The Frequency-Dependent Directivity of a Planar Fabry-Perot … · 2014. 8. 15. · Perot (FP) polymer ﬁlm sensor has been described [6]. However, this model provided the sensor

394 ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 54, no. 2, february 2007

The Frequency-Dependent Directivity of aPlanar Fabry-Perot Polymer Film

Ultrasound SensorBenjamin T. Cox and Paul C. Beard

Abstract—A model of the frequency-dependent direc-tivity of a planar, optically-addressed, Fabry-Perot (FP),polymer film ultrasound sensor is described and validatedagainst experimental directivity measurements made overa frequency range of 1 to 15 MHz and angles from normalincidence to 80�. The model may be used, for example, asa predictive tool to improve sensor design, or to providea noise-free response function that could be deconvolvedfrom sound-field measurements in order to improve accu-racy in high-frequency metrology and imaging applications.The specific question of whether effective element sizes assmall as the optical-diffraction limit can be achieved wasinvestigated. For a polymer film sensor with a FP cavityof thickness d, the minimum effective element radius wasfound to be about 0�9d, and that an illumination spot ra-dius of less than d�4 is required to achieve it.

I. Introduction

The use of thin film Fabry-Perot interferometers (FPI)for the detection of ultrasound in liquids at megahertzfrequencies provides a practical alternative to broadbandpiezoelectric receivers for a variety of measurement andimaging applications [1]–[14]. The sensing element com-prises a thin film spacer (

cox and beard: frequency-dependent planar fp polymer film ultrasound sensor 395

cal models that fully describe their acoustic performancein terms of frequency response and directivity have yet tobe developed. Previously, an experimentally validated an-alytic model of the frequency response of a planar Fabry-Perot (FP) polymer film sensor has been described [6].However, this model provided the sensor response due tonormally incident plane waves only. In this paper, a moregeneral analytic model is developed that describes thecomplex, frequency-dependent, directional response (mag-nitude and phase) of the sensor. As well as providing in-sight into the physical mechanisms that underlie the acous-tic characteristics of the sensor, such a model provides avaluable simulation tool for developing optimized sensordesigns by careful selection of the geometric and materialparameters. Additionally, it can be used to deconvolve forthe directional response of the sensor to reduce spatial av-eraging errors in quantitative ultrasound metrology andblurring and artefacts in imaging applications.

One specific application of the model that is ofwidespread significance to the practical application of thetechnique, and a focus of this paper, is its use to determinethe limits to the commonly held assumption that the ele-ment size is defined by the dimensions of the optical fieldthat addresses the sensor. Although intuitively reasonablefor a laser spot radius much larger than the FPI thick-ness, there will be some limit to this assumption as thespot radius is reduced, and the question as to whether itreally is possible to obtain element sizes down to the op-tical diffraction limit of a few microns (as one might hopefor) remains unanswered. Because the directional responseis directly related to element radius, this limit can be de-termined by using the model to compute the directivityand comparing it to that of an ideal ultrasound receiver toobtain an effective element radius as a function of the illu-mination spot radius. It then should be possible to identifythe smallest element size that can realistically be achievedfor a given sensor geometry and material properties. Thishas important implications for the design of the optical in-terrogation instrumentation. For example, there is little tobe gained in undertaking the nontrivial task of designingan optical system that provides for truly optical diffrac-tion limited spot sizes, if the effective element radius for aspecific sensor configuration is significantly larger.

Although the model described in this paper is generallyapplicable to planar FPI ultrasound sensors, we considerby way of an example, the specific case of those that usea polymer film spacer. Thus in Section II, the sensor fab-rication, typical material and geometric properties of thisconfiguration, are outlined. The model and its experimen-tal validation are described in Sections III and IV. Twoapplications of the model are discussed in Section V. First,it is used to demonstrate how the directional response canbe improved by appropriate selection of the material prop-erties of the backing substrate. Second, to investigate therelationship between the dimensions of the region of thesensor that is optically addressed and the effective elementradius in order to determine the minimum attainable valueof the latter.

