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Three-branch optical fibre interferometer for simultaneous measurement of two physical measurands
1 August 1994
OPTICS COMMUNICATIONS EL‘EVIER
Optics Communications
110 (1994) 55-59
Three-branch optical fibre interferometer for simultaneous measurement of two physical measurands M.C. Navarrete, E. Bernabeu Departamento de Optica, Universidad Complutense de Madrid, Facultad de Ciencias Fisicas, Ciudad Universitaria s/n, 28040 Madrid, Spain
Received 20 September 1993; revised manuscript received 25 January 1994
Abstract
A three-branch all-fibre optic interferometer (3B-FOI) based on a Mach-Zehnder configuration is proposed. We have studied the feasibility of using it to measure two phase perturbations. Results are presented which show the linearity of the detection scheme. It is shown that the 3B-FOI can be used to separate the induced phase changes produced by two measurands acting simultaneously on a single sensing tibre.
1. Introduction
In this work we present a three-branch all-fibre opto be used as a sensor of two physical magnitudes. The detection scheme used makes it possible to separate the phase shifts produced by two separate measurands acting on the sensing fibre at the same time. tic interferometer
The phase of a light wave propagating in an optical fibre is more sensitive to external influences than any other optical propagation parameters. Fibre optic versions of the classical Mach-Zehnder and Michelson interferometer configurations are frequently used to determine the induced phase changes. The fibre optic Mach-Zehnder interferometer utilises two single-mode fibres as branches to detect the relative phase shift of light [ I]. Typically interferometers of this type are made from directional couplers which are spliced together [ 21. Major applications for allfibre optic interferometers are as transducers in sensing systems. The basis of the interferometric fibre optic sensor is the measurement of a physical parameter through the phase modulation which it produces in an optical beam via an auxiliary sensing element bounded to the fibre or direct interaction with the sensing libre. The detection scheme to recover the phase shift can be based upon a servo feedback loop incorporating a piezoelectric cylinder in the reference branch [ 3 1.
2. Experimental set-up The experimental arrangement is shown in Fig. 1. It consists of a tibre optic Mach-Zehnder interferometer configuration with one of its branch replaced by another Mach-Zehnder interferometer. It was made with single-mode fibre (supplied by YORK), and four directional couplers (3 dB couplers, supplied by SIFAM Ltd.). The components were connected together with mechanical splices. Three polarization controllers were used for the alignment of the polarization states in order to increase the fringe visibility of the interference signals [ 41. About forty turns of libre in every branch of the interferometer were wound on a piezoelectric cylinders supplied by
0030-4018/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SsDrOO30-4018(94)00202-6
M.C. Navarrete, E. Bernabeu /Optics Communications I1 0 (1994) 55-59
56
Morgan Matroc Ltd. The cylinders had an external diameter of 5 1 mm, wall thickness of 5 mm and length of 76 mm, the first resonant frequency was approximately 14 kHz. The electronic circuit used for the servo feedback control is based on the scheme proposed by Lobo [ 5 1. The light source was a single mode HeNe laser with mean wavelength of 632.8 nm. The output signals were amplified and converted to a proportional voltages through appropriate circuitry. These signals were acquired using a 100 MHz digitizer scope, HP 54501A. The signals sampled were transferred to a personal computer via a HPIB board for later analysis.
1 -k)
E3=E0[ik”2( +ik”2(
1 -k)
exp( -iti2)
exp( -io3)]
,
(1)
where k is the intensity coupling coefficient and @r, ti2, o3 are the phase changes experienced by the light beams passing through branches 1, 2 and 3 respectively and are proportional to the length of each branch. Then, the intensities are
-~cos(~2-@1)
>
3. Theoretical background 13 =A3
The topology of the interferometric network shows in Fig. 1 allows beams from more than two fibre optic paths to interfere [ 5 1. Referring to Fig. 1, and setting E0 as the amplitude of the input light beam, the total amplitude of the combined beam impinging on detectors D,, Dz and D3 are respectively
+(l-k)‘exp(--%)I, 1 -k)‘/2exp(
-i@,)
-ik3/2(
1 -k)1/2
exp( -ig2)
+ik”2(
1 -k)3’2
exp( -i&)]
cos(@3
-@2)
>
(2)
with A,=Zo[(l-k)3kp:+(1-k)k3p:+(1-k)kp:], A2=Z0[(1-k)4p:+(l-k)2k2p;+k2p:], B,I& =kl(
1 -k),
c/D=P,/P,
>
where p,, p2, p3 are the losses in every branch, a ~12 phase occurs each time light couples across a coupler. In our experimental situation, it is possible to assume k= l/2, so A, =A2 and B, =B2, Thus
E,=&[-kexp(-i$,)-k(l-k)exp(-i@,)
E2 =E0[ik’/2(
+B3
,
-Dd@2-$1)
branch
>
2
Fig. 1. Three-branch all-fibre optic interferometer. DC’s: directional piezoelectric hollow cylinders, Servo: servo feedback circuit.
