Download Basic Optics

Basic Optics
Laura Kranendonk
ERC, UW – Madison
2004
Table of Contents
Basic Optics
1. Introduction…………………………………………………
2. Properties of Light…………………………………………..
a. Color………………………………………………
b. Spectral Power………………………………………
c. Polarization………………………………………….
3. Optical Fundamentals……………………………………….
a. Fermat’s Principle……………………………………
b. Snell’s Law………………………………………….
c. Fresnel Equations…………………………………….
d. Etendue………………………………………………
e. Coherence……………………………………………
f. Scattering…………………………………………….
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2
4
4
5
6
7
9
11
13
Optical Laboratory Toolbox
1. Introduction……………………………………………….
2. Creating Light…………….……………………………….
a. Lasers……………………………………………..
b. LEDs……………………………………………..
3. Directing Light……………………………………………..
a. Lenses/Mirrors…………………………………….
b. Fibers……………………………………………..
4. Controlling Wavelength…………………………………….
a. Diffraction Grating………………………………….
b. Etalons……………………………………………..
c. Harmonic Generation………………………………
5. Measuring Light……………………………………………..
a. Detectors……………………………………………..
b. Cameras/Dyes………………………………………
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Appendix
1.
2.
3.
4.
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29
NA
30
SI Prefixes……………………………………………..
Laser Classifications………………………………………..
Corning SMF-28 Optical Fiber Product Data sheet………
Helpful websites……………………………………………..
Basic Optics
Introduction
In 1960, the first Ruby laser was built [1], opening the door for many new optical
inventions and research topics. Mechanical engineers in particular are currently working
on optical methods for measuring fluid and gas properties, measuring mechanical
stresses, and manufacturing techniques. In order to understand these new developments,
a basic understanding of light and light properties is required. These notes are meant
only as a review of light topics in order to understand principles commonly used in the
mechanical engineer’s optical laboratory. The physics behind the materials being
presented are generally covered in introductory physics courses.
Properties of Light
Visible light is a small part of the electromagnetic spectrum. Electromagnetic waves
have electric and magnetic fields perpendicular to the direction of motion. They are able
to travel through a vacuum or a medium. There are several properties commonly used to
describe light such as color (wavelength, wavenumber, frequency, energy), and power
(intensity).
Figure 1.1: Electromagnetic spectrum, http://hyperphysics.phy-astr.gsu.edu/hbase/ems1.html
 ,
Wavelength (): measured in nanometers (nm), micrometers (m), or angstroms ( 
 = 0.1 nm), the distance from one peak to the next.
1
Frequency (f): measured in Hertz (Hz, 1Hz = 1s-1). Frequency is the inverse of the time
it would take a wave to travel 1 wavelength. To calculate frequency from wavelength:
c  f
Speed of light in a vacuum (c)  3 x 108 m/s
(1.1)
Wavenumber (): measured in inverse centimeters (cm-1). The wavenumber is how
many waves fit in the distance of 1 cm.
1
Energy (E): measured in J/mole, J/photon, or electron volt (eV, 1eV=1.6 x 10-19 J per
photon or per mole of photons). Energy of the wave can be calculated directly from the
wavelength or frequency using the following equation:
c
E  hf  h
(1.2)

Planck’s constant, h=6.626 x 10-34 Js.
________________________________________________________________________
Example 1.1:
a. Consider a 1550 nm wave. What are the frequency, wavenumber and
energy (in a vacuum)?
Frequency:
f 
 3 *10 8 m   1   1nm   1Hz 
 * 
 
 1.94 *1014 Hz
 *  9  * 

1
 
s
1550
nm
10
m


 

 
 s 
c
 
= 194 THz
Wavenumber:
1   1nm   1m 

 *  9  * 
 = 6451.61cm-1
 1550nm   10 m   100cm 
 
Energy:


1

E  hf  6.626 x10 34 Js * 1.94 x1014  = 1.285x10-19J/photon
s

b. Consider the diagram below. What is , , and f?
P
[mW]

600
700
 [nm]
When calculating differences, it is important convert into common units
before taking the difference.
 = 700nm-600nm = 100nm
 1   1   1nm   1m 
 = 

  *  9  * 
 = 2380.95cm-1
 600nm   700nm   10 m   100cm 
2
(Note: realize that directly converting 100nm to cm-1  2380.95cm-1)
1
1   3x10 8 m   1   1   1nm   1Hz 
 * 
f  c *     

  *  9  * 


s
600
nm
700
nm



  10 m   1 



2 
 1
 s 
 
= 7.14x1013Hz
(Note: again, converting 100nm to Hz does not equal 7.14x1013Hz)
________________________________________________________________________
Spectral Power: measured in W/nm (power per wavelength). Light bulbs have much less
spectral power than lasers since light bulbs emit many wavelengths – this is known as
broadband light, as opposed to laser light, which is essentially a single wavelength.
Laser  10 W/nm
Light Bulb  0.2 W/nm
P
400 nm

