Technical Papers
» Non-contact fiber optic vibrometer
Allen Cekorich
Optiphase Inc.
7652 Haskell Ave.
Van Nuys, CA 91406
Abstract
A new type of demodulator is described which uses direct sampling of the optical power to measure the sine and the cosine of the signal phase in the presence of a sinusoid modulation. The modulation sinusoid is also held at the optimum modulation depth and sampling phase.
Keywords: demodulator, interferometry, fiber optic, sensors
1. INTRODUCTION
An interferometer is an optical system where a light source is divided into two beams, which propagate independently along some path and then are recombined before being detected by an optical receiver. The key feature of the interferometer is that the two beams from the common source experience different phase shifts before recombining on the optical detector. In general, the process of demodulation will measure the phase shift between the two beams. An additional complication arises due to the need to impose a known phase shift between the two light beams. This external modulation is required to equalize the scale factor for all phase shifts from zero to 2p. Figure 1 below will be used to explain the operation of an interferometer.
The amplitude of two optical beams is shown on the left where the lower beam has a phase shift of 45 degrees from the top beam. The lower beam has an imposed external phase modulation of plus and minus pradians. The modulation is in the form of sinusoid, which can be expressed by the following symbolic formula.
M sin( t + W)
The symbol M is called the modulation depth and in this case is equal to p radians. The radian time t of the modulation continuously increases and produces one modulation cycle every 2 pradians. The phase of the modulation W is the initial phase when the radian time t is equal to zero. One 2p modulation period of the detected optical power from the combination of the two beams is shown on the right. The detected power is the averaged square of the sum of the amplitudes of the two beams. The voltage from the detector can be expressed symbolically by the following equation.
V(t) = Pdc + Pac cos{ R + M sin(t + W) }
The symbol R has been introduced to represent the phase shift between the two beams, which in this case would be 45 degrees. The amplitude of the modulated optical power is represented by the symbol Pac and the constant level optical power is represented by the symbol Pdc. The goal of demodulation is to use V(t) to provide a measurement of R in the presence of a known modulation: M sin( t + W). The measurement of R must be independent of the signal strength Pac and the noise background Pdc. The demodulator presented in this paper meets all of these requirements.
2. ORTHOGONAL DEMODULATION
2.1 Orthogonal modulation functions
The motivation for the present demodulator can be appreciated by considering the form of the detected optical power when the two input beams from the interferometer have zero phase shift and 90 degree phase shifts. The two cases are shown in figure 2 below.
The detected optical power from the two cases shows an interesting property. The zero phase shift optical power is an even function with respect to points separated by pradians. This means that for a zero phase shift the relation: V(t+p)=V(t), always holds. The 90 degree phase shift optical power is an odd function with respect to points separated by p radians which means that the relation: V(t+p )= –V(t), always holds.
This orthogonal property suggests that the amplitude of the even optical power could be measured in the presence of the odd optical power by adding two points that are separated by p radians and that the amplitude of the odd optical power could be measured in the presence of the even optical power by subtracting two points that are separated by p radians. In other words, the addition of two points separated by p radians rejects the odd optical power and the subtraction of two points rejects the even optical power. With this property in mind, the voltage from the detector can be expanded using a standard trig identity to give the following expression.
V(t) = Pdc + Pac cos(R) cos{ M sin(t+W) } – Pac sin(R) sin{ M sin(t+W) }
= Pdc + Pac cos(R) E(t) – Pac sin(R) O(t)
The orthogonal property of the modulation functions can be used to measure the amplitude of the even function, which is proportional to cos(R), and the amplitude of the odd function which is proportional to sin(R). The orthogonal property is also independent of the modulation depth M or the initial phase of the modulation W.
2.2 Twelve sample orthogonal demodulation
The orthogonal property can be optimally used in a system where the voltage from the optical detector is sampled twelve times in one modulation period. The graph below shows the two orthogonal modulation functions over one modulation period and with a depth of modulation equal to pradians. The functions are plotted against the sample number in figure 3.