couplers,
D’s: photodetectors,
PC’s: polarization
controllers,
PZT’s:
MC. Navarrete, E. Bernabeu /Optics Communications 110 (1994) 55-59
is kept locked. This servo circuit ensures that the following condition is fulfilled:
+Dcos(92-@,).
(3)
The signals Ii and Iz were combined amplifier to produce a signal
in a difference
1,,=2Cc0s(~,-~,)-20c0s(f$,-4,). The mately splices. and so
(4)
branches 2 and 3 were chosen to be approxiequal length and with the same number of For these conditions we may assume p2=p3 C= D. Thus, we found the Zi2 signal to be
(5) Later we will verify the agreement of this signal with experimental results. A phase perturbation was induced in branch 3 by wrapping a piezoelectric cylinder (PZT 3) with about forty turns of fibre. In this situation 1, has the form
where 03* =& -& and asp is the phase shift introduced by the PZT when a voltage is applied. In order to determine a change in this phase shift a similar tibre wrapped PZT cylinder was incorporated into branch 2 of the tibre interferometer and used as part of a servo feedback loop (Servo B ). With this detection scheme the two-branch interferometer was maintained locked at its point of maximum sensitivity. The feedback loop B ensures that the condition @~+o3~-@~=(2m+l)n/2
sin(&,
+a/4+K,V,)
,
where&=&-@,. When a shift in 03p occurs, this produces &, with the same value: MB = A&P .
@*=@2] +d4+K2
(10)
V2 -h,
where gA=KAVA, with KA the volts to radians conversion factor of the PZT of branch 1, VA the output of Servo A (in volts), and @ip the phase shift when branch 1 is subject to a perturbation. As a consequence if a shift in oIp occurs A&, = A& - A&P .
(11)
Therefore, if a simultaneous change of two external stimuli occurs, expressions (9 ) and ( 11) should be used (branch 1 is subject to the variation of both stimuli and branch 3 is only subject to the variation of one of them).
4. Experimental validation In order to verify the result obtained in Eq. ( 5 ) the following experiment was made: an AC voltage was applied to PZT 3 from a signal generator at a frequency of 172 Hz and with servo B operative the behaviour of the output I,, was observed in the scope. The phase shift induced on branch 3 had the form x3 sin(ot), with small amplitude, x3 co.28 rad. With servo B loop closed, a phase shift was induced by PZT B on branch 2 with the form x2 sin( wt+q), where $ is the delay between servo output and perturbation signal. Bearing in mind Eqs. (5) and (7) the output Zi2 is
(7)
is maintained, where fiB is the phase shift induced by PZT B, and $B = KB VB with KS the volts to radians conversion factor of the piezoelectric fibre stretcher cylinder and Va the output of servo B (in volts). Thus, the three-branch interferometer output has the form Z,*a sin(lc/4)
57
(8) a shift in
(9)
In order to induce a second phase perturbation, a PZT cylinder was introduced in branch 1. By incorporating a second servo feedback loop (Servo A), Ilz
Zr2asin{7c/4+
[x3 sin(ot)-x,
Xsin{@2z,+@a +7r/4+ +x, sin(ot+p)]/2},
sin(ot+r)]/2}
[x3 sin(wt) (12)
where & is the phase shift induced by the DC voltage component of servo B output. The parameters o and v, were measured directly with the digitizer scope and x3, x2 and gB by the scope and the radians to volts conversion factor of the PZT’s. The theoretical signal I,, was measured for values of 02r between - 71and n radians. Fig. 2 shows the theoretical and experimental output Ii2 for some values of fi2i. From the figure it can be seen, these results are in a good agreement. The next step was to study the linearity of the servo circuits. A DC voltage was applied to PZT 3 and the
MC. Navarrete, E. Bernabeu /Optics Communications 110 (1994) 55-59
58
b __J ,-. . .._c -> _._ /“-%__A-‘.._/ ,.,, ;-\ ,7.,,~ a /-..