700 nm
Figure 1.2: Spectral power comparisons of lasers versus light bulbs.
________________________________________________________________________
Example 1.2:
The diagram in figure 1.2 shows the difference in spectral power of a 1
mW laser pointer versus a typical light bulb. Considering that an average
laser pointer has a bandwidth of 100MHz (this is f) centered at 650 nm,
compute the spectral power.
First, we need to convert the bandwidth (the wavelength range of the
laser) into nm:
1
1


f  c * 


   0.5 *    0.5 *  
f  100MHz
  650nm
Then,  = 0.0001nm
Then, we can compute the spectral power:
P
1mW
=10 W/nm
P  in 
 0.0001nm
________________________________________________________________________
3
Polarization: When the electromagnetic waves composing a light beam vibrate in the
same direction, the light is said to be polarized. Polarized light is generally classified into
2 groups depending on how the electric waves are aligned with the plane of incidence.
The plane of incidence is the plane composing the incident, reflected, and transmitted
rays. Transverse magnetic (TM), parallel (||), P, and O are all ways to refer to waves that
are polarized such that the electric field is in the plane of incidence. Transverse electric
(TE), perpendicular (  ), S, and E are all ways to refer to polarization perpendicular to the
plane of incidence. The polarization of light can affect many aspects of optics. Materials
can have different properties for different polarizations, such as indexes of refraction
(birefringents) or reflectivity. In addition to these polarized cases, light can be randomly
polarized (also called natural light), or circularly or elliptically polarized.
When the electric field
(E) is parallel to the
paper (as drawn), it is
called: TM, P, O, or ||.
The parallel lines are 1
representation of this
polarization.
Incident ray
Reflected ray
Transmitted ray
Figure 1.3: Parallel polarization diagram.
When the electric field
(E) is perpendicular to
the paper plane, it is
called: TE, S, E, or .
The dots are 1
representation of this
polarization.
Incident ray
Reflected ray
Transmitted ray
Figure 1.4: Perpendicular polarization diagram.
Optical Fundamentals
In this section, fundamental optical principles will be reviewed in order to understand
light behavior. Once these basic principles are understood, complex optical applications
and tools can be studied. Therefore, it is very important to have a good understanding of
these fundamental principles.
4
Fermat’s Principle
Fermat’s principle states that light will take path with the shortest travel time to go from
one point to another, as shown in figure 4.
Figure 1.5: Illustration of Fermat’s principle, light will travel in a straight line from one point to another
since a straight line will be the quickest path.
Fermat’s principle can be used to show the behavior of reflection. Depending on the
incident angle of the light onto a surface (say a mirror), light will only reflect at that same
incident angle. Therefore:
i  r
(1.3)
1
2
1= 2
Figure 1.6: Fermat’s principle on a mirror. The beam from the dashed path is impossible because it is a
longer travel time than the solid line path.
Fermat’s principle can also be used to show how a lens focuses light. In order to
understand this phenomenon, it is important to know that light travels at different speeds
depending on the index of refraction (n) of the material. Glass typically has an index of
refraction of about 1.5, whereas air has an index of refraction of essentially 1 (c = speed
of light in a vacuum). The speed that light travels through a medium is:
c
vlight 
(1.4)
n
In figure 6, it can then be deduced that light will travel from point A to point B in the
same amount of time (and therefore the shortest amount of time) for all of the lines.
(Hint: the thickest part of the lens has the overall shortest path length, but since the light
will travel slower in the glass, the overall time to travel from A to B will be the same for
the centerline).
5
A
B
Figure 1.7: Light pathways through a lens, point A and B are at twice the focus length of the lens.
Snell’s Law
Snell’s law predicts the direction of light as it travels through different mediums. Snell’s
law can be stated by the following equation where n is the index of refraction, and i, t,
and r stand for incident, transmitted, and reflected.
ni sin  i  nt sin  t
i
(1.5)
r
nair = 1
nglass = 1.5
t
Figure 1.8: Diagram of Snell’s law, showing light traveling from air to glass.
Total internal reflection can occur when light in a high index medium reaches a low
index medium (ni > nt). In these conditions, when t=90o, the corresponding incident
angle is called the critical angle. Any incident ray at the critical angle or higher will
cause total internal reflection, meaning all of the light is reflected and none is transmitted.
This will become important when dealing with fiber optic cables.
6
______________________________________________________________________________
Example 1.3:
a.
Consider a beam traveling from air (n=1) to glass (n=1.5) at a 30o
incident angle. Sketch the system, calculating the reflection and
transmission angles.
30o
30o
nair = 1
nglass = 1.5
19.5o
 i   r = 30o (Fermat’s Principle)
ni sin  i  nt sin  t (Snell’s Law)
ni  1; nt  1.5; i  30 o
 1

* sin 30 =19.47o
 1.5

 t  sin 1 
b.
What is the critical angle (onset of total internal reflection) for a
glass/air interface? Zinc-Selenide/air (n=2.5)?
To find the onset of total internal reflection (TIR), set the transmission
angle equal to 90o. Therefore:
Glass:
n

 1.5 
i  sin 1 t sin 90  sin 1 *1 = 41.8o
1 
 ni

Zinc-Selenide:
 nt

 2.5 
sin 90  sin 1 
*1 = 23.58o
n
1


 i

______________________________________________________________________
 i  sin 1 
Fresnel’s Equations
The amount of light that is transmitted or reflected can be determined by the Fresnel
equations. The ratio of reflected to incident power (or the fraction of power reflected)
depends on the polarization of the light, and can be written as:
7
 n cos  i  ni cos t
R||   t
 ni cos  t  nt cos i
 n cos i  nt cos t
R   i
 ni cos i  nt cos t







