SYSTEM AND METHOD FOR MAGNITUDE AND PHASE
RETRIEVAL BY PATH MODULATION
FIELD OF THE INVENTION
Exemplary embodiments of present invention generally relate to systems and
methods of propagating electromagnetic signals and, more particularly, systems and
methods of magnitude and phase retrieval by path modulation.
BACKGROUND OF THE INVENTION
Spectrometry using continuous wave (CW) tunable sources with narrow spectral
linewidth and long coherence lengths has well-known advantages associated with high
spectral contrast, frequency selectivity and excellent sensitivity. Scanning CW terahertz
(THz) spectrometers are a prime example of this technology. In such systems, phase
stability in the transmitter-to-receiver demodulation processing may be required to obtain
an accurate measurement of the transmitted electric-field intensity and to characterize any
resulting absorption losses from samples in the spectrometer. However, spurious thermal
and mechanical disturbances may undesirably generate variations in path length that
modulate the received field intensity. It would therefore be desirable to design a system
and method of controlling and modulating the path length to improve demodulation signal
to noise ratio and stability.
SUMMARY OF THE INVENTION
In light of the foregoing background, embodiments of the present invention
provide an improved system and method of magnitude and phase retrieval by path
modulation. According to one aspect of the present invention, the system includes a
transmitter configured to transmit an electromagnetic signal through a sample cell
(including a sample medium and a base medium) to a receiver at each of one or more
selectable frequencies, where the receiver is configured to receive the electromagnetic
signal and another electromagnetic signal for mixing therewith. The system also includes
an arrangement located along either of first or second propagation paths of signals to the

transmitter or receiver, respectively, or along each of the first and second propagation
paths, for altering the length of respective propagation path(s).
The system also includes a processor configured to recover an amplitude and phase
of the transmitted electromagnetic signal, and configured to calculate a complex index of
refraction of the sample medium as a function of the amplitude and phase of the
transmitted electromagnetic signal. The processor may be configured to receive a
sequence of samples of the received electromagnetic signal, and Discrete Fourier
Transformation process the sequence of samples. In addition, the system may include a
modulator configured to modulate the electromagnetic signal transmitted by the
transmitter, where the modulator may be configured to modulate the electromagnetic
signal at a frequency (e.g., ωm ), which may be above the 1/f noise region of the receiver.
The arrangement is configured to alter the length of a respective propagation path
such that the difference of the lengths of the first and second propagation paths is altered
at a pre-selected rate during transmission of the electromagnetic signal from the
transmitter to the receiver, and receipt of the electromagnetic signal and the other
electromagnetic signal at the receiver. The arrangement may comprise a pair of
arrangements each of which is located along a respective one of the first and second
propagation paths. To effectuate an increase in the difference in the lengths of the first
and second propagation paths at the pre-selected rate, one of the arrangements may be
configured to increase the length of one of the propagation paths, while the other of the
arrangements may be configured to decrease the length of the other of the propagation
paths. The arrangement may include, for example, a spool and actuator. In such
instances, an optical fiber propagation path may be wound about the spool, and the
actuator may be configured to alter the diameter of the spool, and thereby alter the length
of the respective propagation path.
The processor being configured to calculate a complex index of refraction of the
sample medium may include being configured to calculate a real part n5 and an imaginary
part k of the complex index of refraction. The real part of the complex index of refraction
may be calculated as a function of the recovered phase SAMPLE of the transmitted
electromagnetic signal, and as a function of a recovered phase REF of an electromagnetic
signal transmitted through the sample cell including the base medium but without the
sample medium. Similarly, the imaginary part of the complex index of refraction may be
calculated as a function of the recovered amplitude Es of the transmitted electromagnetic

signal, and as a function of a recovered amplitude E0 of an electromagnetic signal
transmitted through the sample cell including the base medium but without the sample
medium. For example, the real part of the complex index of refraction may be calculated
according to the following:

where X (e.g., ΛTHZ) represents a wavelength of the transmitted electromagnetic signal, Ls
represents a propagation path length through the sample medium, and n0 represents the
free-space index of refraction. And the imaginary part of the complex index of refraction
may be calculated according to the following:

The pre-selected rate may comprise a rate selected as a function of the frequency at
which the electromagnetic signal is transmitted. More particularly, the pre-selected rate
may comprise a rate selected to span one or more wavelengths of the electromagnetic
signal transmitted at a respective frequency over a dwell time. In one instance, for
example, the pre-selected rate may comprise a rate ΩFS selected to effectuate a path length
modulation at a frequency:

In the preceding, nF represents the index of refraction of a propagating medium of the
propagation paths, and SF represents the pre-selected rate. In this regard, the processor
may be configured to DFT process the sequence of samples at a first frequency of interest
ωm- ωFS, and DFT process the sequence of samples at a second frequency of interest ωm +
ΩFS. And as the pre-selected rate may comprise a rate selected as a function of frequency,
the pre-selected rate may comprise a rate selected for each of the one or more selectable
frequencies, where the rate selected for one of the frequencies may differ from the rate
selected for another of the frequencies.
According to other aspects of the present invention, a method of magnitude and
phase retrieval by path modulation is provided. Exemplary embodiments of the present
invention therefore provide an improved system and method of magnitude and phase
retrieval by path modulation. As indicated above, and explained below, exemplary

embodiments of the present invention may solve problems identified by prior techniques
and provide additional advantages.
BRIEF DESCRIPTION OF THE DRAWINGS
Having thus described the invention in general terms, reference will now be made
to the accompanying drawings, which are not necessarily drawn to scale, and wherein:
FIG. 1 is a schematic block diagram of a spectrometer system in accordance with
one exemplary embodiment of the present invention;
FIGS. 2 and 3 are flowcharts illustrating various steps in methods of sweeping a
spectrometer system through a frequency spectrum, according to exemplary embodiments
of the present invention;
FIG. 4 is a graph illustrating the measured noise density spectrum of a photomixer
receiver, according to exemplary embodiments of the present invention; and
FIG. 5 illustrates spectral diagrams illustrating frequency down conversion in the
receiver of exemplary embodiments of the present invention.
DETAILED DESCRIPTION OF THE INVENTION
The present invention now will be described more fully hereinafter with reference
to the accompanying drawings, in which preferred embodiments of the invention are
shown. This invention may, however, be embodied in many different forms and should
not be construed as limited to the embodiments set forth herein; rather, these embodiments
are provided so that this disclosure will be thorough and complete, and will fully convey
the scope of the invention to those skilled in the art. In this regard, reference may be made
herein to a number of mathematical or numerical expressions that may be related by
equality. It should be understood, however, that this equality may refer to an absolute or
approximate equality, such that exemplary embodiments of the present invention may
account for variations that may occur in the system and method, such as those due to
engineering tolerances. Further, although a number of variables may be reflected by
mathematical symbols including subscripts at various instances, it should be understood
that these symbols and subscripts are presented solely for illustrative purposes, and should
not be construed as limiting the scope of the invention. Like numbers refer to like
elements throughout.
FIGS. 1 and 2 illustrate a spectrometer system and method that may benefit from
exemplary embodiments of the present invention ("exemplary" as used herein referring to

"serving as an example, instance or illustration"). It should be understood, however, that
the spectrometer system and method illustrated and hereinafter described are merely
illustrative of one type of system and method that may benefit from exemplary
embodiments of the present invention and, therefore, should not be taken to limit the scope
of the present invention. In this regard, while several embodiments of the spectrometer
system and method are illustrated and will be hereinafter described for purposes of
example, other types of systems and methods of propagating electromagnetic signals may
readily employ the present invention. Moreover, the system and method of the present
invention will be primarily described in conjunction with signals in the THz (or mmW)
region of the electromagnetic spectrum. But the system and method of embodiments of
the present invention may be utilized in conjunction with a variety of other applications,
both within and outside the THz region of the electromagnetic spectrum.
As shown, a spectrometer system 10 of one exemplary embodiment of the present
invention includes a transmitter 12 configured to transmit a beam of coherent radiation
(electromagnetic wave) at a given frequency. The transmitter can comprise any of a
number of different transmitters known to those skilled in the art. In one exemplary
embodiment, for example, the transmitter comprises a photomixer transmitter. In such
instances, the transmitter includes a high-speed photoconductive diode (i.e., photomixer),
which may be pumped with two laser sources 14a, 14b via a beam combiner/splitter 16
and an optically coupled first optical path 18 (e.g., optical fiber). In this regard, the laser
sources may be configured to emit signals with electric fields having offsetting
frequencies at ω1 and ω2 (i.e., Eω1 and Eω2), which at the photomixer transmitter may be
represented as follows:

where Ej and E2 represent the electric-field amplitudes of the beams from the first and
second sources, respectively; and ^and fa represent phase constants introduced by
virtue of propagation of the beams through the first optical path. Also note that
frequencies a>\ and a>2 may be expressed as angular frequencies, or as corresponding
temporal frequencies (f= co/2n).
The inherently quadratic nature of the cross-gap absorption creates a difference
(i.e., transmission) frequency (i.e., a>2 - co\) in the photocurrent induced in the diode of the
transmitter 12, where the corresponding electric field may be represented as follows:


where rjr represents the photomixer transmitter conversion efficiency, CDJHZ = a>2-co\, and
]2T = fa- IT- The transmitter 12 may be coupled to a transmitter bias modulator 20
including a voltage source 22 configured to generate a sinusoidal modulated voltage with
which the photomixer of the transmitter may be biased, the modulator producing an
electric field EM = Vmcos(comt), although it should be understood that the system need not
frequency modulate (at frequency aym) the signal. By locating the photomixer at the
driving point of an antenna, such as a spiral, dipole or slot antenna, the difference-
frequency current is converted to difference-frequency photons. The result is a highly-
tunable, continuous-wave (CW), highly-coherent source of radiation contained in a single
(quasi-Gaussian) spatial mode, and having a transmitted electric field EJM- This
transmitted electric field may be represented as the product of ET and EM, as follows:

In equations (4) and (5), ^represents the sum of §m and some phase delay related to the
photomixer and antenna transfer function, which may be significant and may cause large
signal variability when detected at a receiver. For more information on such a transmitter,
see U.S. Patent No. 6,348,683 entitled: Quasi-Optical Transceiver Having an Antenna
with Time Varying Voltage, issued February 19, 2002.
Thus, the method of one embodiment includes selecting a transmission frequency,
thereafter transmitting a beam of radiation (i.e., source beam) at that frequency from the
transmitter 12, as shown in blocks 42 and 48 of FIG. 2. The transmission frequency can
be selected in any of a number of different manners. To detect a sample based upon a
measured absorption signature, however, the transmission frequency may be typically
selected within a range of frequencies over which the absorption signature is defined. In a
photomixer transmitter, then, the photomixer can be pumped with tunable laser sources at
a frequency a>2, and a frequency a>\ that are selected to thereby select the difference, or
transmission, frequency (i.e., a>2 - &>i).
The beam of radiation from the transmitter 12 may pass through a collimating lens
24 to produce a collimated beam of radiation. The beam may then pass through a sample
cell 26 that may be bounded by reflectors 26a and 26b through which the beam passes,
and that may include a sample medium to be analyzed and a base medium, such as

ambient air. As will be appreciated, the sample and base medium can have any of a
number of different forms through which the beam of radiation is at least partially
transmissive. For example, the sample and base medium can comprise a solid, liquid, gas,
plasma or aerosol. More particularly, in various advantageous embodiments, the base
medium of ambient air may be in gas form, while a sample may be in gas or aerosol form.
As the beam of radiation passes through the sample cell 26, the sample and base
medium in the sample cell absorb at least a portion of the beam, or more particularly at
least a portion of the electric field of the beam. The remaining, unabsorbed portion of the
beam of radiation (i.e., received signal) then exits the sample cell. The sample signal then
propagates to a focusing lens 28, from which the focused signal is picked up or otherwise
received by a receiver 30. This received signal ERP, which may include an additional
phase delay added during propagation of the signal from the transmitter 12 to the receiver,
may be represented as follows:

where L represents the propagation distance between the transmitter and receiver, XTHZ
represents the wavelengths of the signal at the frequency COTHZ, n and k represent the real
part and the imaginary part (extinction coefficient), respectively, of the complex index of
refraction of the sample in the path length L.
The receiver obtains a measurement representative of the received electric field
ERP, as shown in block 50 of FIG. 2. Similar to the transmitter 12, the receiver may
comprise an electric-field detector such as a photomixer receiver (homodyne receiver).
The photomixer receiver may include an antenna configured to receive the electric field
and generate a corresponding voltage in response thereto, which may be directed to a high-
speed photoconductor. The photoconductor is also electrically coupled to a second optical
path 32 for pumping the photoconductor with beams from the same two laser sources 14a,
14b pumping the photomixer transmitter 12. In this regard, the beam combiner/splitter 16
may separate each of the signals from the laser sources into the aforementioned first
optical path 18, as well as another, second optical path (e.g., optical fiber) for pumping the
receiver photomixer. These signals, then, may modulate a conductance of the photomixer
as described by the following:


where 77^ represents the photomixer receiver conversion efficiency, and ]2R = 2R - (J>IR,
IR and 2R representing phase constants introduced by virtue of propagation of the beams
through the second optical path.
The voltage generated by the receiver antenna may be applied to the photomixer
active material, and produce a current through the modulated conductance that is the
product of equations (6) and (7). The difference frequency result of the product is the
down-converted signal current /DOW„, which may be represented as follows:

A corresponding down-converted electric-field (or signal) ER, then, may be calculated as
follows:

where RLoad represents the receiver 30 electronic load resistance. For more information on
such a receiver, see the aforementioned '683 patent.
The down-converted signal current Iuown and/or electric-field (or signal) ER may be
applied to receiver signal conditioning circuitry 34 including, for example, an anti-aliasing
filter 36. The output of the signal conditioning circuitry may then be input to a processor
38, such as for performing digital signal processing operations thereon. In this regard, the
processor can comprise any of a number of different processing devices capable of
operating in accordance with exemplary embodiments of the present invention. For
example, the processor can comprise a computer (e.g., personal computer, laptop
computer, server computer, workstation computer), microprocessor, coprocessor,
controller, a specialized digital signal processor and/ or various other processing devices
including integrated circuits such as an ASIC (application specific integrated circuit),
FPGA (field programmable gate array) or the like.
If the spectrometer system 10 frequency modulates (at frequency com) the signal,
the signal processing operations performed by the processor 38 may include recovering

the amplitude of the down-converted signal ER such as by an analog-to-digital converter
(A/D) 39 direct sampling of the signal at the modulating frequency a>m, and the processor
Discrete Fourier Transformation (DFT) processing of the sampled data. Alternatively, for
example, the spectrometer system may further include a synchronous demodulator such as
a lock-in amplifier (not shown) for further processing the down-converted signal ER. In
this regard, such a synchronous demodulator may include a local oscillator operating at the
modulating frequency com to thereby recover the amplitude of the down-converted signal.
In operation as a spectrometer, the system 10 scans through a number of
transmission frequencies in a range of frequencies, such as by pumping the photomixers of
the transmitter 12 and receiver 30 with tunable laser sources at frequency a>2, and
frequency a>\ that are scanned through a number of frequencies, as shown in blocks 54 and
56 of FIG. 2. For each transmission frequency in the range of frequency, and thus each
beam of radiation having a different transmission frequency, the processor 38 may
measure the amplitude and/or phase of the down-converted signal current Ioown- The
resulting collection of transmissions amplitudes and/or phases, and associated
transmission frequencies, may define a measured absorption or dispersion signature for the
sample in the sample cell 26, from which the sample may be identified, as shown in block
58 of FIG. 2.
As explained in the background section, in certain optical transmission systems
(e.g., spectrometer systems), spurious thermal and mechanical disturbances may
undesirably generate variations in path length that modulate the received field phase and
down-converted amplitude. Exemplary embodiments of the present invention therefore
provide an apparatus and method of modulating the path length of either or both of the
first or second optical paths 18, 32 to at least partially reject path length instabilities that
may produce transmitted signal amplitude errors. The applied path length modulation may
allow recovery of the transmitted signal amplitude during a relatively short dwell time of
the measurement of each spectral point (selected frequency) of the scanning spectrometer.
In this regard, phase modulation by spurious path variations may be mitigated through the
intentional path stretching that spans multiple waves of the transmitted frequency. And
with multiple wave modulation, the full amplitude of the transmitted signal may be
observed and errors in signal amplitude may be suppressed.
According to exemplary embodiments, the spectrometer system 10 further includes
a path length modulation arrangement 40 along either the first optical path or the second
optical path, or as shown, or along each of the first and second optical paths. Exemplary