/‘._..I ._A* ._’ -0.15 vm rrrmm r/-m
0.00
‘-1
0.01
‘k,,“
I-.. ‘1 ‘-, . _1
‘
TV' rrni~ 1 0.02
0.03
Time (s)
Experimental response Zu of the system for an AC perturwithf= 172 Hz (x,=0.298, x,=0.287, yl=O.107,all the in radians) and theoretical output for some values of 421: (a) @*,=2.2, @,=1.333; (b) &=-1.75, &=6.119; (~1 @2,=1.5,&,=6.17;(d) @,,=2.07,@,=4.86. Fig. 2. hation, values
Applied voltage (V 1 Fig. 3. Linearity of the servo systems when a DC voltage is applied to PZT 1 and PZT 3. Correlation coefficient of the linear
regressionanalysis for the output of servo A and B: r(A) =0.9995, r(B)=0.9979. outputs of the servo circuits were measured. The experimental results were adjusted to straight lines, and their slopes were compared with the corresponding theoretical values giving an error rate of less than 2%. Applying a DC voltage to PZT 1, the linearity of servo A related to this perturbation was also determined with an error rate of about 1%. Fig. 3 shows the results obtained when the same DC voltage was applied to PZT 3 and PZT 1. The results are in a good
agreement with the linearity expected. By making use of Eqs. (9) and ( 1 1 ), the theoretical slopes were calculated and their comparison with the corresponding experimental results gave an error rate of less than 2%. This scheme can separate two perturbations (with the same frequency) acting simultaneously on branch 3 if one of them is acting on branch 1. In order to verify this, voltages were applied to PZT 1 and 3. Measurands were made in which V3 was chosen to be equal to I’, / 5. For V, = 0.05 V the perturbations acting only in branch 1 was V,, = 0.19 V (calculated by using Eqs. (9) and ( 11) ). Comparing this result with the theoretical one gave an error rate of 3%. Two other values for V, were 1.O and 1.5 V; the resulting values for V,, gave an error rate of about 2 and 6%, respectively.
5. Conclusion A three-branch all-fibre optic interferometer for measurements of two phase perturbations has been presented. This device could be used as an optical sensor for two physical parameters acting on the fibre, such as mechanical pressure and temperature. If the two perturbations act simultaneously, with this scheme one phase shift would be calculated via relation (9) and the second one by using this value and relation ( 11). In this way, the proposed detection scheme can eliminate the problem of an undesirable physical magnitude influence on the measurement of the magnitude of interest. Beside, this scheme seems to assure highly accurate results: the examples given in this work present error rates of 1% or 2%, and the possibility of improving them is under study.
Acknowledgements We would like to express our gratitude to Prof. D.A. Jackson (Physics Laboratory of the University of Kent, U.K. ), A.B. Lobo Ribeiro (Optoelectronics Group, INESC, Porto, Portugal), and Dr. H. Guerrero (Departamento de optica, Universidad Complutense de Madrid, Spain) for their support. We would also like to thank researchers from Phys-
MC. Navarrete, E. Benuzbeu /Optics Communications 110 (1994) 55-59
its Laboratory of University of Kent for their help to M.C. Navarrete during her two stays there. This project has been partially supported by the project TAP-92/0087 of the “Comisi6n Interministerial de Ciencia y Tecnologia” of Spain.
References [ 1 ] EC. Allard, Fiber Optics Handbook, Optical and Electrooptical Engineering Series, eds. R.E. Fisher and W.J. Smith (McGraw-Hill, New York, 1989).