2
(Parallel polarized light)
(1.6)
2
(Perpendicular polarized light)
(1.7)
2
 n  ni 
 when i=90. R is also called the reflected
which can be simplified to R   t
 ni  nt 
power coefficient. If no light is absorbed in the material, transmitted power can then be
calculated by initial power multiplied by:
T  (1  R)
(1.8)
________________________________________________________________________
Example 1.4:
a.
If a 1mW laser beam is directed through a glass microscope slide,
what is the transmitted power? (Assume there is no power absorption in
the glass)
Light detector,
able to measure
power of light
beam.
air/glass interface
1mW laser
glass/air interface
Using the Fresnel equations, it is possible to calculate the reflected power
coefficient for both the air/glass interface, and the glass air interface. It
turns out that these are the same:
2
 n  ni 
 0.5 
  
R   t
  0.04
 2.5 
 ni  nt 
Since there is no absorbance, the transmitted power coefficient is then just
1-R. Therefore, 96% (1-0.04) of the power is transmitted, or 0.96mW.
2
b.
What would be the transmitted power if there were 50 microscope
slides between the laser and detector? (Assume an air gap between each
slide)
Realizing that at each interface, 96% of the power is transmitted, and there
are a total of 100 interfaces, it can be deduced that:
T  (0.96)100  0.0169
Therefore, 1.69% of the original power or 0.0169mW is transmitted.
________________________________________________________________________
8
Reflection versus incident angle (air-glass interface)
1
Reflectance
0.8
0.6
Brewster’s
angle=56.4o
0.4
R
0.2
R||
0
0
20
40
60
80
100
Incident Angle
Figure 1.9: Reflection versus incident angle for an air/glass system.
Equations (1.4) and (1.5) can be plotted versus incident angle, as shown in figure 1.9. At
56.4o, there is no reflection for one of the polarizations. This is a special incident angle
known as Brewster’s angle. By manipulating the incident angle, it is possible to polarize
unpolarized light.
Example 1.5:
If there were the same 50 slides as in example 1.3, but now the incident
angle was at Brewster’s angle, what would the transmitted power be?
(Assume the light is unpolarized, and equal amounts of perpendicular and
parallel-polarized light)
Answer:
At Brewster’s angle, none of the parallel-polarized light will be reflected
(100% transmission, or 50% net transmission). From equation 7 (or from
the graph in figure 8), the reflection for 1 air/glass and glass/air = 0.149.
Therefore, the perpendicular polarized light transmits 85.1% (1-0.149) at
each interface, so over all 100 interfaces, 0.851100=9.84x10-8 is transmitted
(essentially 0).
Overall, the total transmission is then 50%.
Etendue
Etendue, also known as the parameter product, can be thought of as the quality of light,
and is quantified by size multiplied by divergence. The smaller the etendue, the better
collimated the light can be. Etendue is the optical equivalent of entropy; at best it will
remain constant, however it will never improve. Figure 1.10 shows how etendue is
calculated and graphically represented.
9
2
1
a2
a1
2
Etendue = best case:
a1sin(1)=a2sin(2)
Light source
f
1=tan-1(a2/f)
2=tan-1(a1/f)
Figure 1.10: Graphical representation of etendue.
The smallest etendue possible is called the diffraction limit. The diffraction limit is
wavelength dependent, and can be calculated from the following equation, where d is the
“spot size”, the diameter of the light source:

d *  half  
(1.8)

The diffraction limit therefore leads to the minimum spot size that a beam could produce
(rather than an infinitely small point). Lasers are able to produce light that is near
diffraction limited. A genuine increase in etendue is called beam steering.
Example 1.6:
a. Light at 610 nm, leaves a laser at the diffraction limit. The
laser has a 1m diameter emitting area. What is the divergence
angle of the laser?

d *  half 

 610nm 1
1m
10 6 m 

 = 11.125o
 half  
*
* 9
*

1

m
1
m
10
nm


b. If the laser beam were collected in a 1 cm diameter lens at the focal
length, how large would the projected spot be 10 km from the lens?
First, calculated the divergence angle leaving the lens:
r1 sin 1   r2 sin  2 
r1  1m   0.5m
2

10
1  22.25o
r2  0.5cm


 0.5m * sin 22.25 o
 r1 sin(1 ) 
1m
100 cm 
  sin 1 
= 0.002o
* 6
*


r2
0.5cm
1m 
10 m



 2  sin 1 
Next, using trigonometry, figure out the size of the spot:


r3  L * tan  2   10000 * tan 0.002 o = 0.379 m
A   * r32 = 0.45 m2
Coherence
When 2 or more light beams are superimposed on each other, they have the potential to
be either coherent or incoherent. When coherent light has the same peaks and valleys, it
constructively interferes, whereas if a peak lines up with a valley, there is destructive
interference, which weakens the signal.
Destructive Interference
Constructive Interference
1.5
2
0
0
0
0
-1.5
250
-2
Incoherent Light
Figure 1.11: Examples of destructive and
constructive interference, and incoherent light.
1.2
0
0
-1.2
When waves of different wavelengths are traveling along the same path, it is useful to
know at what length the waves become incoherent. This ‘coherence length’ can be
calculated with the following equation:
Lc 
2