embodiments may apply path length modulation to either or both of the optical paths, and
in equal or differing amounts, to thereby effectuate a total system path length stretch. In
this regard, when simultaneously applying modulation to both of the optical paths, the
resulting system path modulation or stretch may correspond to the difference of the
modulation applied to the first and second optical paths, and may require contraction
(decreasing the length) of one of the paths as the other path is stretched (increasing the
length).
The path length modulation arrangement 40 may comprise any of a number of
apparatuses configured to dynamically stretch or contract an optical path length. In one
exemplary embodiment in which an optical path includes an optical fiber, the path length
modulation arrangement may comprise a spool about which the fiber may be wound, and
an actuator (e.g., piezoelectric actuator) coupled to the spool configured to stretch the
diameter of the spool and thus the length of the fiber wound thereabout. In such instances,
contraction of the optical fiber may be effectuated by reducing a previously-applied stretch
to the spool and thus the fiber.
According to exemplary embodiments of the present invention, then, before the
laser sources 14a, 14b pump the photomixer transmitter 12 to thereby transmit a beam of
radiation at a selected frequency (see FIG. 2, block 48), a path length rate scale factor 5>
may be selected, such as by the processor 38, as shown in block 46 of FIG. 3. The path
length rate scale factor represents the rate of applying a system stretch (stretch of one or
both optical paths, or stretch of one path coupled with contraction of the other path) during
the dwell time at each frequency sample point of the scanned spectrum (i.e., the amount of
time the system operates at each frequency sample point before moving to the next point).
The path length rate scale may be selected in any of a number of different manners
to effectuate a desired path length modulation, such as in a manner so as to span one or
more waves of the pump signal (at the difference frequency) within the optical paths 18,
32 over the dwell time at each frequency sample. More particularly, for example, the path
length rate scale may be selected as an integer multiple of the period of the pump signal,
such as in accordance with the following:

where a represents a selectable integer multiple (e.g., 3), XTH2 represents the wavelength of
the pump signal at the difference frequency, and D represents the dwell time (e.g., 0.03

sec). Written in terms of the difference frequency/THZ, the path length rate scale may be
selected as follows:

where riF represents the index of refraction of the propagating medium of the optical path
(e.g., approximately 1.5 for optical fiber). Consider for example, an instance in which a =
3,D = 0.03 s,fmz = 650 GHz, and KF =1.5. In such an instance, given c = 3 x 108 m/s, the
path length rate scale factor SF may be selected as approximately 30.77 mm/s.
As relatively low frequencies of the path length modulation may result in increased
noise in the spectrometer system 10, before, as or after the path length rate scale factor is
selected, a transmitter modulating frequency com may be selected so as to elevate the signal
carrier above the 1/f noise region of the receiver electronics, as shown in block 44. This
selection of the modulating frequency may permit the system to at least partially avoid
increased noise at relatively low frequencies of the path length modulation. The
transmitter modulating frequency may be selected in a number of different manners, such
as from analysis of a measured noise density spectrum of the receiver. One example of a
measured noise density spectrum is shown in the graph of FIG. 4. As shown, the 1/f noise
region of the receiver electronics is at approximately 1 kHz. And from this exemplary
noise density spectrum, it may be shown that a transmitter modulation frequency com at or
above 10 kHz may be needed to at least partially avoid excess 1/f noise.
Having selected the path length rate scale factor Sp and transmitter modulating
frequency com, the method may proceed similar to before, including transmitting a beam of
radiation (i.e., source beam) at a selected transmission frequency, as shown in block 48 of
FIG. 3. As the beam of radiation is transmitted during the dwell time of the selected
transmission frequency, the processor 38 may control the path length modulation
arrangement(s) 40 (or more particularly, for example, the actuator(s) of the arrangements)
to stretch and/or contract the first optical path 18 and/or the second optical path 32 to
effectuate a total system path length stretch. In such an instance, the emitted signals Emi
and Ea2 may be represented as follows:

The difference (i.e., transmission) frequency (i.e., a>2 - (o\) in the photocurrent induced in
the diode of the transmitter 12, then, may have a corresponding electric field:

A system includes a transmitter is configured to transmit an electromagnetic signal
through a sample cell (including a sample medium) to a receiver, which is configured to
receive the electromagnetic signal and another electromagnetic signal for mixing
therewith. Propagation paths of the signals to the transmitter and receiver include a first
propagation path of the electromagnetic signal to the transmitter, and a second propagation
path of the other electromagnetic signal to the receiver. The arrangement, which is located
along either or each of the propagation paths of signals to the transmitter and receiver, is
configured to alter the length of a respective propagation path. And the processor
configured to recover an amplitude and phase of the transmitted electromagnetic signal,
and calculate a complex index of refraction of the sample medium as a function of the
amplitude and phase of the transmitted electromagnetic signal.