59
[2] J.P. Goure, I. Verrier and J.P. Meunier, J. Phys. D: Appl. Phys. 22 (1989) 1791. [ 3 ] D.A. Jackson, R. Priest, A. Dandridge and A.B. Tveten, Appl. Optics 19 (1980) 2926. [ 41 H.C. Lefevre, Electron. Lett. 16 ( 1980) 778. [ 51A.B. Lobo Ribeiro, White-light Interferometty: Applications to tibre optic sensors for AC and DC measurands, M.S.C. Thesis, Physics Laboratory of the University of Kent at Canterbury, U.K., 1992. [ 6 ] A.H. Cook, Interference of electromagnetic waves (Clarendon Press, Oxford, I97 1)
OPTICS COMMUNICATIONS EL‘EVIER
Optics Communications
110 (1994) 55-59
Three-branch optical fibre interferometer for simultaneous measurement of two physical measurands M.C. Navarrete, E. Bernabeu Departamento de Optica, Universidad Complutense de Madrid, Facultad de Ciencias Fisicas, Ciudad Universitaria s/n, 28040 Madrid, Spain
Received 20 September 1993; revised manuscript received 25 January 1994
Abstract
A three-branch all-fibre optic interferometer (3B-FOI) based on a Mach-Zehnder configuration is proposed. We have studied the feasibility of using it to measure two phase perturbations. Results are presented which show the linearity of the detection scheme. It is shown that the 3B-FOI can be used to separate the induced phase changes produced by two measurands acting simultaneously on a single sensing tibre.
1. Introduction
In this work we present a three-branch all-fibre opto be used as a sensor of two physical magnitudes. The detection scheme used makes it possible to separate the phase shifts produced by two separate measurands acting on the sensing fibre at the same time. tic interferometer
The phase of a light wave propagating in an optical fibre is more sensitive to external influences than any other optical propagation parameters. Fibre optic versions of the classical Mach-Zehnder and Michelson interferometer configurations are frequently used to determine the induced phase changes. The fibre optic Mach-Zehnder interferometer utilises two single-mode fibres as branches to detect the relative phase shift of light [ I]. Typically interferometers of this type are made from directional couplers which are spliced together [ 21. Major applications for allfibre optic interferometers are as transducers in sensing systems. The basis of the interferometric fibre optic sensor is the measurement of a physical parameter through the phase modulation which it produces in an optical beam via an auxiliary sensing element bounded to the fibre or direct interaction with the sensing libre. The detection scheme to recover the phase shift can be based upon a servo feedback loop incorporating a piezoelectric cylinder in the reference branch [ 3 1.
2. Experimental set-up The experimental arrangement is shown in Fig. 1. It consists of a tibre optic Mach-Zehnder interferometer configuration with one of its branch replaced by another Mach-Zehnder interferometer. It was made with single-mode fibre (supplied by YORK), and four directional couplers (3 dB couplers, supplied by SIFAM Ltd.). The components were connected together with mechanical splices. Three polarization controllers were used for the alignment of the polarization states in order to increase the fringe visibility of the interference signals [ 41. About forty turns of libre in every branch of the interferometer were wound on a piezoelectric cylinders supplied by
0030-4018/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SsDrOO30-4018(94)00202-6
M.C. Navarrete, E. Bernabeu /Optics Communications I1 0 (1994) 55-59
56
Morgan Matroc Ltd. The cylinders had an external diameter of 5 1 mm, wall thickness of 5 mm and length of 76 mm, the first resonant frequency was approximately 14 kHz. The electronic circuit used for the servo feedback control is based on the scheme proposed by Lobo [ 5 1. The light source was a single mode HeNe laser with mean wavelength of 632.8 nm. The output signals were amplified and converted to a proportional voltages through appropriate circuitry. These signals were acquired using a 100 MHz digitizer scope, HP 54501A. The signals sampled were transferred to a personal computer via a HPIB board for later analysis.
1 -k)
E3=E0[ik”2( +ik”2(
1 -k)
exp( -iti2)
exp( -io3)]
,
(1)
where k is the intensity coupling coefficient and @r, ti2, o3 are the phase changes experienced by the light beams passing through branches 1, 2 and 3 respectively and are proportional to the length of each branch. Then, the intensities are
-~cos(~2-@1)
>
3. Theoretical background 13 =A3
The topology of the interferometric network shows in Fig. 1 allows beams from more than two fibre optic paths to interfere [ 5 1. Referring to Fig. 1, and setting E0 as the amplitude of the input light beam, the total amplitude of the combined beam impinging on detectors D,, Dz and D3 are respectively
+(l-k)‘exp(--%)I, 1 -k)‘/2exp(
-i@,)
-ik3/2(
1 -k)1/2
exp( -ig2)
+ik”2(
1 -k)3’2
exp( -i&)]
cos(@3
-@2)
>
(2)
with A,=Zo[(l-k)3kp:+(1-k)k3p:+(1-k)kp:], A2=Z0[(1-k)4p:+(l-k)2k2p;+k2p:], B,I& =kl(
1 -k),
c/D=P,/P,
>
where p,, p2, p3 are the losses in every branch, a ~12 phase occurs each time light couples across a coupler. In our experimental situation, it is possible to assume k= l/2, so A, =A2 and B, =B2, Thus
E,=&[-kexp(-i$,)-k(l-k)exp(-i@,)
E2 =E0[ik’/2(
+B3
,
-Dd@2-$1)
branch
>
2
Fig. 1. Three-branch all-fibre optic interferometer. DC’s: directional piezoelectric hollow cylinders, Servo: servo feedback circuit.