(1.9)
11
Example 1.7:
Two 10 mW lasers emitting very nearly the same color are arranged as
shown below. The top laser is a free-running laser, and the bottom is a
stabilized laser; the stabilized laser has a significantly smaller linewidth.
The ‘full width half max’ (FWHM) of the top laser is 0.1 cm-1, and the
FWHM of the lower laser is 300 kHz. FWHM is the bandwidth at half the
maximum, since most lasers and other optical sources produce waves on a
Gaussian curve FWHM is a good approximation of bandwidth.
Detector
Lasers:
Free running
beam splitter
mirror
stabalized
30 cm
Free running laser: fr = 700.01 nm, FWHM = 0.1cm-1
Stabilized laser: s = 700 nm, FWHM = 300 kHz
a. What is the coherence length of the 2 lasers?
Free running laser:
1m  
1
 100cm
cm 1 fr * 
* 9

10 nm    fr  0.5 *  fr
 1m
 
1

    0.5 * 
fr
  fr




fr = 0.0049 nm
Lc , fr 
2 700.012 nm 2
1m
= 0.1 m

* 9

0.0049nm 10 nm
Stabilized laser:

1
 1000 Hz 
kHz * 
  c * 
 1Hz 
  s  0.5 *  s
 
1
  
   s  0.5 *  s



s=4.9x10-7 nm
Lc , s
s 2
700 2 nm 2
1m


* 9
= 1,000 m
7
 s 4.9 x10 nm 10 nm
b. The detector is not reading a signal. Why could this be?
12
Answer:
Since the coherence length in the free-running laser is shorter than the path
it is traveling, the light becomes incoherent, and therefore cannot align
with the stabilized laser.
Scattering
When light hits molecules, it can become scattered in several ways. When light is
scattered by very small molecules (d << ) elastic (Rayleigh scattering), inelastic with
increased energy (Anti-Stokes Raman scattering), or inelastic with decreased energy
(Stokes Raman scattering) scattering can occur. When light is scattered by larger
molecules (d>>), Mie scattering occurs.
The incident power, density of molecules, volume of interest, and collection angle of the
optics can determine the scattered power for each type of scattering. The equation for
scattered power is as follows:
 Ps 
  
   nL
(1.10)

  
 PI 
 Ps

 PI

 = ratio of scattered power to incident power



cm 2
  
=
differential
scattering
cross-section




  
 molecules * Sr 
 = solid angle of collection [Sr], 1 Steradian = 4*radians
n = molecular density
(at standard temperature and pressure, air has a molecular density of:
2.7*1019 molecules/cm3)
Rayleigh:
2
  
2 ( n  1)

4



2
N o 4
   Rayleigh
(1.11)
 cm 2

  
 8.4 *10 28 



   Rayleigh
 molec  Sr  N 2 at 500nm, STP
for eqn. (1.11), n = refractive index of molecule
No = number density at STP
This inverse relationship with wavelength can be used to explain why during the day, the
sky is blue and not red. As the sun sets, the sun’s rays travel through more of the
atmosphere, allowing more scattering, causing the sunset to be a redder color.
Raman:
  
  
 10 3 



   Raman
   Rayleigh
13
Mie:
2
  

  2d
   Mie
d = diameter of scattering molecule
Wavelength changes due to Raman scattering can be calculated by the following
equations:
E
E  E;   '
Anti-Stokes Raman:
(1.12)
h
E
E  E;   '
Stokes Raman:
(1.13)
h
Example 1.8:
A 1 W laser at 500nm is directed into air at room temperature and
pressure. If a detector could pick up 10% of the area of a sphere, and 1 cm
long, how much power would be detected due to Rayleigh scattering?
L
Pin
Detector
Answer:
 Ps 
  
