-- Lei Huang, 08/2005 --
Advisor: Dominik Schneble
Scattering of light from sound waves affords a convenient means of controlling the frequency, intensity, and direction of an optical beam.
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Taking particle picture of diffraction process, assume,
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Approaching sound wave (phonons): momentum ħks, energy ħws |
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Incident light wave (photons): momentum ħki, energy ħwi |
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Diffracted light wave (photons): momentum ħkd, energy ħwd |
From conservation law,
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momentum conservation:
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energy conservation:
(frequency shift) |
Some approximations,
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wi>>ws ⇒ wi=wd and |ki|=|kd|=k. |
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⇒
(same as Bragg Diffraction conditions)
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Detailed physical and mathematical processing will give out the diffraction efficiency (Idiffrected/Iincident) dependence on acoustic energy (Iacoustic) as,
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where in our case n is the index of refraction, p is the photoelastic constant of the medium, ρ is the mass density, νs is the velocity of sound in the medium. |
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By defining 
, 
In our experiment, PbMoO4 is used, so n=2.3, vs=3.75km/s, p=0.28, ρ=6.95g/cm3, with M=73 in MKS units. Besides, the wavelength of the laser beam is 780nm. |
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Deflection of optical beams can be achieved by changing the sound frequency while operating near Bragg-diffraction condition.
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 (beam
deflection vs. acoustic frequency)
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VCO (voltage controlled oscillator) is an electronic oscillator specifically designed to be controlled in oscillation frequency by a voltage input. The frequency of oscillation, or rate of repetition, is varied with an applied DC voltage, whilst modulating signals may be fed into the VCO to generate frequency modulation (FM) or phase modulation (PM).
For high-frequency VCOs, like those will be used in our experiment, the voltage-controlled element is usually a varicap diode connected as part of an LC tank circuit. A varicap diode is basically a type of diode of which the width of its depletion layer, or the insulating region in the semiconductor where the charge carriers have been swept away through recombination, varies with applied bias voltage. Therefore the capacitance of varicap diode can also be made to vary. Generally, the capacitance is inversely proportional to the square root of applied voltage.
--definition from wikipedia.com
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By varying the input voltage of the VCO, the amplitude of output AC voltage can be changed. To calibrate their relationship, we connect an oscilloscope to the attenuated output end of VCO, using DC coupling, and finally record the upper calibration curve, which is very useful for the following experiments.
Now connect the AOM to VCO output. First turn off the external power supply, so that the VCO applys no energy into AOM, and thus no acoustic wave is generated. use Photo Diode equipped with attenuator to measure the beam intensity behind AOM, which is regarded as total output beam intensity. Then turn on the control voltage and set it to be 10 volts. 0th diffraction component will remain in position with reduced intensity, while additional orders of diffraction appear. Rotate the AMO within the horizontal plane so that the 1st order diffraction is optimized. Use Photo Diode to measure the beam intensity of 1st order component. Diffraction efficiency of 10 Volts control voltge thus can be calculated by dividing 1st order intensity by total output intensity recorded previously. Repeat this process while turning down control voltage level step by step, and the upper curve can be recorded.
Combining the first two graphs, upper
calibrated curve can be produced. Now let us look at the original
formula. By assuming the acoustic wave intensity equals to input power,
which is
, and also considering
,
the original formula can be modified as
,
where
.
This exactly corresponds to the parameters placed on the upper graph, thus gives us a way for data fitting.
Fitting Formula:
. The two curves are fairly close
to each other, which proves the desired relation.
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By moving the AOM with in a small range in the horizontal plane, the angular dependence of diffraction efficiency can be measured. It features a symmetric distribution, and fits well with Gaussian shaped profile as indicated.
In principle, such a dependence should bear a δ-function profile, providing that a perfect crystal is presented. This is because if path difference between rays scattered by first two planes differs slightly from an integral number of wavelengths, then another plane deep within the crystal will scatter a ray exactly out of phase with the ray from the first plane and would completely cancels one another. Similarly, the rays scattered by subsequent plane from this one will cancel scattering rays from second plane, and so on and so forth. But such a "destructive interference" mechanism will broke once the crystal is so small that the required plane does not exist, and complete cancellation of all the scattered rays will not result for small deviation from strict Bragg condition. Detailed calculation of this "broadening effect" will give out a Gaussian Shape Distribution, and the width of diffraction peak is a measure of crystal size.
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By tuning the voltage applied on VCO to change the acoustic frequency, we can measure the diffraction angle displacement, as compared with theoretical prediction. The result features a linear relation, with slope of about 0.0002 rad/MHz. While in our experiment, Δθdiff=2Δθbeam, so that based on previous formula, slope should be 2λ/(nνs)=0.00018 rad/MHz.
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As we can see in the picture, the period of the blue interfere pattern is about 12.57 ns (88 ns/7 periods), i.e. 79.55 MHz. This value is in accordance with the RF frequency recorded with RF frequency counter of 80 MHz to within the experiment reading error.
In summary, all the desired relations of optic-acoustic interaction, as well as several other calibrations, are measured and fit theoretical expectation to a favorable extent. Also we can see that AOM indeed provide a very convenient approach for tuning the laser beam frequency, without losing much of its energy.