An analog to digital converter can be used to sample the voltage every p/6 radians which is twelve times during one modulation cycle. The twelve samples are labeled S0 through S11. The amplitude of the odd function O(t) can be measured by the addition of two orthogonal sample pairs as follows.
SR = (S7 – S1) + (S11 – S5) = 4 Pac sin R
The differences S7–S1 and S11–S5 each return the peak to peak of the odd function. The constant optical power Pdc is also rejected by each difference. The four points are on flat portions of the odd function so the measured peak to peak is insensitive to small shifts in the sampling time. The amplitude of the even function E(t) can be measured by the addition of two orthogonal sample pairs as follows.
CR = (S0 + S6) – (S3 + S9) = 4 Pac cos R
The first sum S0+S6 measures twice the maximum of the even function plus 2Pdc. The second sum S3+S9 measures twice the minimum plus 2Pdc. The constant optical power drops out when the pairs are subtracted. The four points are on the flat portions of the even function so the measured peak to peak is insensitive to small shifts in the sampling time.
2.3 Computing the phase
The demodulation numbers SR and CR measure the sine and the cosine of the phase R and they have the same gain 4Pac. Their ratio will be the tangent of R. The inverse tangent is used to solve for the actual phase R. The approach taken here is to reduce the computation to the inverse tangent within one of eight octants. The octant containing the phase R can be found without doing an inverse tangent computation by looking at the sign bits of CR, SR and the difference of their absolute values. The following symbols are defined.
Y = abs( SR ) X = abs( CR ) D = X – Y
The octant, which contains R, is identified as shown in the table below.
Octant sign CR sign SR sign D
0 0 0 0
1 0 0 1
2 1 0 1
3 1 0 0
4 1 1 0
5 1 1 1
6 0 1 1
7 0 1 0
The computation of the phase R can now be reduced to computing the inverse tangent within an octant. The computed value of R will be named ANGLE and will be restricted to the unit circle where: 0<=ANGLE<2p. The computation is shown below.
octant phase angle condition
0 ANGLE = atan (Y/X) X > Y
1 ANGLE = p/2 – atan (X/Y) Y > X
2 ANGLE = p/2 + atan (X/Y) Y > X
3 ANGLE = p– atan(Y/X) X > Y
4 ANGLE = p + atan(Y/X) X > Y
5 ANGLE = 3p/2 – atan(X/Y) Y > X
6 ANGLE = 3p/2 + atan(X/Y) Y > X
7 ANGLE = 2p– atan(Y/X) X > Y
The argument of the inverse tangent is always less than or equal to one, which means that it need only be evaluated within the first octant. The domain is further reduced to the first half octant between 0 and ð/8 radians by noting the following mathematical identity.
atan(s) =p/4 – atan{ (1–s) / (1+s) } where tan(p/8) <= s < tan(p/4)
This identity can be used to generate the following mathematical procedure, which finds the replacement value of atan (b/a) within the first octant:
IF {b–a(2½–1)} <= 0 THEN atan(b/a) ELSE p/4–atan{(a–b) / (a+b)}
The argument of inverse tangent is now restricted to the first half octant so is less than tan( p/8) which is equal to 0.41. The inverse tangent can be found by using a lookup table over this limited domain or by using the Taylor’s series expansion for the inverse tangent in the first half octant.
arctan(s) = s – s3/3 + s5/5 – s7/7 + ... where 0 <= s < tan(p/8)
This can be rewritten as:
arctan(s) = s [ 1 + s2( –1/3 + s2( 1/5 + s2( –1/7 + ... ] where 0 <= s < tan(p/8)
The resulting inverse tangent computation is reduced to one divide and a table lookup or series evaluation.