   nL


   Rayleigh
 PI 
n=2.7*1019 molecules/cm3
 = 0.1*4 = 1.26
L=1 cm

cm 2
  
 28 
 8.4 x10 



   Rayleigh
 molecule * Sr 
Ps = 2.86x10-8 W
Conclusions
In this section, we have developed the tools necessary to understand phenomena dealing
with optics. With these fundamental principles, we can now move ahead to learning
about the instruments used in optical laboratories, and other optical applications.
14
Optical Laboratory Toolbox
In the optical laboratory, it is important to be able to control the direction, wavelength,
and spectral power of light. Using the principles developed in the previous section, it is
possible to do this in many ways. Lasers and LEDs can be used to create light at specific
wavelengths and powers. Lenses, mirrors, and fiber optics can be used to direct light.
Diffraction gratings and etalons are examples of ways to control the wavelength. Finally,
various detectors can be used to measure light properties.
Creating Light
When we think of artificial light sources, the first thing that comes to mind is probably a
light bulb. Light bulbs give off incoherent light over a wide range of wavelengths.
Heating metal filaments creates light in standard light bulbs. When specific wavelengths
or coherent light is required, lasers or LED’s are the usual method.
Lasers – Light Amplification by Stimulated Emission of Radiation
Lasers produce coherent electromagnetic waves at narrow wavelength bands, either
continuously or pulsed. There are three parts to a laser: gain medium, pump, and
feedback. The basic idea behind a laser is that first atoms in the gain medium are excited
to higher than usual energy states. Once enough atoms are excited, a photon causes them
to drop to a lower energy state. In going from a higher energy state to a lower energy
state, the atoms give off another photon at a specific wavelength. These photons cascade
throughout the medium between mirrors, causing more the process to continually repeat.
Finally, one of the mirrors will have only partial reflection, allowing a beam to exit the
cavity (see figure 2.1). Lasers have the potential to have very high power, and could be
dangerous if proper safety precautions are not used. Appendix 3 breaks down the types
of lasers into classifications based on power and potential harmfulness.
Lamp “pump”
Gain medium
Ruby
mirror
mirror
Figure 2.1: Simplified diagram of a Ruby laser. Ruby (Al2O3 + Cr) is the gain medium; lamps are used to
pump the ruby to an excited state with mirrors on either end of the cavity.
Lasers are classified according to the medium that is being excited. The three basic
categories are solid (i.e. Ruby), gas (i.e. He-Ne), and liquid (dye) lasers. The medium
15
determines the wavelength that the laser produces. Gain mediums can be very different,
but essentially they all need to do the same thing. First, a “population inversion” needs to
occur. Individual atoms normally are found in a ground energy state. When 1 electron is
excited and moves to a different orbiting shell, the atom has an overall higher energy
state. Population inversion occurs when there are more atoms in the excited state than the
ground state. An example of a noble gas (Ne) excitation is shown in figure 2.2, and
population inversion is shown in figure 2.3.
Ne
Ne
Ground State
Excited State
Figure 2.2: In naturally occurring neon, there are 2 electrons in the inner shell (K), and 8 in the outer shell
(L). In one excited state, one electron is moved to the third shell (M).
Naturally occurring
Population Inversion
N2
N2
Energy
N1
Energy
N1
No
Population
No
Ground State
Population
Figure 2.3: Population inversion occurs when more atoms in a sample are excited rather than in the
ground (low energy) state.
Thermodynamically, it is very difficult to raise atoms to higher energy states, however
optically it is much easier. The 3 main transitions of photons important to laser
understanding are shown in figure 2.4. Stimulated absorbance pumps the atoms to the
higher energy states. Stimulated emission produces the desired electromagnetic wave.
Spontaneous emission occurs naturally. In a laser, relative to simulated emission,
spontaneous emission is rare.
16
Energy
Transmissions
Stimulated Absorbance
Excited State
photon
Ne
Ground State
Spontaneous Emission
Excited State
photon
Ground State
Stimulated Emission
Excited State
Ground State
Figure 2.4: The three main transitions, stimulated absorbance, spontaneous emission, and stimulated
emission.
Some atoms have metastable states, which are stable, high-energy states. Metastable
states are crucial for lasers because they allows for population inversion. Gain media is
limited to molecules that have metastable states (generally a rare occurrence). As an
atom goes from the metastable state to a lower state, it releases a photon with specific
wavelength. There may be a small range of energies at the metastable state that cause the
laser to have a range of wavelengths. Depending on the application, this can be either
beneficial or not. We will see later in this chapter how to manipulate wavelengths.
Probably the best way to understand how lasers work is to look at an animation from one
of the websites listed in appendix 4, however figure 2.5 gives a general overview of the
process. A list of some available lasers and their wavelengths is listed in appendix 2.
17
Metastable State
State
Ground State
Population Inversion
Stimulated Emission
Figure 2.5: Steps in creating a laser. The gray dots represent atoms as they travel from naturally occurring
ground states, to higher metastable states when stimulated by photons. This figure is simplified for clarity.
In actuality, each atom being stimulated requires 1 photon, whereas only 1 photon total is shown in this
diagram. The photons bounce back and forth between the surrounding mirrors (not shown), allowing the
stimulated emission to continue. The odd atom that went to a higher state is a rare occurrence; it is shown
here to simply to illustration what can happen.
As mentioned before, lasers can be either pulsed or continuous. Pulsed lasers have a
limit to how short they can be based on the Heisenberg Uncertainty Principle, which is
stated as follows:
(2.1)
tE  h
4
where t is the pulse duration, E is the energy (remember E=hf, therefore E is a
measure of linewidth), and h = Planck’s constant (6.626x10-34 Js). A consequence of this
principle is that the pulse duration and linewidth cannot both be small. It follows that:
10.5
(2.2)
 cm 1 
t p  ps 


where tp is the pulse duration in picoseconds and  is the linewidth in wavenumbers.
When equation 2.2 is an equality, the pulse said to be transformed limited.
________________________________________________________________________
Example 2.1:
What is the coherence length of a transform limited, 600 nm pulse of 10 ns
duration? Compare this length to the length of 1 pulse in air.
Answer:
First recall the equation for coherence lengths:
2
Lc 

We can calculate  from equation 2.2:
10.5
10.5
= 0.00105 cm-1
 cm 1 

t p  ps 10,000 ps


18
600nm 
1
   16666.7cm 1

    16666.7cm 1  0.00105cm 1 = 16666.70105 cm-1
1
1
1  
1


  


