Optical fiber acoustic sensor utilizing mode-mode
interference
M. R. Layton and J. A. Bucaro
A method for detecting sound using a single step-index multimode fiber is presented. The detected signal
results from differences in acoustically induced phase shifts between two different waveguide modes propagating in the fiber. The relative sensitivity of this technique compared with a two-path interferometer was
experimentally determined and agreed with that calculated using the fiber parameters. Because the sensitivity of this approach is proportional to the difference in propagation constants for modes in the fiber, it is
approximately 10-3 less sensitive than the single-mode interferometer arrangement.
Introduction
Recently several workers1-3 have demonstrated the
feasibility of using optical fibers to detect acoustical
signals. Their detection scheme employs a length of
single-mode fiber in one arm of an optical interferometer. When the fiber is placed in a suitable sound field,
pressure-induced optical pathlength variations lead to
where i is the z-component of the propagation constant
for the waveguide mode, and 4ik is the phase upon entering the fiber.
The amplitude of the acousticallyinduced phase shift
for each mode when the fiber is immersed in a uniform
sound field of pressure P and frequency c is
i(t)=
a phase modulation of the propagating waveguide mode,
which, when properly recombined with the reference
beam on a photocathode, results in a modulation of the
photocurrent. Although this method of sound detection is highly sensitive, in some applications the need
for a reference arm is a disadvantage.
In this paper we
present a detection scheme that employs a single multimode fiber standing alone, eliminating the need for
a reference light path. The principle of operation, as
in the two-arm system, depends upon acoustically induced phase modulation of the light propagating in the
fiber. However, this technique requires a multimode
fiber, for the intensity modulation of the light exiting
the fiber results from the difference in the induced
phase shifts between two or more propagating modes.
Analysis
Consider the ith mode propagating in a step-index
optical fiber of length I with core/cladding indices n 0
and n 1, respectively. Upon exiting the fiber, the phase
of this mode is given by
hi = il + q'i,
1
The authors are with U.S. Naval Research Laboratory, Washington,
D.C. 20375.
Received 1 March 1978.
0003-6935/79/030666-05$00.50/0.
© 1979 Optical Society of America.
666
APPLIEDOPTICS/ Vol. 18, No. 5 / 1 March 1979
M
+j
fiT,
]P sinwt
i sinwt.
(2)
If we now consider a fiber which supports a number
of distinct waveguide modes, [see Eq. (2)] the acoustically induced phase shift for each mode is related to the
value of for that mode. This suggests the possibility
of using two modes in a single fiber in a manner analo-
gous to using two fibers in an interferometer arrangement. Since both propagating modes will be phaseshifted by the sound, strictly speaking, there is no longer
a reference beam, and the net phase shift is given by the
difference between the phase shifts for the respective
modes.
We show below that for a judicious choice of
modes, the intensity of the light impinging upon the
output endface of the fiber will be modulated at the
sound frequency.
In the following discussion we shall restrict ourselves
to the case for which only two modes propagate down
the fiber, as it is straightforward to generalize from the
results derived below to any number of modes.
The transverse components of the electric and magnetic fields in the fiber core are derived by Snitzer 4 and
for the two modes considered can be expressed
El(p,O,t) = EI(p,O) exp[i( 11 - wt) + i]
HI(p,O,t) = H1 (p,O) exp[i(fll - wt) + i 1J
E 2(p,O,t) = E2 (p,O) exp[i(132 1 - Ot)+ i4 21
H 2 (p,O,t) = H2(P,O)exp[i( 21 - Ot)+ 421
(3)
Here p, 0, z represent the appropriate cylindrical coordinates, the optical angular frequency, and p1and 2
the propagation constants for the two modes. The
above transverse field components are then those that
exist at the output endface of the fiber under static (no
sound) conditions. When the fiber is placed in the
sound field, the additional phase shifts given by Eq. (2)
must be included in Eq. (3).
A generalized transverse field component becomes
Am(p,O,t)= Am(pO) expi(Oml- wt +
m + ebmsinwot).
(4)
The fiber length can now be adjusted (e.g., with temperature) or the phase of one mode varied with respect
to the second at the input, 6 forcing cos(A4'+ A3) -> 0
and sin(AV/+ A(3)- 1. The sensitivity to sound detection will then depend only on Ark and the field
structures of the respective modes. First, consider the
factor AO. Returning to Eq. (2), AO can be expressed
AO = AO 1 611lp,
I TP)
The intensity resulting from the combination of the two
modes at the fiber output endface is then the timeaverage of the real part of the z-component of. the
complex Poynting vector.5 The resulting expression
is
(7)
where we have neglected the difference [/(bP)]Afl as
a second-order effect. For a given step-index fiber, 3
is restricted between nclk <( < nc0k, where k is the free
space wavenumber, and thus an upper limit on AOis
(p,O,t) = Il(P,O) + I2(P,O)
A4Imax=
+ '/21[El(p,O) X H2(p,o)] expi(A (l + At'
+ [E 2(p,O) X H1(p,0)I exp
.
