Reflective intensity modulated (RIM) fiber-optic sen-sors have been extensively reported [1,2]. Most appli-cations of the RIM fiber sensors mainly focus ondisplacement measurement. A typical RIM fiber dis-placement sensor consists of a reflector and a two-fiber probe (one as a transmitting fiber and the otheras a receiving fiber). Based on the two-fiber config-uration, some complicated probes were designedfor axial rotation and angular displacement mea-surement [3–5]. Displacement measurements havealso been used in other applications such as vibra-tion, velocity, pressure, temperature, and electro-magnetic fields [5–9]. In these applications, therewas not any other medium between the sensing fiberand the reflector. Light emitted from a transmittingfiber arrived at the reflector directly and was re-flected by the reflector, and then the reflected lightwas collected by the receiving fiber.The fiber-optic dipping liquid analyzer (FDLA) is a
novel instrument that can simultaneously detectconcentration, refractive index, surface tension, con-tact angle, and viscosity [10]. The configuration of
the optical fiber probe of a FDLA consists only oftwo fibers positioned parallel in a thin metal tube.When the fiber probe is vertically pulled out of thetested liquid, a liquid drop is formed at the end faceof the probe. A photograph and a diagram of the fiberprobe with a distilled water drop are shown in Fig. 1.The drop can be regarded as a plane-convex lens. Thetransmitting and receiving fibers are imaged by theliquid drop, and the locations of the images are deter-mined by properties of the drop. According to para-xial spherical refraction, there is an angle betweenthe images of transmitting and receiving fibers,and the angle depends on the refractive index ofthe drop. In addition, the light passing throughthe liquid drop is reflected by the undisturbed planeliquid surface; thus the reflective intensity modula-tion method with two inclined fibers can be employedfor analysis. This work mainly discusses the applica-tion of the FDLA on the measurement of refrac-tive index.
2. Mathematical Model
The liquid drop can be regarded as a plane-convexlens whose refractive index is equivalent to that ofthe tested liquid. Thus the analysis can be simplified
into two processes: paraxial refraction imaging andreflective intensity modulating.
A. Reflection and Transmission Coefficients
The light from the transmitting fiber passes throughthe liquid-to-air spherical interface and is thenreflected by the air-to-liquid plane interface andconsequently transmitted through the air-to-liquidspherical interface. Therefore, we have to calculatethe reflection and transmission coefficients at thethree interfaces. The refractive indices of fiber core,tested liquid, and air are defined as n1, n2, and n3(equal to 1), respectively. The sensing fibers em-ployed in the FDLA are commercial silica single-mode optical fibers. The light ray along the fiber axisis used for calculating the reflection and transmis-sion coefficients since the core diameter is relativelysmall.As shown in Fig. 2, the distances between the axis
of the fiber probe and transmitting and receivingfibers are l1 and l2, respectively. The height of the li-quid drop is h, and the distance from the apex of thedrop to the plane reflective surface is d2. The radiusof the liquid drop curvature is R. The chief ray ar-rives at point A at the liquid-to-air drop interface. As-suming the incident light is unpolarized, according to
Fresnel’s law, the transmission coefficient at the in-terface of the liquid drop can be written as
T ¼ 12
�4n2 cos θ × cos θtðn2 cos θ þ cos θtÞ2
þ 4n2 cos θ × cos θtðn2 cos θt þ cos θÞ2
�;
ð1Þ
where θ ¼ arcsinðl1=RÞ and θ1 ¼ arcsinðn2 sin θÞ arethe angles of incidence and refraction of the chiefray from the transmitting fiber at the liquid-to-airinterface, respectively.
After the chief ray passes through point A, it ar-rives at point D at the plane reflective surface.The reflection coefficient R1 at the plane reflectivesurface is
R1 ¼ 12
��cos θ1 − n2 cos θt1cos θ1 þ n2 cos θt1
�2
þ�cos θt1 − n2 cos θ1cos θt1 þ n2 cos θ1
�2�; ð2Þ
where θ1 ¼ θt − θ and θt1 ¼ arcsinðsin θ1=n2Þ are an-gles of incidence and refraction of the chief ray atthe plane reflective interface, respectively.
