Using Fourier Domain Techniques to Characterize String Instruments, Nano-Scale Quantum Devices, and Human Brain Waves

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Using Fourier Domain Techniques to Characterize
String Instruments, Nano-Scale Quantum Devices,
and Human Brain Waves
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Problems Addressed in Our Researches

• How can we characterize the quality of a string
  instrument independent of human subjectivity?
• How can we evaluate the quality of a quantum
  cascade laser wafer prior to the fabrication and
  testing processes?
• How can we typify the mental health of
  psychiatric patients by looking at brain waves?
• How can Fourier Analysis simplify the evaluation
  process and help to extract key features from all
  the above signals?
• How can these seemingly unconnected subject areas
  be correlated through Fourier Analysis?

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Fourier Transform (Time ? Frequency or Spatial
? K )
x(t)
X(f)
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Fourier Transform
FREQUENCY DOMAIN
TIME DOMAIN
Amplitude
Amplitude
Amplitude
Amplitude
Amplitude
Amplitude
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Fourier Analysis
Inverse Fourier transform is converting frequency
domain signals into time domain signals a time
domain signal can be synthesized by adding
together different frequency components. Individua
l harmonics in the frequency domain can be
selectively added together to synthesize a time
domain signal.
X(f)
x(t)
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Applications to String Instruments

• Using Fourier Analysis to evaluate the quality of
  string instruments
• Time domain signals -gt Frequency domain spectra
• Analyze both static and dynamic spectra (vibrato
  efficiency)
• Ref Real time spectrum analyzer analysis and
  evaluation of string instruments, Acoustic
  Research Institute, The 16th International
  Congress on Sound and Vibration, Kraków, Poland,
  5-9 July 2009.

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Introduction- Modes

• Fundamental Frequency longest wave in a family
  of standing waves determined by the string length
  and resonates with the cavity the original
  pitch.
• Higher (Nth) Harmonic has a frequency that is an
  integer multiple (N) of the fundamental pitch
  (and with a wavelength N times shorter than the ?
  of the fundamental)

The degree to which the harmonics are expressed
depends on the instrument material and pattern
(Chladni patterns for violins).
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Static Spectra Comparison of Different Violins
G-string X-axis in unit of (Hertz), Y-axis in
unit of (dB)

• What to look for
• Presence of harmonics no missing harmonics
• Number of harmonics the more the better
• Narrow peaks no extra noise
• Spectrum of one note at one instance a whole
  family of harmonics is shown

Higher Price gt18,000
Lower Price lt2,000
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Extracting Statistic Data From Dynamic Spectra
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Frame 0
Frame 1
In Frame -1 All harmonic peaks Shifted to higher
frequency compared with frame 0
Total gt50 frames here
9 frames here
First 3 of the 9 frames
Frame -2
Frame -1
Frame 0
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Statistical Data of Frequency Modulated Harmonic
Peaks (Oscillating up and down in Unit of Hz)
(a)
(b)
Single tone data generated from (a) a less
expensive violin and (b) a higher price violin.
Bottom red numbers are the Standard Deviations of
their data above The higher price violin
apparently has a higher FM efficiency
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Applications to Nano-Scale Quantum Devices
(Quantum Cascade Lasers)

• Use Fourier Analysis to evaluate quality of grown
  laser materials (very useful to manufacturers)
• Spatial Domain Structure -gt k(angle) -domain
  x-ray spectra
• Correlate x-ray diffraction patterns to various
  growth problems understand what has happened
  during growths, decide whether to proceed on to
  fabrication
• Ref
• 1. Evaluation of nano-scale quantum devices
  using x-ray diffraction and Fourier analysis,
  Material Research Society Spring Meeting, Paper
  HH8.3, San Francisco, CA, April 13 17, 2009.
• 2. X-ray diffraction analysis of quantum
  cascade lasers, IEEE, LEOS, The 21st
  International Conference on Indium Phosphide and
  Related Materials, paper WP2, Newport Beach, CA,
  May 10 14, 2009.

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Quantum Cascade Laser Applications
Gas sensing and chemical detection for chemical
industry, health, environmental
Infrared counter measure
See-through Imaging for homeland security and
health
Breath Analysis Portable (equivalent) blood
testing, Drug dynamics study
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QCL Processing and Characterization
Testing
Compare X-ray, PL, data with final laser L-I, I-V
results Identify design and growth problems
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Quantum Cascade Lasers X-Ray Analysis and Bragg
Diffraction
Madmax

• Can be used to monitor the growth results of
  Quantum Cascade Lasers (QCLs)
• The gradual change in the superlattice period can
  be seen in the Fourier domain as a reduction of
  the convolution length.
• The degradation of the superlattice periodicity
  will affect the line-shape, width, and height of
  satellite peaks in a x-ray scan diffraction
  pattern.