-
- ncl)(
(8)
-p.
If we now divide Eq. (1) by Eq. (7), the amplitude of the
i(AfI + A01)
cos(Ah6+ lA3)1Jo(\A) + 2 a
k(n..
induced phase shift for the mode-mode beat effect
compared to the interferometer described by Bucaro et
J 2 k(AO) cos(2kcost)
al.' is
- sin(AG4+
1AO)12L
J2k+k(AO) sin[(2k + 1)cost]})'
(5)
1 al
zt
IUP
AS
where
_ =
p
(9)
T1
I2 = /2 (E 2 X H2),
I = /2 (E1 X H*),
At = 4'1 - k2,
1
/vO = 01 -02,
For pure compressional pressures on the silica glass
A: = 01 - 1S2,
are Bessel functions of the first kind. The
fibers, the second term in the denominator of Eq. (9) is
time-dependence in Eq. (5) is typical of sinusoidal phase
modulation phenomena, where the amplitude of the kth
This is on the order of 10-3 for a typical step-index fiber.
and
Jk
harmonic is proportional to
Jk.
Note that the inter-
ference term in Eq. (5) is a product of a spatial term and
a term depending upon AO5and t.
For small differences AO<< 1, the expression reduces
considerably to
I(pOt)
0
= Il(P,O) + I2(P, )
1
X
+ /2 [E1(p,O) H2(p,O)] expiAl
lA
+ [E2(p,O) X HI(p,0)] exp - i
- [cos(Ao + lAf) - AO sin(A4' + lAf) sinwst].
(6)
about 2/3of the first term. Thus (A)/k
-
0.6 (Ai3)/3.
Thus, the sensitivity of the two-mode fiber method is
several orders of magnitude lower than that of the two
arm interferometer arrangement.
The value of AO is limited by the index difference
between the core and cladding, but in order to approach
this limit, a proper choice of modes must be made. The
dependence of ( on the mode type is shown in Fig. 1,
where the propagation constant (3is plotted against V,
a parameter characterizing the fiber.7 Two modes are
desired that (a) exhibit a large difference in ( and (b)
possess transverse field components for which the
magnitude of the interference term in Eq. (5) is maximized. If, in addition to these restrictions we demand
that the spatial part of the interference term depends
in a simple way on p and 0, and also that the phase of the
interference terms remain invariant in the 0-coordinate,
one is reduced to considering only a small fraction of the
possible mode combinations that exist. The HEm
modes satisfy the above conditions for a number of
reasons. HEm modes are all linearly polarized (for n, 0
nci), and, for two modes polarized in the same di_
rection, intensity modulation will result at all points on
the fiber output endface where the modes have finite
field amplitudes. This can be seen by inspection of Fig.
2, where the transverse electric field is sketched for four
of the modes. From this figure it is also obvious that
Fig. 1. The relationship between the propagation constant O and
7
the mode type, as a function of the V-value of the fiber.
no beat signal would result for a combination of TEO,
and TMo, modes. Also, if the HE,, mode is combined
with the TMo, mode, beating occurs over specific regions of the fiber endface. In this case, the phase of the
interference term will differ by 7r in opposite semicircular regions defined by a diameter perpendicular to the
1 March 1979 / Vol. 18, No. 5 / APPLIEDOPTICS
667
-1
HE-11
Fig. 2.
HE-12
Schematic drawing shovwingthe direction of the transverse
electric field for four of t he lowest waveguide modes.
Fig. 3. Calculated spatial variation of the magnitude of the interference term on the fiber endface for a combination of HE and HE
11
12
modes in a V = 4 fiber. The intensity in the image is proportional
to the magnitude. The dashed line indicates the core radius.
1.0
TE-01
TM-01
Fig. 5. Spatial variation of the magnitude of the interference term
on the fiber endface for a combination of HE11 and TMo modes in
1
a V = 4 fiber. The intensity is proportional to the magnitude, and
the dashed line indicates the core radius.
a
1.0
I~~~~~~~~~~~~
I-
z .5
z.
z
0
0
01
RADIAL
DISPLACEMENT
-.