Consequently, the chief ray is reflected by theplane reflective surface, and arrives at point E atthe air-to-liquid spherical interface. The angles of in-cidence and refraction θ2 and θt2 at pointE depend onthe position of point E, which can be derived from d2and DO in polar coordinates by introducing a param-eter t. The transmission coefficient at point E iswritten as
T2 ¼ 12
�4n2 cos θ2 cos θt2
ðcos θ2 þ n2 cos θt2Þ2þ 4n2 cos θ2 cos θt2ðcos θt2 þ n2 cos θ2Þ2
�:
ð3Þ
B. Imaging of Fibers
In the present work, the diameter of the fiber probe is800 μm, and the diameter of fiber core is 9:2 μm. Theheight of a liquid drop is about 220 μm, which is ob-tained from a drop picture. The liquid drop is as-sumed to be a sphere, and then the radius of thedrop R is about −480 μm. Therefore, the analysiscan be approximately done under the conditions ofparaxial refractive imaging [11]. The schematic oftransmitting and receiving fiber imaging by the li-quid drop and plane reflective surface is describedin Fig. 3.
First, we have to determine the locations of imagesof sensing fibers. Points O1 and O2 are centers oftransmitting and receiving fiber end faces, respec-tively, and their image distances sO1
0 and sO20 can
be calculated according to paraxial refraction ima-ging at a spherical surface.
To determine directions of the fiber images axes,we take a point A close to pointO1 and a point B closeto point O2. Unlike points at the end face of fibers,
Fig. 1. Image and schematic illustration of a fiber probe with adrop of distilled water.
Fig. 2. Scheme of light transmission and reflection.
points A and B are located beyond the liquid drop;thus they are imaged twice, once by the plane inter-face between the fiber end and the liquid drop, andthe other by the liquid drop. After obtaining the loca-tions of images of A and B, the angles of images oftransmitting and receiving fiber axes drifting offthe fiber probe axis can be respectively obtainedas α1 ¼ arctanðjlA0
− l10j=jdA0− d10jÞ and α2 ¼
arctanðjlB0− l20j=jdB
0− d20jÞ; here l10, l20, lA0, lB0, d10,
d20, dA0, and dB
0 can be derived from the refractionimaging equation [11].Second, we have to determine the image locations
U10 andU2
0 of the upper edges, and L10 and L2
0 of thelower edges to acquire the effective area of the twofibers’ end faces. For approximation, the images offiber end faces are considered as a plane circular sur-face, and the diameters of the circles are U1
0L10 and
U20L2
0. The angle of line U 01L
01 from line U1L1 is
ε1 ¼ arctanðjd1U − d1Lj=jl1U 0− l1L0jÞ, and the angle
of line U20L2
0 from line U2L2 can be expressed asε2 ¼ arctanðjd2U − d2Lj=jl2U 0
− l2L0jÞ, where l1U 0, l2U 0,l1L0, l2L0, d1U , d2U , d1L, and d2L are as shown in Fig. 3
and can also be obtained from the refraction imagingequation.
Consequently, the effective radii of the transmit-ting and receiving fiber images can be expressed as
a1effect ¼ U10L1
0 cosðε1 þ α1Þ=2; ð4Þ
a2effect ¼ U20L2
0 cosðε2 þ α2Þ=2: ð5Þ
C. Reflective Intensity Modulation
Now, the collected power can be analyzed by a reflec-tive intensity modulation method with two inclinedfibers. The reflector is the undisturbed plane testedliquid surface. As shown in Fig. 3, O1
0 and O20 are
images of O1 and O2 under the effect of a liquiddrop, respectively. O1
00 is the image of O10 by the un-
disturbed plane surface. The distance between pointO1
00 and O20 is O1
00O200 ¼ ½ðl10 þ l20Þ2 þ ðd10
0þd20
0Þ2�1=2, d100 ¼ z − ½d10 − ðr − hÞ�, and d20
0 ¼ z−½d20 − ðr − hÞ�, where z is the distance from the end
Fig. 3. Schematic illustration of fiber imaging with the liquid drop.
face of the probe to the reflector. The angle betweenlineO1
00O20 and line A00O1
00 is χ ¼ π=2 − γ − α1, where γis shown in Fig. 3. Thus the distance between lineA00O1
00 and B0O20 is p ¼ O1
00O20 · sin χ.