Bragg diffraction is a Fourier Transformation
from the spatial domain to the k domain
(represented by angles)
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Straining and Hetero-Interface Defects
Compressive Straining Indium Rich In atoms
are larger than Ga or Al ones grown material
has larger lattice constant than that of the
wafer material lattice mismatch between the
wafer and substrate peaks shift left Tensile
Straining the opposite of compressive
straining grown material has a smaller lattice
constant than that of wafer material peaks shift
right Blue Arrow
hetero-interface defects
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Change in Growth Rate
Double Peaks wafer growth rate changes suddenly,
resulting in two different lattice constants
Double Peak simulation using Bede By suddenly
changing to a 2nd growth rate
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Chirping
X-ray spectra of a high-power room temperature CW
QCLs (blue) and X-ray scans from the same layer
structure but with dramatically worse device
performance (red).
Chirping gradually changing chemical
composition asymmetric satellite peaks
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Relaxation in Strain Balanced Superlattices

• Strain Balanced Superlattice layers alternately
  compressed and tensile strained. Overall strain
  is balanced - the average lattice constant close
  to the substrate lattice constant.
• Relaxation Occurs in the thicker layers - layers
  are strained too much, chemical bonds between
  atoms begin to break, degrading the periodicity.

Fig. 1 (a) Simulated layer relaxation with around
0.3 relaxation in some of the thicker InGaAs
layers note how peaks tilt upwards from left to
right (b) 0.3 relaxation in some of the thicker
InAlAs layers note how peaks tilt downwards
from left to right. (c) experimental results of a
seriously relaxed superlattice material. (d)
Simulation - .2 relaxation in some of the
thicker InAlAs layers
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Table of QCL Performance Degradations
No chirping
Parameters for evaluating QCL performance Thresho
ld the lower the better Temperature Range the
larger the better P max the greater the
better, best is to achieve continuous wave (CW)
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Applications to Human Brain Waves

• Use Fourier Transform to evaluate the health of
  human brain waves
• Signals are less periodic (compared with music
  signals and superlattice structures) and more
  noisy (needs averaging and filtering).
• Comparison of brainwaves of psychiatric patients
  to typical brainwaves
• Is the gating effect observed?
• Publication for new results (new gating effect at
  Fourier domain) is in preparation.

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Human Brain Waves Setup

• EEG patterns measured for both normal people and
  psychological patients
• Comparison of excitement from first stimulus with
  excitement from second stimulus for various brain
  wave rhythms.
• Observation of the brains gating effect is
  there suppression of response to repeated audio
  stimuli?
• Time domain EEG waves must be Fourier transformed
  into the frequency domain in order to compare
  power graphs for normal vs. abnormal brain
  functioning.

Rhythm Freq (Hz) Amp(µV)
alpha 8-13 20-200
beta 13-30 5-10
delta 1-5 20-200
theta 4-8 10
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Human Brain Wave Fourier Domain Comparison Graphs
Blue-1, Red-2
Control
Patient
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Discussion and Conclusions
Highly Periodic Signals String Instruments Quantum Cascade Lasers
Symmetry of Satellite Peaks - The more symmetric the better string density and diameter is uniform and constant - The more symmetric the better satellite peaks become asymmetric when chemical composition is not constant
Number of Peaks -The more harmonics - the more colorful and vibrant the sound The more peaks- the better the growth and the more angles there are for good constructive interference the better the growth periodicity and laser gain
Width of Peaks The narrower the better the sound is clear and pure no extra noise -The narrower the linewidth the better- the less defects and the less relaxation damages in strain balanced superlatticess
For less periodic and noisy signals like brain
waves, Fourier analysis is still helpful to
identify patients from normal people. Our new
study results (publishable) clearly show that the
patients frequency peaks of the two auditory
responses are shift instead of aligned (normal).

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Acknowledgements

• Professor Jacob Khurgin, the Johns Hopkins
  University, Department of Electrical and Computer
  Engineering
• QCL Crystal Growers
• Dr. Jiaxing. Chen - Princeton University,
• Mr. Liwei Cheng - UMBC,
• Dr. Xiaojun Wang Adtech Optics Inc.
• Dr. Elliot Hong, Maryland Psychiatric Research
  Center, UMAB

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Questions?
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Laser Growth Problems Interpreted with Madmax
and Bede Ray Simulations
Indium Rich In atoms are larger than Ga or Al
ones, so lattice constant increases, resulting in
lattice mismatch between the wafer and
superlattice compressive straining shown in red
Blue Arrow
hetero-interface defects
Double Peaks wafer growth rate changes suddenly
resulting in two different lattice constants
Double Peak simulation using Bede Rays
Chirping gradually changing chemical
composition asymmetric satellite peaks
X-ray spectra of a high-power room temperature CW
QCLs (blue) and X-ray scans from the same layer
structure but with dramatically worse device
performance (red).