LAB 8 TEMPORAL COHERENCE
FALL 2009
Objective
Background
(1)
To understand the concept of coherence as applied to interference and the relationship
between the temporal and spectral properties of light.
(2)
To measure the power spectral content of a He-Ne laser.
So far, we have been assuming that the laser is purely monochromatic. This week we depart from
the assumption that the laser’s power spectral density (S(ν)) is a delta function. Instead, we
realistically consider it to be a function with a finite width, such as a Gaussian or Lorentzian. This
has profound implications on the propagation of optical waves. We will study the effect of this
finite spectral width on the ability to form a stationary interference pattern.
PRELAB QUESTIONS
1)
All photodetectors detect intensity, which is the time average of the magnitude squared of the field.
(Refer to your lecture notes or text.) Let the incident field on a photodetector be written as
u(t )  e j 1t  e j 2 t
(Sum of harmonics.)
where 1  2 << 1 or 2.
ut u * t  )
i)
Determine the frequency of the intensity on the photodetector. (Hint: determine
ii)
If 1 and 2 are adjacent modes of a multimode laser, determine the beat frequency as a function of
the size of the cavity. (Hint: the mode spacing (Hz) on a laser is co / 2d where d is the cavity
length.)
Many He-Ne lasers have a power spectral density S() that can be modeled (simply) as
2)

  ( )  (  m r )
S( ) 
m
In this equation, r = c/(2d) is the free spectral range (i.e. mode spacing) of the cavity. The function
() is called the lineshape and is modeled for the He-Ne laser as
 ( ) 
 / 2
(  vo ) 2  ( ) 2
where  is the operating frequency of the laser and  measures the frequency spread. (The
function is called a Lorentizian.)
i)
Using the shift property of Fourier transforms and an appropriate Table from the Appendix,
determine the Fourier Transform of . (Hint: this is two exponentials back-to-back at the origin.)
ii)
Using the results of part i), determine the coherence function G(). (Hint: S and G are Fourier
transform pairs.)
1
Using the grating equation and differentials, determine the relationship between  the differential
angular difference and the differential wavelength difference as a function of the grating period
. Assume the incoming light is at normal incidence.
3)

l
Grating
d
(i)
(ii)
If a screen is placed a distance d from the grating, determine the relationship between the
differential difference on the screen l and .
Using =c and differentials, show that / = /and thus determine d in order to resolve a
frequency  = 3 Ghz for a grating with m at = 632 nm. (This will be a large number.)
PART I
SIMPLE GRATINGS
Discussion
A basic method to determine the spectral content of a signal is to use a diffraction grating. Here we
will use reference wavelengths to determine the period of a grating and then use this information to
determine the wavelength of a laser pointer.


