive cones in the eye
response of rods
response of red cones
response of green cones
What color are the center pieces?
response of blue cones
1.26
Spectral Characteristics
Power spectral density!
“blackbody radiation”
The Fourier transform of the time
dependent power produced by a
I(f)
light source
Wavelength
The wavelength associated with the
LED output
intensity weighted average
frequency of the power spectral
density
Bandwidth
The full-width half-maximum of
the power spectral density
f
I(f)
f
1.27
“Luminous” vs. “Radiant”
Quantities such as intensity, flux,
etc can be preceded by the words
Luminous or Radiant.
Luminous refers to the
amount of a quantity
perceived by human vision,
that is weighted by
frequency to acount for the
spectral sensitivity of the
human eye
Radiant refers to the total
amount of a quantity
radiating as an
electromagnetic wave
The response of a typical human eye to light,
as standardized by the CIE in 1924, and used
as the luminous weighting function
1.28
“Brightness”
The lumen measures brightness of an object. A lumen is the unit
of total luminous flux emitted by a 1 candela light source
(approximately the output of a wax candle) into 1 steradian of solid
angle
Object
luminous
output (lm)
luminous
efficiency
Typical wax candle
13
0.04%
100 W incandescent bulb
800
2%
15 W Compact Fluorescent Bulb
800
8%
15 mW LED
1
10%
10 W Sodium Lamp
1800
27%
1 W Laser at 555 nm
683
100%
1.29
Radiant Quantities and Units
Term
Symbol
SI Units
Electric Field
E
V/m
Magnetic Field
H
A/m
Frequency
f or ν
Hz
Angular Frequency
ω
rad/s
Power or Radiant Flux
P or Φe
W
Irradiance or Intensity
I
W/m2
Spectral Intensity
I(f)
W/m2√Hz
Relevant relations
B=μH
ω=2πf
I=EH
1.30
Irradiance Example
Find an expression for the irradiance of a point
source emitting 1 W of radiant power.
Use your expression for the irradiance to
determine the functional form for the electric
field wave traveling away from this point
source.
1.31
Summary
Optical waves are electromagnetic waves and
have behavior governed by the same laws that
govern all electromagnetic radiation
Unlike other electromagnetic waves, optical
waves are at too high a frequency for the
waveform to be directly observed, and are at
too low a frequency to ionize typical materials
1.32
References
Pedrotti, “Introduction to Optics” Ch 1
Hecht “Optics” Ch 1
1.33