couplers,
D’s: photodetectors,
PC’s: polarization
controllers,
PZT’s:
MC. Navarrete, E. Bernabeu /Optics Communications 110 (1994) 55-59
is kept locked. This servo circuit ensures that the following condition is fulfilled:
+Dcos(92-@,).
(3)
The signals Ii and Iz were combined amplifier to produce a signal
in a difference
1,,=2Cc0s(~,-~,)-20c0s(f$,-4,). The mately splices. and so
(4)
branches 2 and 3 were chosen to be approxiequal length and with the same number of For these conditions we may assume p2=p3 C= D. Thus, we found the Zi2 signal to be
(5) Later we will verify the agreement of this signal with experimental results. A phase perturbation was induced in branch 3 by wrapping a piezoelectric cylinder (PZT 3) with about forty turns of fibre. In this situation 1, has the form
where 03* =& -& and asp is the phase shift introduced by the PZT when a voltage is applied. In order to determine a change in this phase shift a similar tibre wrapped PZT cylinder was incorporated into branch 2 of the tibre interferometer and used as part of a servo feedback loop (Servo B ). With this detection scheme the two-branch interferometer was maintained locked at its point of maximum sensitivity. The feedback loop B ensures that the condition @~+o3~-@~=(2m+l)n/2
sin(&,
+a/4+K,V,)
,
where&=&-@,. When a shift in 03p occurs, this produces &, with the same value: MB = A&P .
@*=@2] +d4+K2
(10)
V2 -h,
where gA=KAVA, with KA the volts to radians conversion factor of the PZT of branch 1, VA the output of Servo A (in volts), and @ip the phase shift when branch 1 is subject to a perturbation. As a consequence if a shift in oIp occurs A&, = A& - A&P .
(11)
Therefore, if a simultaneous change of two external stimuli occurs, expressions (9 ) and ( 11) should be used (branch 1 is subject to the variation of both stimuli and branch 3 is only subject to the variation of one of them).
4. Experimental validation In order to verify the result obtained in Eq. ( 5 ) the following experiment was made: an AC voltage was applied to PZT 3 from a signal generator at a frequency of 172 Hz and with servo B operative the behaviour of the output I,, was observed in the scope. The phase shift induced on branch 3 had the form x3 sin(ot), with small amplitude, x3 co.28 rad. With servo B loop closed, a phase shift was induced by PZT B on branch 2 with the form x2 sin( wt+q), where $ is the delay between servo output and perturbation signal. Bearing in mind Eqs. (5) and (7) the output Zi2 is
(7)
is maintained, where fiB is the phase shift induced by PZT B, and $B = KB VB with KS the volts to radians conversion factor of the piezoelectric fibre stretcher cylinder and Va the output of servo B (in volts). Thus, the three-branch interferometer output has the form Z,*a sin(lc/4)
57
(8) a shift in
(9)
In order to induce a second phase perturbation, a PZT cylinder was introduced in branch 1. By incorporating a second servo feedback loop (Servo A), Ilz
Zr2asin{7c/4+
[x3 sin(ot)-x,
Xsin{@2z,+@a +7r/4+ +x, sin(ot+p)]/2},
sin(ot+r)]/2}
[x3 sin(wt) (12)
where & is the phase shift induced by the DC voltage component of servo B output. The parameters o and v, were measured directly with the digitizer scope and x3, x2 and gB by the scope and the radians to volts conversion factor of the PZT’s. The theoretical signal I,, was measured for values of 02r between - 71and n radians. Fig. 2 shows the theoretical and experimental output Ii2 for some values of fi2i. From the figure it can be seen, these results are in a good agreement. The next step was to study the linearity of the servo circuits. A DC voltage was applied to PZT 3 and the
MC. Navarrete, E. Bernabeu /Optics Communications 110 (1994) 55-59
58
b __J ,-. . .._c -> _._ /“-%__A-‘.._/ ,.,, ;-\ ,7.,,~ a /-..