 = 0.0000378 nm
     16666.7   16666.70105 
2
600 2  1m 
Lc 


 = 9.5 m
 0.0000378  10 9 nm 
The length of the pulse is just speed of sound multiplied by the pulse duration:
 
 1s 
L pulse  c * t p  3x10 8 m *10ns*  9  = 3 m
s
 10 ns 
________________________________________________________________________
LEDs – Light Emitting Diodes
LEDs are similar to lasers, however they use only spontaneous emission rather than
stimulated. Removing the mirrors from a laser gets rid of the stimulated emission. The
main difference therefore between lasers and LEDs is that LEDs produce incoherent
light. Since LEDs are more efficient than light bulbs, and less harmful to the human eye
then lasers, LEDs have found many practical applications such as brake lights and stop
lights.
Directing Light
Once we have light from a light source, we need to be able to accurately control where it
travels. There are several ways to do this. Lenses can be used to collimate or focus light.
Mirrors are used to reflect light. Finally, fiber optic cables are very useful for
transferring light to specific locations.
Lenses
Lenses can be used to either focus or collimate light. A lens can be concave, convex, or
combinations of the 2. Collimated light coming in through a concave lens will focus at
the focal length (f). In general, the Gaussian lens formula is:
 1 1 1
   
f
 s o si
and the magnification from the lens is:
s
M  i
so
(2.3)
(2.4)
19
Bi-convex lens
so
si
Figure 2.6: Diagram for Gaussian lens formula: (1/so) + (1/si) = (1/f)
f
Figure 2.7: Lens special case, a point source at the focal length collimates light. A point source not on the
center plane (yet still at the focal length) will collimate the light at an angle to the center plane.
d
2*f
-2*f
-f
f
Figure 2.8: An image at twice the focal length will give 1:1 imaging.
20
The F-number (referred to as: F/# or F# in industry) of a lens is f/d, as shown in figure
2.8. Other behaviors of lenses can be determined from Snell’s law. Analytically, this can
be quite rigorous, however graphical solutions or ray tracing software programs can
simplify the computational work.
________________________________________________________________________
Example 2.2:
Given an LED with a 100 m diameter emission area, and half-angle divergence
of 20o, what F# lens would you need to have a divergence half angle of a typical
laser pointer, 0.01o?
Answer:
First calculate the minimum diameter of the lens from basic etendue principles:
d1 sin 1, 1   d 2 sin  2, 1 
 2
 2
d1 sin  1, 1 
6
o
 2   100 x10 sin 20  = 0.196 m
d2 
sin 0.01
sin  2, 1 
2

Next, calculate the focal length of the lens, it is probably easiest to understand this
by drawing the system, considering the LED source essentially a point source:
20o
LED
d
f
tan 20 o  
d
0.196m
2 f 
2 = 0.27 m
o
f
tan 20 
Finally, calculate F#:
d 0.197
= 0.73

f
0.27
_____________________________________________________________________
F#
Deviations from theoretically perfect performances in lenses are called aberrations.
There are 6 basic types of aberrations: spherical aberrations, astigmatism, coma, field
curvature, distortion, and chromatic aberrations. Spherical aberrations are caused by lens
shape, orientation, conjugate ratio, or index of refraction purities. Spherical aberrations
will cause the light to not focus to a point (infinitely small), but rather a finite area.
Astigmatism can occur when an off axis object is focused by a spherical lens. The
system will then appear to have 2 focal lengths, since the object will focus at 2 different
spots. When different parts of a spherical lens exhibit different degrees of magnification,
21
blurring of the image will result, which is known as coma aberration. Field curvature
occurs when a lens tends to focus on a curved, rather than flat plane. When image shapes
do not actually correlate with the actual shape of the object, distortion is occurring.
Distortion is usually increased as the object is placed further off the center axis. Finally,
chromatic aberration occurs when different wavelengths have different behaviors through
the lens, causing different colors to focus at different points. It is important to know that
all of these aberrations can be either eliminated or minimized with various techniques
such as using multiple lenses or the use of aspherical mirrors.
Mirrors
Mirrors are made with highly reflective surfaces. Fermat’s principle can be used to
deduce that the reflection angle is always equal to the incident angle. Aside from flat
mirrors, parabolic shaped mirrors are commonly found in optical laboratories. Parabolic
mirrors focus light to a point.
Fibers
A very useful tool in today’s optical toolbox is the fiber. Fibers are essentially drawn
glass (or some transparent medium) with a protective coating. A sample diagram of a
fiber is given in figure 2.9.
Core, n1
Cladding, n2
Buffer
Jacket
Figure 2.9: Diagram of fiber optic cable.
When light is directed into the core at small enough angles, total internal reflection will
occur between the core and cladding interface. The numerical aperture is the sine of the
maximum half angle that can be directed at the core and total internal reflection will still
occur (see figure 2.10). The following equation can therefore be calculated from Snell’
law:
2
2
NA  sin  1   ncore
 ncladding
(2.5)
 2
22
Light within this cone
is guided along fiber
1/2=12.2o
1/2
cr
Core, n=1.5
1/2
Cladding, n=1.485
Figure 2.10: When light is guided into a fiber at an angle less then the half angle, total internal reflection
will occur, improving light transmission.
When the diameter of the fiber core is small enough, waves only can travel in one path.
These fibers are called single-mode fibers. At larger diameters, waves coming in at
different angles can travel in different paths, and are therefore termed multi-mode paths.
The number of modes a fiber has can be determined from:
dNA 