Fig. 1. A FP ultrasound sensor comprising a polymer layer (10–50 µm) with two thin reflective coatings, forming the FP interfer-ometer, overlying a solid glass or polymer substrate (1–2 cm thick).The substrate has an angled back-face to prevent optical reflectionswithin it acting as a second interferometer. The shape and size ofthe acoustically sensitive region is determined by the profile of theinterrogating laser beam and the acoustic properties of the materials.The width is not to scale and is typically of centimeter order.

II. Fabry-Perot Polymer FilmUltrasound Sensor

A schematic of a FP polymer film sensor is shown inFig. 1. It consists of a layer of polymer sandwiched be-tween two thin, optically reflective coatings and overlyinga transparent solid substrate, typically several centime-ters thick. The polymer layer is typically 10–50 µm thickand forms the interior of the FP cavity, with the reflectivelayers forming the mirrors of the cavity. Several methodshave been investigated for fabricating FPIs with polymerspacers. The first involves forming an FPI by depositinga reflective aluminium coating on either side of a discretepolyethylene terepthalate film that then is bonded to aglass or polymer substrate (typically a centimeter or twothick) using an optically transparent adhesive [6]. Morerecently, spin coating [8] and gas phase vacuum depositiontechniques, such as the Parylene process [4], [12], have beenused to directly deposit the polymer spacer on to a sub-strate precoated with metallic or dielectric layers that formthe first FPI mirror. The second mirror then is depositedon to the polymer spacer. The use of the Parylene process,in particular, to form the polymer layer, holds a numberof advantages. It provides a high degree of thickness uni-formity and excellent surface finish, both of which are re-quired to form a high-quality FPI and allows the sensorsto be inexpensively batch fabricated with a high degree ofrepeatability. For these reasons, sensors fabricated in thisway provide an example of practical importance to studyand are considered in this paper.

As indicated in the previous section, the transductionmechanism may be considered to consist of two parts: thesensitivity of the light-intensity modulation to a change in


optical phase (the intensity-phase transfer function men-tioned above), and the sensitivity of the optical phase tothe incident acoustic wave. The former can be describedusing the well established descriptions of multiple beaminterferometry used to model transfer functions of FPIs[15]. The latter is the subject of this paper, as its depen-dence on the direction and frequency of an incoming planeacoustic wave determines the directionality and frequencyresponse of the sensor.

III. Model of Directivity andFrequency Response

As described above, the FP sensor can detect ultra-sound because an acoustic wave traveling through the sen-sor changes the thickness of the polymer layer. This resultsin a change in the phase between the light reflected fromthe surfaces of the polymer layer, which alters the reflectedlight intensity. It is this modulated light intensity that ismeasured and from which the acoustic pressure is inferred.The frequency and angle of incidence of the incident wavewill affect the amount by which the thickness of the poly-mer layer changes. Therefore, the sensor response exhibitssome directionality. This section describes a model to cal-culate this frequency-dependent directional response.

A. Optical Phase Change

In the absence of an acoustic wave, the difference inphase between the light reflected from the two sides of theinterferometer is φ = 4πnd/λ0 radians, where n is the re-fractive index, d is the thickness of the polymer film, andλ0 is the in vacuo wavelength of the interrogating lightbeam. A change in this phase could be due to a physicalchange in the thickness of the film or a change in its refrac-tive index. As the passage of an acoustic wave through thesensor could, in principle, produce either effect, there aretwo possible mechanisms by which the sound wave couldcause a modulation of the intensity of the reflected beam.In this work, the second mechanism—the refractive indexchange—was assumed to be negligible; the reasons for thisare discussed in the Appendix. The optical phase change,therefore, is modeled as due only to the first mechanism: achange in the physical thickness of the sensor as the acous-tic wave propagates through it. To a linear approximation,the change in phase, ∆φ, may be written as:

∆φ =∂φ

∂d∆d =

(4πnλ0

)∂d

∂p∆p, (1)

where ∆d is a change in the thickness due to a changein the external acoustic pressure ∆p. ∂φ/∂d = 4πn/λ0 isthe sensitivity of the phase to a change in the thickness,and ∂d/∂p is the sensitivity of the thickness to a change inpressure. As the change in thickness, ∆d, is the differencein the changes in the vertical particle displacements, uz,on the two sides, z = 0 and z = d, of the FP layer, the

sensitivity of the thickness to a change in pressure may bewritten:

∂d

∂p=

uz(d) − uz(0) − d∆p

. (2)

In other words, for a unit amplitude incident wave, thesensitivity of interest, ∂d/∂p, is just the difference in theparticle displacements at the two sides of the polymer filmminus its unperturbed thickness. In Section III-B, a modelof elastic waves in solids is used to calculate this sensitiv-ity as a wave travels through the sensor. ∂d/∂p, averagedover the illuminated area of the sensor, weighted by thelaser beam profile S(x), gives the frequency-dependent di-rectional response of the sensor:

D(f, θ) =

∫ ∞−∞

(∂d/∂p)S(x)dx∫ ∞

−∞S(x)dx

. (3)

B. Elastic Wave Model

The sensor was treated as a linear, elastic, layeredmedium characterized by two elastic constants per layer,e.g., Lamé’s parameters λ and µ, or Young’s modulus Eand Poisson’s ratio σ. It was assumed to be infinitely wide,with physical properties unvarying in the horizontal direc-tion, and with a semi-infinitely thick substrate. By takingadvantage of the planar geometry of the sensor in this way,an analytical model can be formulated that provides con-siderably greater computational efficiency than numericalmodels based on less specific approaches, such as finite dif-ference or finite-element schemes [16]. It also was assumedthat the optically reflecting layers forming the FP cavitywere much thinner than an acoustic wavelength and, there-fore, acoustically negligible. This is a good assumption forthe type of sensors and frequency ranges used in the mea-surements, Section IV, and allows for a simpler mathe-matical description. For higher frequencies in which thisassumption becomes invalid, the model may be straight-forwardly extended to include these extra layers.

The equation for the time-varying vector particle dis-placement, u = (ux, uy, uz), in an isotropic, elastic, mate-rial is well known [17]:

ρ∂2u∂t2

= (λ + 2µ)∇(∇ · u) − µ∇ × (∇ × u). (4)

The task here is to calculate the vertical particle dis-placement on the boundaries of the polymer (interferom-eter) layer, uz(0) and uz(d), by solving (4) within eachlayer subject to appropriate matching conditions on theboundaries at which the layers meet. A standard methodof achieving this is described in [17]–[19].

The displacement vector may be written as the sumu = ∇φ + ∇ × ψ, where φ and ψ are scalar and vectorpotentials, respectively. In an isotropic medium, (4) thensplits into two wave equations: one for the scalar potential,


Fig. 2. The coordinate system and notation for the wave approach tocharacterizing the acoustic response of the sensor. The FP layer hasthickness d in the absence of an acoustic wave. The solid lines arecompressional (P) waves and the dotted lines shear waves. Assum-ing an incident wave of known amplitude, there are seven unknownwave amplitudes (A−, B±, C±, D+, E+) and so seven boundaryconditions must be specified.

which describes longitudinal waves traveling with speedcl =

√(λ + 2µ)/ρ, and one for the vector potential, which

describes shear waves with speed cs =√

µ/ρ:

∂2φ

∂t2− c2l ∇2φ = 0, (5)

∂2ψ

∂t2− c2t ∇2ψ = 0. (6)