2.4 Adding a Fringe Counter
The computed approximation to the phase R represented by ANGLE is restricted to the unit circle between 0 and 2p. The range can be increased by adding a fringe counter, which contains the signed multiplier of 2p. Fringe crossings can be tracked by the following logic. Suppose R changed by p radians between two consecutive readings. There is no way to tell
if the phase gained p radians or lost p radians since the result is the same position on the unit circle. Therefore, the phase step between consecutive readings must be less than p radians for the fringe counter to work. When this input condition is met, the direction of rotation of the phase is given by the direction of the shortest arc length from the previous to the current phase. A counter clockwise arc should always increase the phase except when crossing zero when a one must be added to the fringe counter. A clockwise arc should always decrease the phase except when crossing zero when a one must be subtracted from the fringe counter. Let the previous phase be held in the variable: ANGLE0, then do the following logic:
DANGLE = ANGLE – ANGLE0
if DANGLE < 0 then
if p + DANGLE < 0 then FRINGE = FRINGE + 1
else
if p – DANGLE < 0 then FRINGE = FRINGE – 1
ANGLE0 = ANGLE
The complete multiple fringe approximation to the input phase R will be given by:
RANGLE = ANGLE + FRINGE * 2p
2.5 Modulation Control
The modulation sinusoid has been assumed to give a modulation depth M equal to p radians and a phase of the modulation W equal to zero. This assumption hides the real situation. The actual modulation supplied by the demodulator is given by the following expression.
MF sin(t + WF)
The estimated modulation depth MF and the estimated phase of the modulation WF is supplied by the demodulation process when the sine is generated. These two parameters must be derived from the sampled voltage of the optical receiver in a way that guarantees the optimum modulation condition. To this end, consider what happens when the modulation depth goes to 90% of p. The graph is shown below in figure 4.
The odd modulation function has a large deviation at the samples S3 and S9. These samples are separated by six samples which is p radians which satisfies the orthogonality condition so the difference rejects the even modulation function. A measure of the deviation of the modulation depth from p radians would be given by the following expression.
SM = S3 – S9 = –2 Pac (p–M) sin(R)
This error term can be used to adjust the estimated modulation depth MF to insure it is maintained at p radians. A servo with an integral feedback term to the modulation depth estimate would close the value on pas desired. However, there is a problem. The error term depends on the phase R and becomes small and unusable near 0 and p radians. The solution is to derive a measure of the modulation depth deviation from p radians using the even modulation function. Looking very closely at the even function for the 10% decrease in modulation depth, we can see that the sample at S1 gets closer to the sample at S0 and the sample at S5 gets farther from the point at S3. The difference of these differences would measure the modulation depth change since it would be zero at the correct p modulation depth. However, the measure of the modulation must also reject the odd function. By shifting the measure over to the second half of the even function, the response will be doubled but because of the odd function sign change, the odd response will be zero. Putting it all together we get:
CM = (S7+S1) – (S6+S0) + (S11+S5) – (S9+S3) = 2 Pac (p–M) cos(R)
For the correct p modulation depth the error will be zero and so enable the formation of an integral null seeking servo. All the four sums of two samples in the parenthesis satisfy the orthogonal condition and reject completely the odd modulation function. The error term is unusable when the input phase R is near p/2 and 3p/2 radians and so it complements the odd function error term. The modulation depth error used in the integral servo can be selected by noting the octant containing R. The last problem to be addressed is the sampling phase W. Figure 5 below shows the modulation functions with the phase of the modulation decreased by 0.1 radians.
The phase error is clearly measured through the odd function by the sample combination shown below.
SW = S6 – S0 = 2 Pac M W sin(R)
This difference will always be zero for the even function and therefore provides an orthogonal measure of the phase error. The even function measures the phase error through the sample combination shown below.
CW = (S7+S1) – (S11+S5) = 4 Pac (3½/2) M W cos(R)
The sums are again orthogonal so reject completely the odd modulation function. The two measures of the phase of the modulation error are again complementary and can be selected by knowing the octant containing R. The modulation depth and phase of the modulation errors are summarized below.
SM = S9 – S3 = 2 Pac (p–M) sin(R)
CM = (S7+S1) – (S6+S0) + (S11+S5) – (S9+S3) = 2 Pac (p–M)cos(R)
SW = S6 – S0 = 2 Pac M W sin(R)
CW = (S7+S1) – (S11+S5) = 4 Pac (3½/2) M W cos(R)
The modulation depth and phase of the modulation corrections are summarized below.