  
# modes = N m  0.5
2
(2.6)
Due to the generally long lengths, attenuation (loss due to scattering and absorbance) is a
critical property for fibers. Figure 2.11 shows attenuations in decibels per kilometer for a
common fiber material, silica. Power transmission for the length of fiber can be then
calculated.
L
T  10 10
(2.7)
where  is the loss in decibels per length(dB/km), and T is power out over initial power.
Appendix 4 is a product data sheet for Corning’s SMF-28 Optical Fiber (SMF = single
mode fiber). Hopefully by now most of the properties are clear.
________________________________________________________________________
Example 2.4:
Using figure 2.11, what is the transmission (ratio of output to input intensity) of a
10 km length of fiber at 850 nm? At 1550 nm?
Answer:
From the figure, at 850 nm, there is approximately 2 dB/km attenuation loss,
T850nm  10
2*10
10
= 1% transmission
at 1550nm there is approximately 0.2 dB/km attenuation loss
23
0.2*10
10
T1550nm  10
= 63 % transmission
________________________________________________________________________
Figure 2.11: Attenuation in fused silica (common material of fibers). From:
http://www.spme.monash.edu.au/teaching/msc3011/elec_&_opt(3).pdf (probably should find a better graph)
Controlling Wavelength
Many times, it is necessary to work with a specific wavelength or to separate light by
wavelength. This is especially useful in spectroscopy applications. There are many ways
to separate wavelength. This section will describe some of the more straightforward
methods: diffraction gratings, etalons, and harmonic generation.
Diffraction Grating
When polychromatic light is directed at a diffraction grating, it is reflected as a sheet.
The waves spread out into the sheet according to wavelength. Diffraction gratings are
mirrors with very precise grooves cut into them. Figure 2.12 illustrates the behavior of a
diffraction grating.
24
 = incident angle
 = reflected angle

1
Subscripts are the
order of the grating
o
-1
blaze
d
Figure 2.12: Diagram of a diffraction grating. The light is incident at angle , and leaves at discreet
angles .
The grating equation is as follows, where m is the order:
(2.8)
m  d sin   sin  
So for different wavelengths, there are different  angles, causing the light sheet. As a
general rule of thumb, the most efficient use of the grating occurs around the blaze
wavelength.
3
2

For:   Blaze   Blaze   50%efficient
2
3

________________________________________________________________________
Example 2.5:
An infrared (=1.5 m) diode laser has an emitting area of 1 m x 1 m emitting
area. A 9 mm focal length lens is used to collimate the laser. The beam is
directed at a 1200 g/mm (grating per millimeter) grating. At what angle does the
1st order diffracted beam come back towards the laser (this is called the Littrow
angle)?
Answer:
First, calculate d:
1
1
d
 8.3 3 x10 7 m
mm
 1000
1200  g
m
 mm
Then, plug into the grating equation:
m  d sin   sin  
knowing:    for Littrow condition, and m=1 (order 1)
  2 sin 
 = 64.16o
________________________________________________________________________


25
Etalons
An etalon can occur whenever there are pairs of flat plates separated by an optically
contacted spacer (such as a laser cavity, or even just a microscope slide). Due to
constructive and destructive interference, etalons only allow certain wavelengths to be
transmitted. Many times etalons are unwanted, however either using angled surfaces or
shifting the incident light to an angle can eliminate them.
c
Hz 
Free Spectral Range (FSR) =
(2.9)
2nd
n = index of refraction, d = distance between parallel planes
 R
Finesse (F) =
(2.10)
1 R
FSR
Bandwidth = FWHM (full width half max) =
(2.11)
F
FSR
Transmission
(%)
FWHM
Wavelength
Figure 2.13: Etalon diagram.
________________________________________________________________________
Example 2.6:
What is the free spectral range and FWHM of a glass (n=1.5) microscope slide, 1
m wide?
Answer:
FSR 
c
= 100 THz
2 *1.5 *1x10 6
c  f    c
f

 s  = 3m = FSR
100 x10 1 
s
3x10 8 m
12
 n 1
R
 = 0.04
 n  1
2
F
 R
1 R
= 0.654
26
FWHM 
FSR 3mm 
= 4.584 m

F
0.654
________________________________________________________________________
Harmonic Generation
Aside from separating wavelengths with gratings or using interference as in etalons, it is
possible to mix wavelengths to get new wavelengths. These processes must satisfy
energy and momentum conservation (however, not photon conservation). Figure 2.14
demonstrations how differential and summation mixing can occur.
Second Harmonic Generation:
1=1064 nm
2=1064 nm
“nonlinear
crystal”
3 = 532 nm
Energy Balance:
Ein = Eout
hc hc hc


