**2.2 Modulation and demodulation techniques**

Interferometer modulation and demodulation techniques have been broadly classified into homodyne and heterodyne approaches. Homodyne modulation techniques refer to those systems where the two interfering beams of light are of the same optical frequency. The homodyne techniques may be further sub-divided into active homodyne and passive homodyne. In the early days of interferometry, the scientist would wait until the interferometer drifted into the quadrature operating point before starting data collection. Today, active feedback control is used to adjust the reference path and maintain the interferometer at quadrature. Digital signal processing of the modulated signal is applied to compute the feedback control signal.

The optical configuration shown in Fig. 5 is attributed to the French physicist, Hippolyte L. Fizeau. In this configuration, interference occurs between light reflected from the target and the partial reflection from the facet of the fiber probe. High stability (i.e., low phase drift) is a key advantage of this configuration since the fiber/target separation distance is typically very small to accommodate sufficient light coupling back into the fiber. Another advantage of this system is the common path travelled by the majority of the reference and signal beams. In the fiber optic embodiment shown in Fig. 5, this means that the single mode optical fiber does not have to be polarization preserving, since both beams will undergo the same polarization evolution as they travel through the fiber. Each of the interferometric configurations described in this section, encodes a displacement, index change, or rotation rate into a phase shift between two interfering light beams. Practical techniques for extracting the desired parameter

Interferometer modulation and demodulation techniques have been broadly classified into homodyne and heterodyne approaches. Homodyne modulation techniques refer to those systems where the two interfering beams of light are of the same optical frequency. The homodyne techniques may be further sub-divided into active homodyne and passive homodyne. In the early days of interferometry, the scientist would wait until the interferometer drifted into the quadrature operating point before starting data collection. Today, active feedback control is used to adjust the reference path and maintain the interferometer at quadrature. Digital signal processing of the modulated signal is applied to

from the phase shift information form the topics of the next section.

Fig. 4. Sagnac optical configuration.

Fig. 5. Fizeau optical configuration.

compute the feedback control signal.

**2.2 Modulation and demodulation techniques** 

In passive homodyne modulation, techniques are used to generate output signals which after digital processing eliminate the signal fading that occurs as the interferometer drifts from the quadrature condition. In this way, no feedback control loop is needed to maintain the optimum operating point.

Heterodyne modulation techniques refer to those systems where the two interfering beams have differing optical frequencies. A frequency shift in one of the beams may be accomplished by acousto-optic frequency modulation (Culshaw & Giles, 1982), frequency ramping the optical source (Jackson et al., 1982), or piezoelectric-induced path length changes (Jackson et al., 1980; Cole et al., 1982). These heterodyne modulation approaches have been categorized as true-heterodyne, pseudo-heterodyne, and synthetic-heterodyne techniques.

In practical application, the boundaries between the homodyne and heterodyne approaches as well as their sub-divisions become blurred. In the remainder of this section, we describe 3 digital demodulation methods for interferometric sensors.

The phase-generated carrier demodulation technique was first proposed by Dandridge, Tveten, and Giallorenzi in 1982 (Dandridge et al., 1982). It is a homodyne demodulation technique which eliminates signal fading as slow environmental perturbations drift the interferometer's path length difference away from quadrature. In this technique, a sinusoidal modulation with known frequency and amplitude is imposed on the interferometer's phase difference. Detection and mathematical manipulation of interferometer signal allow the desired phase shift to be separated from environmental perturbations. The signal processing involves detection of the signal amplitudes at the carrier's fundamental and second harmonic frequencies. The time derivative of each of these signals is computed and they are cross-multiplied with the original signals. Integration of the difference of these signals results in the desired phase change. We have analyzed the quantum-noise-limited sensitivity of interferometers using a phase-generated carrier modulation scheme (Tayag, 2002).

Dandridge et al. first demonstrated this phase-generated carrier demodulation technique using analog electronics. Since that time, a commercially-available demodulation system based on the phase-generated carrier has come on the market (Bush et al., 1996; Cekorich et al., 1997; Davis et al., 1998). This system is based on real-time digital processing of the interferometer's signal in the time domain. We have used this demodulation system to characterize the contraction of myofibroblast cells within a collagen matrix (Kern et al., 2003). In Section 3 of this chapter, we present a new demodulation algorithm whose kernel is based on a frequency domain analysis of the interferometer's phase-generated carrier signal.