The shear wave propagates as a transverse wave, andψ points in a direction perpendicular to both the dis-placement ∇ × ψ and the direction of travel. Any shearwave may be considered as a combination of horizontallyand vertically polarized shear waves in which the parti-cle motion is in the horizontal or vertical plane, respec-tively. The two-dimensional (2-D) model described below,and shown in Fig. 2, does not include horizontally polar-ized shear waves; therefore bulk shear horizontal (SH) andLove waves are excluded. This is not a limitation, however,as such waves by definition do not cause displacements inthe z direction, which are the only movements of interestfor this application. With this simplification, the vectorpotential can be written, without loss of generalization, asψ = (0, ψ, 0), where ψ is a scalar. In this case, the dis-placement vector becomes:

u =(

∂φ

∂x− ∂ψ

∂z,∂φ

∂y,∂φ

∂z+

∂ψ

∂x

). (7)

Single-frequency, plane wave, solutions to the waveequations (5) and (6) will take the forms:

φ ∝ ei(k·x−ωt), (8)ψ ∝ ei(kt·x−ωt), (9)

where ω is the temporal frequency, t is the time, x =(x, y, z) a position vector, and k = (kx, ky, kz) and kt =(ktx, kty, ktz) are wavevectors for the compressional (lon-gitudinal) and shear waves, respectively. Here we study a

2-D model (Fig. 2) because, for a plane wave incident atany angle θ on another plane, the coordinate axes alwayscan be aligned with the wavefront to reduce the problemfrom three to two dimensions. Here the axes have beenchosen such that ky = kty = uy = 0.

As a plane pressure wave propagates through the sen-sor, part of it will be reflected and part transmitted ateach boundary. No reflections occur other than at theboundaries between the layers because the sensor’s phys-ical properties were assumed to be constant in the hori-zontal direction. In the steady state, all the (multiply re-flected) upward-traveling compressional waves within thepolymer layer can be summed and treated as one wavewith a complex amplitude, C−. Similarly, all the downwardtraveling waves can be lumped together with the com-plex amplitude, C+. In this way, the field in the polymerlayer can be described by four wave amplitudes, B± andC±, corresponding to downward/upward traveling shearand compressional waves, respectively. A similar approachand notation is used to describe the wave amplitudes inthe surrounding media too. Surface waves, such as leakyRayleigh waves, are implicitly included in this model asthey are combinations of shear and compressional motions.The medium above the sensor is fluid, chosen here to bewater, so it does not support shear waves. In the poly-mer layer and the substrate, however, both shear (dotted)and compressional (solid) waves are excited (see Fig. 2).If, with no loss of generality, the incident pressure waveis assigned unit amplitude, there are seven unknown waveamplitudes, denoted in Fig. 2 by A−, B±, C±, D+, andE+ (+ indicates waves traveling in the positive z direc-tion. A, C, and E are amplitudes of compressional wavesand B and D of shear waves). If the displacements in thethree layers of the model are denoted with subscripts 1, 2,and 3 for water, polymer layer, and substrate, respectively,the displacement potentials may be written, for this 2-Dcase as:

φ1 = Ψ(A+eikz1z + A−e−ikz1z), (10)

ψ2 = Ψ(B+eiktz2z + B−e−iktz2z), (11)

φ2 = Ψ(C+eikz2z + C−e−ikz2z), (12)

ψ3 = ΨD+eiktz3z, (13)

φ3 = ΨE+eikz3z , (14)

where the common factor Ψ = exp(i(kxx − ωt)) appearsbecause the wavenumbers in the x direction, kx, alwaysmust be the same as that of the incident wave (Snell’slaw). Eq. (10)–(14) are for displacement potentials, so theknown amplitude A+ = P0/(ρ1ω2), where P0 is the inci-dent pressure amplitude. Here, we set P0 = 1.

To solve for the seven unknown amplitudes, seven con-ditions of continuity at the boundaries are required. It isassumed that the solid polymer layer and solid substrateare in welded contact, but that the fluid can slip over thesolid. This results in the following seven requirements:

• the normal particle displacement is continuous acrossboth boundaries (1 and 2),


• the tangential particle displacement is continuousacross the polymer layer/substrate boundary (3),

• the normal stress is continuous across both boundaries(4 and 5),

• the shear stress is continuous across both boundaries(6 and 7).