1  2 3
Third Harmonic Generation:
Energy Balance:
Ein = Eout
hc hc hc


1  2 3
hc hc hc


3  4 5
1=1064 nm
“nonlinear
crystal”
3 = 532 nm
2=1064 nm
4=1064 nm
5 = 355 nm
Figure 2.14: Illustration of 2nd and 3rd harmonic generation.
________________________________________________________________________
Example 2.7:
Fourth harmonic generation is illustrated in the figure below. Calculate the final
wavelength by performing an energy balance. All of the input beams enter at
1064 nm.
27
Answer:
First, realize the dashed line above can define the system. Then, Ein=Eout:
 hc  hc
4  
 1  out
out = 266 nm
________________________________________________________________________
Measuring Light
Once we finally have light directed in the correct direction and at the desired wavelength,
it is usually important to be able to detect the light. Cameras can be used for visual
detection, and various other detectors can be used to measure power, wavelength,
polarization, and coherence. This section needs more work…
Conclusion
In this section, we have been introduced to some of the basic tools used in an optics
laboratory. Light bulbs, lasers, and diodes can be used to create light, mirrors, lenses,
and fibers can be used to direct light. Gratings, etalons, and harmonic generation are
ways to manipulate the wavelength. Finally, detectors and cameras are used to measure
light intensity. By no way however did this chapter contain a complete list of optical
tools available. In the next section, optical applications will be discussed to realize the
potential of these tools.
28
Appendix 1
Scientific Units
Optics frequently deals with very large or very small numbers (nm, GHz, etc.).
Therefore, it is important to become familiar with the SI prefixes:
Tera
Giga
Mega
Kilo
T
G
M
k
1012
109
106
103
Centi
Milli
Micro
Nano
c
M

n
10-2
10-3
10-6
10-9
Pico
Femto
Atto
p
f
A
10-12
10-15
10-18
Table 1: SI Prefixes
It is also important to remember to always be consistent with units. If an equation calls
for change in wavenumber, but the wave is specified in wavelength (nm), remember to
convert the wavelength into wavenumbers, then calculate the difference (see example
problem).
Appendix 2 – Laser Classifications:
Class I lasers - Lasers that are not hazardous for continuous viewing or are designed in
such a way that prevent human access to laser radiation. These consist of low power
lasers or higher power embedded lasers (i.e., laser printers).
Class II visible lasers (400 to 700 nm) - Lasers emitting visible light which because of
normal human aversion responses, do not normally present a hazard, but would if viewed
directly for extended periods of time. (like many conventional light sources).
Class IIa visible lasers (400 to 700 nm) - Lasers emitting visible light not intended for
viewing, and under normal operating conditions would not produce a injury to the eye if
viewed directly for less than 1,000 seconds (i.e. bar code scanners).
Class IIIa lasers - Lasers that normally would not cause injury to the eye if viewed
momentarily but would present a hazard if viewed using collecting optics (loupe or
telescope).
Class IIIb lasers - Lasers that present an eye and skin hazard if viewed directly. This
includes both intrabeam viewing and specular reflections. Class IIIb lasers do not
produce a hazardous diffuse reflection except when viewed at close proximity.
Class IV lasers - Lasers that present an eye hazard from direct, specular and diffuse
reflections. In addition such lasers may be fire hazards and produce skin burns.
29
Here is another description that relates the laser classifications to common laser types and power levels:
Class I - EXEMPT LASERS, considered 'safe' for intrabeam viewing. Visible beam.
Maximum power less than 0.4 W. This will not cause damage even where the entire
beam enters the eye and it is being stared at continuously.
Class II - LOW-POWERED VISIBLE (CW) OR HIGH PULSE RATE LASERS, won't
damage your eye if viewed momentarily. Visible beam.
Maximum power less than 1 mW for HeNe laser.
Class IIIa - MEDIUM POWER LASERS, focused beam can injure the eye.
HeNe laser power 1.0 to 5.0 mW.
Class IIIb - MEDIUM POWER LASERS, diffuse reflection is not hazardous, doesn't
present a fire hazard.
Visible Argon laser power 5.0 mW to 500 mW.
Class IV - HIGH POWER LASERS, diffuse reflection is hazardous and/or a fire hazard.
Some available lasers and wavelengths
Laser Name
HCN
H2O
Quantum Cascade (QC)
semiconductor
CO2
CO
He-Ne (IR)
HF
Diode
Dye
Nd:YAG
Ti:Sapphire
Alexandrite
Ruby
He-Ne
Argon ion
Violet diode
KrF excimer
ArF excimer
F2 excimer
Xe2 excimer
Type
Gas
Gas
Semiconductor
Wavelength range
744, 676, 545, 337, 311 (m)
120,48,28 (m)
3-20 (m)
Gas
Gas
Gas
Gas
Semiconductor
Dye
Solid state
Solid state
Solid state
Solid state
Gas
Gas
Semiconductor
Gas
Gas
Gas
Gas
9.6-10.6 (m)
5.1-6.5 (m)
3.39 (m)
2.7-3 (m)
0.6-2.3 (m)
0.3-1.5 (m)
1.06 (m)
680-1000 (nm)
700-800 (nm)
694 (nm)
632 (nm)
488-515 (nm)
380-440 (nm)
248 (nm)
193 (nm)
157 (nm)
72 (nm)
30
Appendix 4 – Useful Websites
http://scienceworld.wolfram.com/physics/topics/Optics.html
http://micro.magnet.fsu.edu/primer/lightandcolor/index.html
http://www.howstuffworks.com
http://webphysics.ph.msstate.edu/javamirror/
31