Fig. 4. Calculated variation of the HE 11 mode and HE mode in12
tensity, as well as the magnitude of the interference term across one
diameter: (a) magnitude of the interference term; (b) HE,, mode
intensity; (c) HE 12 mode intensity. The power carried in the HE,,
and HE1 2 modes is equal.
668
APPLIEDOPTICS/ Vol. 18, No. 5 / 1 March 1979
ro
0
-Z--
RADIAL
ro
DISPLACEMENT--
Fig. 6. Calculated variation of the HE11 and TMO mode intensity,
1
as well as the magnitude of the interference terms, across one diameter: (a) magnitude of the interference term; (b) HE mode intensity;
11
(c) TMO mode intensity. The power carried in the HE and TMo
1
modes is equal.
Acoustic signals were generated in a water-filled
HE1 polarization direction. Other possibilities exist
involving EHnm modes and HEn,m(n > 1) modes, but,
due to the complexity of their respective transverse
6).
The spatial dependence of the interference term at
the fiber endface resulting from a combination of the
served in the photocurrent and were identifiable by
fields,4 '8 we exclude these from the present discussion.
HE,, and HE1 2 modes is shown in Fig. 3. The intensity
in the figure corresponds to the magnitude of the interference term and, as expected, is peaked on the fiber
axis, drops to zero where the amplitude of the HE1 2
mode is zero, and then peaks again, corresponding to the
second peak of the HE1 2 mode. The radial variation of
the intensity for the individual modes [corresponding
to I, and I2 in Eq. (5)], as well as the magnitude of the
interference term, is shown in graphical form in Fig. 4.
In this as well as the following example, the power carried by each mode is equal. A phase reversal occurs in
the interference term across the boundary defined by
the zero of the HE1 2 mode. The ratio of the integrated
interference term outside this radius to that inside is
0.99 in this example. Thus, the beat signal occurring
in the outer region is not negligible. If all the light from
the fiber is detected (i.e., no optical mask is used), a
phase shift of 7r would need to be introduced into the
light exiting from one of the two regions. In Figs. 5 and
6 the spatial variation of the interference term for a
combination of the HE1 mode and TM 01 mode is
shown. In this case, a phase reversal in the interference
term occurs across a diameter separating the lobes seen
in part (a) of the figure. The curves in part (b) were
taken across a diameter perpendicular to this boundary.
In addition to exhibiting a simple interference pat-
tank with a piezoelectric disk driven at 23.3kHz. The
fiber coil was positioned in the tank with the plane of
the coil parallel to the plane of the transducer face.
When the fiber was insonified, two signals were obtheir associated frequency spectra.
The first, which can
be eliminated by index matching, is caused by beating
between light passing straight through the fiber and
light reflected back through the fiber from the fiber
terminations. This effect has been discussed by Bucaro
and Carome 9 and can be used as a calibration to mea-
sure . The coefficients determining the relative
amount of signal in each harmonic depend upon Jk (20)
for reflection signal and Jk (A) for the mode-mode
beat signal. Since 20 >> AO at a sound level for which
the first and second orders of the mode-mode beat signal were just beginning to appear, most of the reflection
signal resided in orders above the 10th harmonic. This
allowed for separation of the two signals in the fre-
quency domain. A calibrated attenuator was introduced into the transducer driving circuitry, and using
the spectrum of the above-described reflection signal,
the pressure level was increased until the spectrum
corresponding to 20 = 3.8 appeared. The attenuation
was then reduced by 55 dB, and linearity between the
second pressure and the driving voltage was assumed.
At this pressure level, the fundamental, second, and
third harmonic of the mode-mode beat signal were
visible in the signal spectrum as is shown in Fig. 7.
Although the reflection signal is also in the spectrum,
the lowest orders associated with it are buried in the
noise at this pressure level. From Eq. (5), the timedependent factor in the expression for the intensity is
tern, the HE1 and HE1 2 mode combination as well as
the HE11 and TM0 1 combination possess propagation
constants, which differ by a substantial amount for a low
V fiber, as can be seen in Fig. 1. Furthermore, in any
working device, the modes chosen will need to be se-
lectively excited, and, as Kapany and Burke have
demonstrated,6 this can be done without resorting to
elaborate techniques for these modes.
w
Experiment
An experimental verification of acoustically induced
mode-mode beating within a single fiber was made for
beating of the HE1 and TM0 1 modes. A 28-m length
of step-index fiber with a V value of 4.0 was used, of
which a 26-m section was wound into coil approximately
5 cm in diameter. An argon-ion laser operating at 514.5
nm was used to excite the fiber through a 10X micro-
.