/‘._..I ._A* ._’ -0.15 vm rrrmm r/-m
0.00
‘-1
0.01
‘k,,“
I-.. ‘1 ‘-, . _1
‘
TV' rrni~ 1 0.02
0.03
Time (s)
Experimental response Zu of the system for an AC perturwithf= 172 Hz (x,=0.298, x,=0.287, yl=O.107,all the in radians) and theoretical output for some values of 421: (a) @*,=2.2, @,=1.333; (b) &=-1.75, &=6.119; (~1 @2,=1.5,&,=6.17;(d) @,,=2.07,@,=4.86. Fig. 2. hation, values
Applied voltage (V 1 Fig. 3. Linearity of the servo systems when a DC voltage is applied to PZT 1 and PZT 3. Correlation coefficient of the linear
regressionanalysis for the output of servo A and B: r(A) =0.9995, r(B)=0.9979. outputs of the servo circuits were measured. The experimental results were adjusted to straight lines, and their slopes were compared with the corresponding theoretical values giving an error rate of less than 2%. Applying a DC voltage to PZT 1, the linearity of servo A related to this perturbation was also determined with an error rate of about 1%. Fig. 3 shows the results obtained when the same DC voltage was applied to PZT 3 and PZT 1. The results are in a good
agreement with the linearity expected. By making use of Eqs. (9) and ( 1 1 ), the theoretical slopes were calculated and their comparison with the corresponding experimental results gave an error rate of less than 2%. This scheme can separate two perturbations (with the same frequency) acting simultaneously on branch 3 if one of them is acting on branch 1. In order to verify this, voltages were applied to PZT 1 and 3. Measurands were made in which V3 was chosen to be equal to I’, / 5. For V, = 0.05 V the perturbations acting only in branch 1 was V,, = 0.19 V (calculated by using Eqs. (9) and ( 11) ). Comparing this result with the theoretical one gave an error rate of 3%. Two other values for V, were 1.O and 1.5 V; the resulting values for V,, gave an error rate of about 2 and 6%, respectively.
5. Conclusion A three-branch all-fibre optic interferometer for measurements of two phase perturbations has been presented. This device could be used as an optical sensor for two physical parameters acting on the fibre, such as mechanical pressure and temperature. If the two perturbations act simultaneously, with this scheme one phase shift would be calculated via relation (9) and the second one by using this value and relation ( 11). In this way, the proposed detection scheme can eliminate the problem of an undesirable physical magnitude influence on the measurement of the magnitude of interest. Beside, this scheme seems to assure highly accurate results: the examples given in this work present error rates of 1% or 2%, and the possibility of improving them is under study.
Acknowledgements We would like to express our gratitude to Prof. D.A. Jackson (Physics Laboratory of the University of Kent, U.K. ), A.B. Lobo Ribeiro (Optoelectronics Group, INESC, Porto, Portugal), and Dr. H. Guerrero (Departamento de optica, Universidad Complutense de Madrid, Spain) for their support. We would also like to thank researchers from Phys-
MC. Navarrete, E. Benuzbeu /Optics Communications 110 (1994) 55-59
its Laboratory of University of Kent for their help to M.C. Navarrete during her two stays there. This project has been partially supported by the project TAP-92/0087 of the “Comisi6n Interministerial de Ciencia y Tecnologia” of Spain.
References [ 1 ] EC. Allard, Fiber Optics Handbook, Optical and Electrooptical Engineering Series, eds. R.E. Fisher and W.J. Smith (McGraw-Hill, New York, 1989).
59
[2] J.P. Goure, I. Verrier and J.P. Meunier, J. Phys. D: Appl. Phys. 22 (1989) 1791. [ 3 ] D.A. Jackson, R. Priest, A. Dandridge and A.B. Tveten, Appl. Optics 19 (1980) 2926. [ 41 H.C. Lefevre, Electron. Lett. 16 ( 1980) 778. [ 51A.B. Lobo Ribeiro, White-light Interferometty: Applications to tibre optic sensors for AC and DC measurands, M.S.C. Thesis, Physics Laboratory of the University of Kent at Canterbury, U.K., 1992. [ 6 ] A.H. Cook, Interference of electromagnetic waves (Clarendon Press, Oxford, I97 1)