The J1…J4 demodulation method (Sudarshanam & Srivivasan, 1989) is a passive homodyne technique which requires no feedback for stabilization of the interferometer at quadrature. This technique is applicable to measuring the amplitude of sinusoidally vibrating structures. In this approach, the photodetected interferometer output is expanded using a Fourier-Bessel series. In fact, the "J1…J4" name of this method refers to the *Jn(x)* Bessel functions of the first kind of order *n* in the Fourier-Bessel expansion. Jin et al. (Jin et al., 1991; Jin et al., 1992) modified this method to extend the desired phase modulation index. Marcal et al. (Marcal et al., 2007) have used this digital demodulation technique with a Michelson interferometric configuration to characterize the frequency response of a novel piezoelectric actuator. We have used a similar digital demodulation algorithm to characterize microelectromechanical system (MEMS) structures (Pitt et al., 2003). Our fiber optical configuration for this system was based on the Fizeau interferometer.

A final digital demodulation technique we will review is based on the use of a 3x3 fiber optic coupler in the optical system (Kwon et al., 1999; Sheem, 1981; Sheem et al., 1982). Again, the demodulation method is classified as a passive homodyne technique. The optical configuration is shown in Fig. 6 and is basically a Mach-Zehnder optical configuration, but uses a 3x3 fiber coupler to combine the beams. Koo, Tveten, and Dandridge (Koo et al., 1982) were the first to introduce this passive stabilization technique for fiber optic interferometers. Todd et al. (Todd et al., 1999) later applied digital demodulation techniques to Koo's 3x3 fiber coupler-based system.

The unbalanced path lengths and 3x3 coupler of the Mach-Zehnder interferometer shown in Fig. 6 are necessary for the interferometer to function as a wavelength discriminating sensor. The outputs from the 3 photodetectors are shifted from each other by *2π/3* radians. With *a priori* knowledge of the coupler and detector characteristics, the post-detection digital processing can extract the interferometer's phase shift corresponding to a wavelength shift in the optical signal. This interferometric wavelength discriminator has been used to interrogate and multiplex a network of fiber Bragg grating sensors (Johnson et al., 2000) and as a velocity sensor which measures the Doppler shift in the light reflected from a moving surface (Fabiny & Kersey, 1997). We have also used this system as a Doppler velocimeter to measure the motion of armored plates experiencing ballistic shock (Kumar et al., 2009).

Fig. 6. Interferometric velocimeter based on a 3x3 fiber coupler and homodyne demodulation technique (Kumar et al., 2009).

#### **2.3 Summary**

322 Modern Metrology Concerns

microelectromechanical system (MEMS) structures (Pitt et al., 2003). Our fiber optical

A final digital demodulation technique we will review is based on the use of a 3x3 fiber optic coupler in the optical system (Kwon et al., 1999; Sheem, 1981; Sheem et al., 1982). Again, the demodulation method is classified as a passive homodyne technique. The optical configuration is shown in Fig. 6 and is basically a Mach-Zehnder optical configuration, but uses a 3x3 fiber coupler to combine the beams. Koo, Tveten, and Dandridge (Koo et al., 1982) were the first to introduce this passive stabilization technique for fiber optic interferometers. Todd et al. (Todd et al., 1999) later applied digital demodulation techniques to Koo's 3x3

The unbalanced path lengths and 3x3 coupler of the Mach-Zehnder interferometer shown in Fig. 6 are necessary for the interferometer to function as a wavelength discriminating sensor. The outputs from the 3 photodetectors are shifted from each other by *2π/3* radians. With *a priori* knowledge of the coupler and detector characteristics, the post-detection digital processing can extract the interferometer's phase shift corresponding to a wavelength shift in the optical signal. This interferometric wavelength discriminator has been used to interrogate and multiplex a network of fiber Bragg grating sensors (Johnson et al., 2000) and as a velocity sensor which measures the Doppler shift in the light reflected from a moving surface (Fabiny & Kersey, 1997). We have also used this system as a Doppler velocimeter to measure the motion of armored plates experiencing ballistic shock (Kumar et al., 2009).