IU
scope objective, the lateral and angular orientation of
the fiber end being adjustable to achieve some degree
of mode selection. The light distribution at the output
end of the fiber was magnified and imaged onto a RCA
7265 photomultiplier tube fitted with a 0.05-cm aperture, the image itself measuring approximately 1.9cm
in diameter.
The excited modes were identified by
their spatial intensity patterns together with their polarization states. The aperture was placed near the
center of one lobe of the image where the beat between
the HE 11 and TM 01 modes is maximum (see Figs. 5 and
-4f -3 -2f% -f
f. 2f. 3f. 4f%
0
FREQUENCY
Fig. 7. Experimental frequency spectrum of the mode-mode beat
signal. The sound frequency fs is 23.3 kHz. The signals near zero
frequency are due to electronic and mechanical noise.
1 March 1979 / Vol. 18, No. 5 / APPLIEDOPTICS
669
I(t) = 2 cos(AV6+ Afl)I
-
2 sin(AV/+ AfL)
J2k(AO)
cos2kcostj
J2k+s(AO)
in(2k + 1)wot]
(10)
The relative amount of signal residing in the different
orders is thus a function only of Ak and ( + AO). If
one compares the levels of only odd or only even orders,
however, the ratios depend only upon AO. We were
unable to generate acoustic pressures for which more
than the third-order signal was measurable. However,
mechanical vibration at the excitation end of the fiber
and'temperature changes caused Aipand A(3to fluctuate
in a random fashion, and, using averaged signal spectra,
we could assume that the averaged values of the cos(At
+ AO)and sn(AL + AO)were equal. The ratio of the
average amplitude of the first and second orders can
then be used to determine AO,and the ratio of second
to third can serve as a self-consistent
check.
The
measured ratios fixed AOat 1.8. Taking into account
the 55-dB attenuation factor, 0 is found to be 2140 at
this pressure. From Eq. (9), the measured ratio of Ar0/0
can be compared with 0.6 A(3/(3,and the two should be
equal. For a combination of HE,, and TM0 1 modes in
this fiber, 0.6 A(3/( = 8.9 X 10-4, and from above, A/q5/
= 8.4 X 10-4. This agreement is wellwithin the errors
involved in estimating A: for our fiber. In particular,
the estimate of An (and therefore AO)is probably no
better than 25%.
Conclusions
Although the measured mode beating effect is small
compared with that with the highly sensitive two-fiber
interferometer arrangement, we feel that the sensitivity
of the former is sufficient for many applications, such
as those in which ambient noise sets a low signal
limit.
The absolute sensitivity of the device can be increased
by using longer lengths of fiber. The limitation here is
the laser coherence length 1c, where it is required that
the maximum length of fiber be less than klc/A(3.
It should be pointed out that we have utilized fibers
optimized for communication applications and not for
acoustic detection. Presently available step-index fibers are constructed in such a manner as to reduce differences in waveguide mode phase velocities, which is
directly opposed to the type of fiber desired in a detector
utilizing intermode beating. A fiber with a ratio of A(3/:
of 10-1, with a small V, does not appear impractical.
This would bring the sensitivity of this method of detection to within 20 dB of the two-path interferometer
method.
Note added in Proof. In estimating the magnitude
of the acoustically induced mode-mode beat signal, the
contribution due to (AO)/(P) was neglected. This
term can, however, be of the same order of magnitude
as that calculated in Eq. (7) and should not, in general,
be neglected.
References
1. J. A. Bucaro, H. D. Dardy, and E. F. Carome, Appl. Opt. 16,1761
(1977).
2. J. H. Cole, R. L. Johnson, and P. G. Bhuta, J. Acoust. Soc. Am. 62,
1136 (1977).
3. J. A. Bucaro, H. D. Dardy, and E. F. Carome, J. Acoust. Soc. Am.
62, 1302 (1977).
4. E. Snitzer, J. Opt. Soc. Am. 51, 494 (1961).
5. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,
1964), p. 33.
6. N. S. Kapany and J. J. Burke, Optical Waveguides (Academic,
New York, 1972), pp. 164-179, 205-222.
7. D. B. Keck, in Fundamentals of Optical Fiber Communications,
M. K. Barnoski, Ed. (Academic, New York, 1976), p. 11.
8. E. Snitzer and H. Osterberg, J. Opt. Soc. Am. 51, 500 (1961).
9. J. A. Bucaro and E. F. Carome, Appl. Opt. 17, 330 (1978).
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