The images of the fiber axes are not perpendicularto the plane reflective surface and are inclined toeach other. We assume the light field at the outputof single mode fiber obeys a Gaussian distribution;thus the irradiance of emitted light from the imageof the transmitting fiber can be described as
Iðρ; zÞ ¼ I0
� ω0
ωðzÞ�
2exp
�−
2ρ2ωðzÞ2
�; ð6Þ
where I0 is the intensity at the center of fiber endface,ω0 ¼ a1efect is the effective radius of the transmit-ting fiber image, ρ is the radial distance from the cen-tral axis, and ωðzÞ is the beam waist at position zfrom the reflective surface and is given by
Finally, the light power collected by receiving fibercan be written as
P ¼ZZs
T2 · R1 · T · Iðρ; zÞds: ð8Þ
Taking into account the small size of a single modefiber core, the light intensity can be approximatelyconsidered as uniform. We use the intensity at thecenter of the fiber core for calculation. Thus Eq. (8)can be expressed as
P ¼ T2 · R1 · T · Iðp; zÞ · πa22effect: ð9Þ
From Eq. (9), the collected light power depends onthe refractive index of the tested liquid. The calcu-lated receiving light power curves from the fiberprobe with and without a liquid drop are presentedin Fig. 4. The solid line describes the characteristic ofthe fiber probe with a liquid drop. As a comparison,the model with the same parameters without thedrop was employed, and the simulated result isshown as a dashed line.
From Fig. 4, we can see that the width of the re-sponse curve of the fiber probe with a liquid dropis much narrower than that of the curve without aliquid drop owing to the refracting effect of the drop.
Fig. 4. Receiving light power curves from a fiber probe with and without a liquid drop: (a) theoretical results and (b) experimental results.
Fig. 5. (Color online) Collecting power versus displacement from the reflective interface of the undisturbed liquid: (a) theoretical resultsand (b) experimental results.
Here we approximately chose water’s refractive in-dex of 1.33 for calculation. Because the refractive in-dex of the tested liquid is larger than the air’srefractive index (n ¼ 1), the light passing throughthe liquid drop is refracted toward the receiving fiber.Figure 4 also indicates that the slopes of the twocurves are different. The reason is that, for the fiberprobe with a liquid drop, there is an angle betweenthe images of the transmitting and receiving fibersthat makes the collected light power increase fasterthan the probe without a liquid drop.
3. Results and Discussion
To verify the mathematical model, sucrose-water so-lutions at different concentrations were prepared forexperimental test. The refractive indices of the twosolutions measured with an Abbe refractometer were1.3610 and 1.3783, respectively. The theoretical andexperimental collecting power curves are shownin Fig. 5.Figure 5 implies that the trend of the experimental
curves agreeswell with that of simulation curves. Thewidth of the receiving power curve depends on the re-fractive index of the tested liquid. The larger the re-fractive index, the narrower thewidth of the receivingpower curves, owing to the liquid drop at the fiberprobe end. In addition, the slopes of the receivingpower curve are varied with the refractive index.Therefore, the refractive index of the tested liquidcan be presented by calculating the parameters ofthe response curve such as slope, width, and area.However, some differences between experimental
and calculation results such as width and slopeswere observed due to approximations in theoreticalanalysis. In addition, the profile of the power curveis affected by the locations of transmitting and re-ceiving fibers. This mismatching can be resolvedby introducing a correction factor based on experi-mental data.Sucrose-watermixtures at different concentrations
were also tested. The normalized characteristic cur-ves are shown in Fig. 6(a), and the width at half-peakversus refractive indices is illustrated in Fig. 6(b).
As shown in Fig. 6, the peak width decreases withthe refractive index because the angle between theimages of transmitting and receiving fibers increaseswith the refractive index. The refractive index can beobtained by fitting the width–refractive index curve.In addition, we can see that there are noises in thesignals that are caused by the high sampling rate(10kHz) in the experiment. This sampling rate is ne-cessary for capturing the high frequency signal (ap-proximately 1kHz), as illustrated in Ref. [10], whichis related to the viscosity of the tested liquid.
4. Conclusion
We have described light transmission of a fiber-opticdipping liquid analyzer (FDLA), which is importantfor theoretical analysis of FDLAs. The mathematicalmodel of light power collected by a fiber probe with aliquid drop was established. By comparing the char-acteristic of light power curves of a fiber probe withand without a liquid drop, we can see that a liquiddrop can change the light path and make the collect-ing power curve narrower. The experimental resultsagree well with the theoretical predictions. The the-oretical and experimental results indicate that theparameters of receiving power from a fiber probewith a liquid drop are dependent on the refractive in-dex of the tested liquid. Therefore, refractive indexcan be measured by calculating the characteristicsof the receiving power curve.
The authors gratefully acknowledge Dr. LiangChen for valuable suggestion and Yu Jiang for usefuldiscussion. This work was supported by the NationalNatural Science Foundation of China (NSFC#) undergrants 60577005, 60707013, 60877046, and60807032 to Harbin Engineering University.
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