Fig. 6. Interferometric velocimeter based on a 3x3 fiber coupler and homodyne

demodulation technique (Kumar et al., 2009).

configuration for this system was based on the Fizeau interferometer.

fiber coupler-based system.

In this section, we have reviewed the classical optical configurations and modulation/demodulation techniques of interferometric systems. There exists a close relationship between the optical configuration and the modulation technique. Our focus has been on those systems which exploit digital processing in their demodulation methods. It is interesting to note that all the interferometric systems we have presented are based on passive homodyne demodulation. This is not surprising, since the digital processor and its associated demodulation algorithm are replacing the optical hardware (acousto-optic modulators or swept frequency lasers and their drive electronics) and electronic hardware (piezoelectric transducers, analog differentiators, integrators, and lock-in amplifiers) necessary in active homodyne and heterodyne systems. In this way, the overall interferometric system is greatly simplified with the burden of complexity falling on the software. This results in a more robust and less expensive metrology system.

#### **3. Frequency domain demodulation of a phase-generated carrier**

Since the time and frequency domains represent different ways of describing the same signal, it is reasonable to consider an algorithm based on frequency domain analysis as an alternative to the phase-generated carrier time domain algorithm discussed in the previous section. In this section, we describe the kernel of a novel algorithm for the demodulation of an interferometric signal based on a phase-generated carrier modulation scheme. Following this section, practical implementation of the algorithm is discussed with simulation results verifying the algorithm. Finally, the frequency domain demodulation algorithm is summarized with suggestions for further research.

#### **3.1 Algorithmic development**

Two-beam interferometers produce an optical interference pattern, which varies with optical path length difference, *R t*( ) , between a signal arm and a reference arm. It is wellknown that the optical power, *P t*( ) , of this interference pattern, takes the form

$$P(t) = P\_{dc} + P\_{ac} \cos\left[R(t)\right],$$

where *Pdc* is the dc optical power and *Pac* is the *0*-to-peak modulation index of the interference pattern. This optical power is incident on a photodetector for conversion to an electronic signal. Since *PIN* photodiodes are characterized by a responsivity, which is constant with respect to the incident optical power, the photo-generated current takes the same mathematical form of a raised cosine. If the electronic signal from the receiver is a voltage, then this signal takes the form

$$V(t) = V\_{dc} + V\_{ac} \cos\left[R(t)\right]\_{\prime\prime}$$

where *Vdc* is the dc voltage level and *Vac* is the *0*-to-peak modulation index of the interference pattern. The objective of the demodulator is to sample this electronic signal, process the signal samples in real-time to determine the phase *R t*( ) , and produce an analog signal output, which is proportional to the phase *R t*( ) .

The demodulation algorithm is based on a phase generated carrier modulation technique. A key assumption is that a sinusoidal carrier signal is imposed on the interferometer's optical signal such that the interference pattern becomes

$$V(t) = V\_{dc} + V\_{ac} \cos\left[R(t) + M \sin\left(2\pi Ft + W\right)\right],\tag{1}$$

where *M* is the phase index of the sinusoidal carrier, *F* is the frequency of the carrier, and *W* is the phase of the carrier. Since the sinusoidal carrier appears as a phase term in the interferometer's cosinusoidal interference pattern, this form of modulation is referred to as a "phase-generated carrier."