Screen

Laser Source
Undiffracted
Reflection
(DC)
Grating
(Period
)
Alignment
1)
Set up the tunable He-Ne (6328Å) on a stand and diffract the beam off the grating onto the screen
as shown in the figure above. Set the grating about 50 cm away. Be sure that the screen is parallel to
grating as shown in the figure, and that the beam has normal incidence onto the grating.
2)
Locate the DC spot of the diffracted field and carefully retro-reflect this spot back to the exit port of
the laser. You are now squarely aligned to determine the diffraction angle.
Experiment
1)
Using the line at 6328Å, measure and record the diffraction angle. Also record the estimated
accuracy of the measurement (i.e in degrees) and include a simple error analysis in your answers.
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2)
Repeat for all of the other HeNe lines. Plot the sine of the diffraction angle vs. wavelength and
calculate the fundamental frequency 1/ (lp/mm) of the grating. The other HeNe wavelengths are
612nm, 604nm, 594nm, and 543nm. The green line may not be available for the laser on your
bench.
3)
Now, replace the HeNe with the laser pointer. Using your data from part 2 and the same diffraction
grating, determine the wavelength of the pointer.
PART II
MODE SPACING OF A HeNe USING MODE BEATING ON A
PHOTODETECTOR
Discussion
Most He-Ne lasers 'lase' in several modes separated by the free spectral range (FSR) of the FabryPerot cavity. These cavity modes (frequencies) are all present in the laser beam and will beat at a
receiver. This induces a sinusoidally-varying current, whose frequency spectrum can be displayed
on an electrical spectrum analyzer.
Experiment
Couple the laser light into the fiber using the lens and fiber mount, as shown in the figure. The fiber
should be disconnected from the receiver, and you should hold the fiber up to a white piece of
paper for alignment. Once you have successfully coupled sufficient light into the fiber, connect the
fiber output to the receiver. Send the output of the receiver through the electrical amp (built into
the receiver) and then into the electrical spectrum analyzer. Find the RF frequency resulting from
the beating of two adjacent cavity modes. What is the laser FSR? Center this signal with 3MHz
span. Explain the spectrum you see.
Focusing Lens
Fiber Cable
Laser
Receiver
ESA
PART III
S() FOR AN HE-NE LASER USING FABRY PEROT INTERFEROMETER
Back
Front
Controller
Output
Amp Output
PD Input
Scanning F-P
Trigger
Ch1
Trig
Oscilloscope
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HeNe
Discussion
From question (3) (ii) of Pre-lab we note that a very long length d is required to determine the
frequency structure of a He-Ne laser using a grating. Thus, in order to see the structure in more
detail we will use a Fabry-Perot interferometer.
Alignment
Set the experiment up as shown in the figure above. The TA will help you align the laser into the
interferometer. If you have not done this before, it is easier to show than to explain. This will take
some time. If it was just turned on, the HeNe will require at least 10 minutes to gain thermal
stability. Until stable, you will observe the spectrum drifting around quite a bit. This is not a
trigger issue on the scope.
Experiment
1)
Once the interferometer is aligned, use the controller to produce 1 free spectral range over the entire
scope trace. Calibrate the Fabry-Perot (MHz/ms) knowing that the specification for the free spectral
range (fsr) is 1.5 GHz. The TA will help you if you are confused about this.
2)
Using this calibration, determine the mode spacing for the laser.
3)
Save a copy of the data if this is available.
4)
Turn the laser off for a few minutes and then turn it back on. Set the scope to infinite persistence
(this is the ‘Autostore’ button on HP scopes.) and explain what you see on the trace. As the lines
move, use the cursors to determine the Full Width Half Max (FWHM) of envelope that the modes
sweep out. This envelope is the gain profile of the He-Ne.
PART IV
G() FOR A HE-NE LASER
Discussion
Coherence may be considered the ability of two beams of light to interfere. When two beams
interfere, the intensity of the sum is greater (or less) than the individual intensities. The degree of
coherence can be determined by the contrast or visibility of the fringes.
Experiment
1)
The basic set-up shown in the figure will be provided, and the TA will show you how to use the
setup if you don’t remember from Lab 2. Set the distance from M1 and M2 to the beamsplitter to be
equal. Using only the tilt adjustments on M1, try to produce as few interference fringes as possible.
(i.e one fringe over the screen) Sketch the intensity pattern you achieve. Explain why it is not
uniform.
2)
Adjust M2 until the fringes are vertical and linear. Move M1 in increments of 5 cm and take 8
readings until you reach 40 cm. Measure the following quantities using the cursors on the scope.
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Laser
Scope
Camera
TV
Mirror M1
Spatial
Filter
1" Turning
Mirror
Rail
Collimating
Lens
Iris
Cube
Beamsplitter
Mirror M2
i)
ii)
iii)
iv)
The relative intensity of arm 1 (block arm 2) (I 1).
The relative intensity of arm 2 (block arm 1) (I2).
The maximum value of the intensity of the sum (I max, from the fringes)
The minimum value of the intensity of the sum (I min, from the fringes)
Be sure that the measurements are made at the same place on the trace (Why?) These values will allow you
to calculate the relative visibility function and thus the coherence function G(). Note: you only have to
perform measurements i) and ii) once, while iii) and iv) have to be done for each mirror position.
4)
We will provide you with a complete data set for I max, Imin, I1, and I2 for 40 points via the class Web site. (It
will take you too long to do this yourself.) From this data, plot out the magnitude of the normalized
coherence function |g12|.
QUESTIONS FOR WRITEUP
1)
Instead of measuring the distance with a meter stick or a tape measure, we could have used an array
of 256 detectors each of 50.8 to determine the period of the grating.
i)
Determine the distance from the grating to the array detector in order for the array to intercept all first
order components of the He-Ne.
ii) For this distance and assuming the error l is half the size of the individual array element, how far
must one place a screen in order to achieve the same accuracy as the array detector?
2)
In Part III, why did the spectrum flop around when you turned the laser back on?
3)
Compare the free spectral range of the He-Ne laser measured from the coherence function, the ESA,
and from the Fabry-Perot. Try to explain any differences between the three measurements.
4)
Following a procedure similar to that in Lab 1, calculate magnitude of the Fourier transform of G()
from Part IV, which is S(ν), and compare with the spectrum measured in Part III. Be sure to correctly
get the scaling between t, , and nn where nn is the total number of points in the trace, to be sure
that the relative spectrum is correct.
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