Octant modulation depth servo phase of modulation servo
0 MF = MF + GCM • CM WF = WF + GCW • CW
1 MF = MF – GSM • SM WF = WF – GSW • SW
2 MF = MF – GSM • SM WF = WF – GSW • SW
3 MF = MF – GCM • CM WF = WF – GCW • CW
4 MF = MF – GCM • CM WF = WF – GCW • CW
5 MF = MF + GSM • SM WF = WF + GSW • SW
6 MF = MF + GSM • SM WF = WF + GSW • SW
7 MF = MF + GCM • CM WF = WF + GCW • CW
The four gain constants GSM, GSW, GCM and GCW serve two purposes. One is to set the time constant of the integral servo response. The other is to normalize the gains of the measures. Separated, the gain constants are:
GSM = (1 / ( 2 Pac sinR ) ) • (1 / TGSM)
GCM = (1 / ( 2 Pac cosR ) ) • (1 / TGCM)
GSW = (1 / ( 2pPac sinR ) ) • (1 / TGSW)
GCW = (1 / (4cpPac cosR ) ) • (1 / TGCW) c = (3½/2)
The gains change somewhat with the phase R but setting them to their average value will close the servos to zero error. The time constant of the servos are selected by the values of TGSM, TGCM, TGSW and TGCW.
2.6 Summary of 12 Sample Demodulation
The outline of the demodulation procedure is listed below. The computations are performed once every modulation cycle.
3. OTHER TYPES OF ORTHOGONAL DEMODULATORS
The orthogonal properties of points separated by p radians can be used to generate other types of demodulators.
3.1 Six sample demodulator
Consider the case where six samples per modulation cycle are taken of the optical detector voltage when the initial phase of the modulation W is equal to 0.2908 radians. The graph of the even and odd modulation functions is shown in figure 6 below.
The following sample combinations satisfy the orthogonal condition and return a measurement of the sine and cosine of the phase R.
CR = (S0 + S3) – (S1 + S4) = 3.235 Pac cos(R)
SR = (S3 – S0) – (S5 – S2) = 3.325 Pac sin(R)
This demodulator is interesting since it halves the bandwidth required of the ADC although the depth and modulation phase servos cannot be implemented and the error sources are larger that the twelve sample.
3.2 Low distortion 12 sample demodulator
The twelve sample demodulator is shown below in figure 7 with an additional twelfth sample point.
The last point on the even modulation function is really the first sample point from the next cycle. The expression for CR is modified as shown below.
CR = ( (S0+S12)/2 + S6 ) – (S3 + S9) = 4 Pac cos R
This simple change of replacing S0 with the average value of S0 plus S12 will insure that the even modulation function is measured at the same average time as the odd modulation function. Since the modulation functions are modulated by the signal phase R, this means that the cosine and sine of R will be measured at the same average time giving an accurate tangent function and a low distortion measurement of R when the inverse tangent is computed.
3.3 Optical power compensating demodulator
The twelve sample demodulator is shown again in figure 8 below with the radian time scale shifted by three sample points.
This p/2 shift will change the cosine and sine measures as follows.
SR = (S8 – S2) + (S4 – S10) = 4 Pac sin R
CR = (S3 + S9) – ( (S0+S12)/2 + S6) = 4 Pac cos R
These new sample combinations have the property of averaging linear changes in the optical power that occur during a modulation cycle. This will result in the correct tangent of R, which will give low distortion measurements in the presence of moderate optical power fluctuations.
3.4 Stepped modulation demodulator
The sine modulation function can be conceptualized as providing modulation steps at each of the twelve sample points. The modulation does not have to be continuous. This idea can be used to derive a five step modulation from the twelve step demodulator. The drawback is the requirement for a wide bandwidth modulator, but this method is more adaptable for a multiple channel array with many sensors. The details of this demodulator version can be found in reference one.
REFERENCES
1. DEMODULATOR AND METHOD USEFUL FOR MULTIPLEXED OPTICAL SENSORS, Allen Cekorich and
Ira J. Bush. U.S. Patent 5,903,350. May 11, 1999.
©Adobe Acrobat Reader is required to view product data sheets, product manuals, & technical papers