The output voltage of Eqn. (1) describes the interferometer's output signal. Let us rewrite this equation with the angular frequency of the carrier signal defined by 2 *o F* rad/s:

$$V(t) = V\_{dc} + V\_{ac} \cos\left[R + M\sin\left(\alpha\_o t + \mathcal{W}\right)\right].\tag{2}$$

We may set our sinusoidal carrier signal such that *M* rad and *W* 0 rad. Then Eqn. (2) becomes

$$V(t) = V\_{dc} + V\_{ac} \cos\left[R + \pi \sin\left(\phi\_o t\right)\right].\tag{3}$$

Trigonometric expansion of the cosine term yields

$$V(t) = V\_{dc} + V\_{ac} \left\{ \cos \left[ R \right] \cdot \cos \left[ \pi \sin \left( \alpha\_b t \right) \right] - \sin \left[ R \right] \cdot \sin \left[ \pi \sin \left( \alpha\_b t \right) \right] \right\}. \tag{4}$$

Now, note the Bessel expansions

$$\mathbb{E}\cos\left[\pi\sin\left(\phi\_{o}t\right)\right] = I\_{0}(\pi) + 2\sum\_{n=1}^{\alpha} I\_{2n}(\pi)\cos\left[\left(2n\right)\cdot\phi\_{o}t\right] \text{ and} \tag{5}$$

$$2\sin\left[\pi\sin\left(\phi\_o t\right)\right] = 2\sum\_{m=1}^{\infty} I\_{2m-1}(\pi)\sin\left[\left(2m-1\right)\cdot\phi\_o t\right]\_{\prime\prime} \tag{6}$$

where ( ) *nJ x* is a Bessel function of the first kind of order *n* . Substitution of Eqns. (5) and (6) into Eqn. (4) reveals the harmonic content of the interferometric signal in the frequency domain as

$$\begin{split} V(t) &= \left[ V\_{dc} + V\_{ac} \cos(R) f\_0(\pi) \right] \\ &- 2V\_{ac} \sin(R) f\_1(\pi) \sin(a\_b t) \\ &+ 2V\_{ac} \cos(R) f\_2(\pi) \cos(2a\_b t) \\ &- 2V\_{ac} \sin(R) \sum\_{m=2}^{\infty} f\_{2m-1}(\pi) \sin\left[ (2m-1) \cdot a\_b t \right] \\ &+ 2V\_{ac} \cos(R) \sum\_{n=2}^{\infty} f\_{2n}(\pi) \cos\left[ (2n) \cdot a\_b t \right]. \end{split} \tag{7}$$

The demodulation algorithm is based on a phase generated carrier modulation technique. A key assumption is that a sinusoidal carrier signal is imposed on the interferometer's optical

( ) cos ( ) sin 2 *V t V V R t M Ft W dc ac*

where *M* is the phase index of the sinusoidal carrier, *F* is the frequency of the carrier, and *W* is the phase of the carrier. Since the sinusoidal carrier appears as a phase term in the interferometer's cosinusoidal interference pattern, this form of modulation is referred to as a

The output voltage of Eqn. (1) describes the interferometer's output signal. Let us rewrite

( ) cos sin *Vt V V R M t W dc ac*

( ) cos sin *Vt V V R t dc ac*

 

 0 2 1

2 1

where ( ) *nJ x* is a Bessel function of the first kind of order *n* . Substitution of Eqns. (5) and (6) into Eqn. (4) reveals the harmonic content of the interferometric signal in the frequency

> 

 

 

*ac m o*

*V RJ m t*

*ac n o*

*V RJ nt*

2 1 2

2 2

 

*t J mt*

cos sin ( ) 2 ( )cos 2 *o no n*

1 sin sin 2 ( )sin 2 1 *o m <sup>o</sup> m*

*ac o ac o*

*V RJ t V RJ t*

2 sin sin 2 1

*m*

 

2 cos cos 2 .

*n*

0 1 2

 

*Vt V V R* ( ) *dc ac* cos cos sin sin sin sin 

this equation with the angular frequency of the carrier signal defined by 2

 *<sup>o</sup>*

and (5)

, (6)

 *ot R* 

> 

> >

> > >

 

 

 *tJ J nt* 

, (1)

*o* 

rad and *W* 0 rad. Then Eqn. (2)

*<sup>o</sup>* . (2)

. (3)

  *F* rad/s:

*ot* . (4)

(7)

signal such that the interference pattern becomes

We may set our sinusoidal carrier signal such that *M*

Trigonometric expansion of the cosine term yields

( ) cos

 

*dc ac*

*Vt V V R J*

 2 sin sin 2 cos cos 2

 

Now, note the Bessel expansions

"phase-generated carrier."

becomes

domain as

The first and second harmonics of *V t*( ) are

$$V\_{a\mathbf{o}\_{\boldsymbol{\vartheta}}}(t) = -2V\_{a\mathbf{c}}\sin\left(R\right)I\_{1}\left(\pi\right)\sin\left(o\boldsymbol{\vartheta}\_{\boldsymbol{\vartheta}}t\right) \text{ and} \tag{8}$$

$$V\_{2a\_\flat}(t) = 2V\_{ac} \cos(\aleph) I\_2\left(\pi\right) \cos\left(2a\_\flat t\right). \tag{9}$$

These two frequency components contain sufficient information to determine the desired phase, *R* . By taking the ratio of the amplitudes of these frequencies, the dependence on the interference pattern's modulation index *Vac* is removed. The Bessel functions are known constants, which may be removed from the ratio as follows:

$$-\frac{\int\_{\mathcal{I}\_2} \left(\pi\right) \left| \begin{array}{c} V\_{a\flat\_\flat}(t) \ \text{amplitude} \end{array} \right|}{\int\_{\mathcal{I}\_1} \left(\pi\right) \left[ \begin{array}{c} V\_{2a\flat\_\flat}(t) \ \text{amplitude} \end{array} \right]} = \frac{\sin\left(R\right)}{\cos\left(R\right)} = \tan\left(R\right) \,. \tag{10}$$

The phase *R* may be computed from this quotient over an entire 2 rad phase range using an inverse tangent approximation. This forms the kernel of the frequency domain demodulation algorithm. A fringe counting approach similar to that described by Cekorich (Cekorich, 1999), may be used to extend the range of demodulation. In the next section, we describe practical DSP techniques to implement the frequency domain demodulation algorithm.

#### **3.2 Simulation results**

We have verified the frequency domain demodulation algorithm through computer simulation. In this section, we present those simulation results and describe our DSP implementation approach. The frequency spectrum of the interferometer's output signal is shown in Fig. 7. The magnitude spectra are shown for the five different values of the desired phase *R* . The periodic nature of the time domain signals correctly results in a set of discrete, harmonically-related sinusoids in the frequency domain. The open circles in Fig. 7 indicate the fundamental and second harmonic frequencies which are oscillating at *o* and 2*o* , respectively. Note the consistency between the magnitude spectra of Fig. 8 and Eqns. (8) and (9). Specifically, the zeros are apparent at *<sup>o</sup>* for *R* 0 and rad and at 2*<sup>o</sup>* for *R* <sup>2</sup> rad. As described by Eqn. (10), these two frequency components will be used to demodulate the interferometer.

A significant aspect of the frequency domain algorithm is the conversion of the sampled time domain signal into a frequency domain representation. Computation of the frequency spectrum is conveniently performed via the well-known fast Fourier transform (FFT) algorithm. Since only 2 spectral components are required for this algorithm, the Goertzel algorithm provides a slightly more computationally efficient approach. We implemented both algorithms, but only present the results for the FFT algorithm here since the Goertzel results are equivalent.

The FFT algorithm results in complex-valued signals at each frequency bin. The magnitudes of these complex signals are shown in Fig. 7. Fig. 8 illustrates how the magnitudes of the fundamental and second harmonics vary with the phase, *R* . These plots are consistent with the mathematical form given in Eqns. (8) and (9). The difference in peak amplitudes results from the Bessel function weighting factors, i.e. <sup>1</sup>*J* 0.5692 and <sup>2</sup>*J* 0.9709 . Note that the magnitude spectrum contains only positive values. Therefore, the magnitude spectrum alone is insufficient to unambiguously determine *R* . To resolve *R* over the full 2 phase interval, the phase spectrum is also needed. Fig. 9 shows the phases of the fundamental and second harmonics as a function of *R* .

Fig. 7. Normalized frequency spectra of the interferometer signal described by Eqn. (7) at five different values of the phase *R .* The open circles represent the values of the fundamental and second harmonic frequency components.

fundamental and second harmonics vary with the phase, *R* . These plots are consistent with the mathematical form given in Eqns. (8) and (9). The difference in peak amplitudes results

that the magnitude spectrum contains only positive values. Therefore, the magnitude spectrum alone is insufficient to unambiguously determine *R* . To resolve *R* over the full

Fig. 7. Normalized frequency spectra of the interferometer signal described by Eqn. (7) at

*<sup>o</sup>* 3

*<sup>o</sup>* 3

*<sup>o</sup>* 3

*<sup>o</sup>* 3

*<sup>o</sup>* 3

> *<sup>o</sup>* 4

*<sup>o</sup>* 4

*<sup>o</sup>* 4

*<sup>o</sup>* 4

*<sup>o</sup>* 4

five different values of the phase *R .* The open circles represent the values of the

**Frequency [rad/s]**

*<sup>o</sup>* 2

*<sup>o</sup>* 2

*<sup>o</sup>* 2

*<sup>o</sup>* 2

*<sup>o</sup>* 2

fundamental and second harmonic frequency components.

*o*

*o*

*o*

*o*

*o*

phase interval, the phase spectrum is also needed. Fig. 9 shows the phases of the

0.5692 and <sup>2</sup>*J*

0.9709 . Note

*R*= 0 rad

*R*= /4 rad

*R*= /2 rad

*R*= 3/4 rad

*R*= rad

from the Bessel function weighting factors, i.e. <sup>1</sup>*J*

fundamental and second harmonics as a function of *R* .

2

**Normalized Magnitude Spectrum [V/V]**

0 0

0 0

0 0

0 0

0 0

0.5

0.5

0.5

0.5

0.5

1

1

1

1

1

Fig. 8. Magnitudes of the fundamental and second harmonic frequencies as a function of the phase *R* .

Fig. 9. Phases of the fundamental and second harmonic frequencies as a function of the phase *R* .

Fig. 10. Reconstructed amplitudes of the fundamental and second harmonic frequencies as a function of the phase *R* .

A key to reconstructing the amplitudes of the fundamental and second harmonic frequencies is to utilize both the magnitude and phase information provided by the signal's frequency spectrum. This information shown in Figs. 8 and 9 may be used to determine the amplitudes of Eqns. (8) and (9). Fig. 10 shows the amplitudes of the fundamental and second harmonics reconstructed from the data of Figs. 8 and 9. Notice that the amplitude of the fundamental harmonic varies sinusoidally as a function of *R* and the second harmonic varies cosinusoidally as a function of *R* . This data applied to Eqn. (10) may be used with an inverse tangent approximation to determine the desired phase *R* over a full 2 phase interval. This forms the kernel of the frequency domain demodulation algorithm. We have incorporated a fringe counting routine and verified the frequency domain demodulation algorithm through computer simulation.

In this section, we have verified the theoretical development of the previous section through computer simulation and described how to implement the algorithm in practice. A key step in the kernel of the frequency domain algorithm is use of both the magnitude and phase components of the frequency spectrum.

#### **3.3 Discussion**

We have presented a frequency domain approach to the demodulation of an interferometer, which uses a phased-generated carrier modulation scheme. Through simulation analysis, we have shown that the frequency domain algorithm produces correct demodulation results by assuming fixed operating parameters (i.e., *M* and *W* 0 ). These are the same operating conditions as used in commercially-available time-domain demodulators.

Unfortunately, the frequency domain algorithm suffers from the same limitations as the time domain algorithm with regard to the carrier's modulation depth and phase. Drift of the carrier's modulation depth from *M* produces a change in the amplitudes of the fundamental and second harmonic frequencies. This dependence manifests itself in the Bessel functions in Eqn. (7). If the value of this drift is unknown, it cannot be corrected in the computation of Eqn. (10) and errant demodulation results. Similarly, a drift of the carrier's modulation phase from *W* 0 produces an irreparable change in the amplitudes of the harmonic frequencies.

As a consequence of these limitations, the frequency domain algorithm (like the time domain algorithm) requires careful initialization of the carrier's modulation depth and phase. Practical implementation of interferometric systems often use hardware, such as piezoelectric transducers, whose modulation depth drifts over time. Therefore, closed-loop active feedback is necessary to maintain the requisite modulation depth at *M* rad.