Conditions 4–7 are written in terms of stress. Thedisplacement-strain and strain-stress relationships arewell-known and allow the seven conditions to be rewrit-ten as:

uz1(0) = uz2(0), (15)uz2(d) = uz3(d), (16)ux2(d) = ux3(d), (17)

λ1

(∂ux1∂x

∣∣∣∣0+

∂uz1∂z

∣∣∣∣0

)= λ2

∂ux2∂x

∣∣∣∣0

+ ν2∂uz2∂z

∣∣∣∣0,

(18)

λ2∂ux2∂x

∣∣∣∣d

+ ν2∂uz2∂z

∣∣∣∣d

= λ3∂ux3∂x

∣∣∣∣d

+ ν3∂uz3∂z

∣∣∣∣d

,(19)

µ2

(∂ux2∂z

∣∣∣∣0+

∂uz2∂x

∣∣∣∣0

)= 0, (20)

µ2

(∂ux2∂z

∣∣∣∣d

+∂uz2∂x

∣∣∣∣d

)= µ3

(∂ux3∂z

∣∣∣∣d

+∂uz3∂x

∣∣∣∣d

),(21)

where ν = 2µ + λ. Substituting expressions for the poten-tials obtained from (10)–(14) and (7), into these match-ing conditions results in seven equations in the seven un-known amplitudes A–E. These can be written in matrixform as (22) (see next page), where e1 = exp(iktz2d),e2 = exp(ikz2d), e3 = exp(iktz3d), and e4 = exp(ikz3d).The following shorthand also was used: λ̄2 = λ2 − ν2,λ̄3 = (λ3 − ν3), γ2 = λ2k2x + ν2k2z2, and γ3 = λ3k2x + ν3k2z3.This matrix equation may be solved efficiently using stan-dard matrix techniques [18], [19] to yield the unknownwave amplitudes A−, . . . , E+. Using (7), (11), and (12),the vertical component of the displacement vector withinthe polymer layer may be written:

uz2(x, z) =∂φ2∂z

+∂ψ2∂x

(23)

= ikz2Ψ(x)(C+eikz2z − C−e−ikz2z)+ ikxΨ(x)(B+eiktz2z + B−e−iktz2z). (24)

Once the system of equations has been solved for the sevenunknown amplitudes using (22) the difference in verticalparticle displacements uz2(0) − uz2(d) can be ascertained,and the required sensitivity ∂d/∂p can be calculated from(2) as a function of frequency ω and incidence angle θ =sin−1(kx/k) where k = ω/c. The final task in calculatingthe sensor response is to average over the area of the spotilluminated by the interrogating laser, taking into accountthe laser beam profile.

C. Spatial Averaging Over the Beam Profile

The interrogating laser beam has a finite spot radius,so rather than measuring at one point on the sensor, it

measures the spatial average over the spot. This averagingwill affect the directional response of the sensor, i.e., theresponse to a wave incident at an oblique angle θ �= 0.

In (23) the factor Ψ(x) contains all the x-dependence,so the integral in the numerator of (3) becomes the inte-gral of Ψ(x)S(x) over x, where S describes the spot sizeand profile. S could describe any beam profile, such as aGaussian, but here a top-hat beam of radius a is used:

S(x) = 1 for − a ≤ x ≤ a= 0 elsewhere.

(25)

Averaging over x gives the normalized directivity factorΨ as:

Ψ =

∫ ∞−∞

Ψ(x)S(x)dx∫ ∞

−∞S(x)dx

. (26)

For the top-hat case in 2-D this becomes:

Ψ =12a

∫ a−a

eikxxdx =sin(kxa)

kxa, (27)

where the time-dependent phase factor exp(−iωt) has beenomitted for convenience.

In 3-D Cartesian coordinates (x, y, z), the profile S, asgiven in (25), represents a line source of width 2a, infinitelylong in the y direction. In order to model a circular spot,rather than a line source, the beam profile must be de-scribed as a function of a radial coordinate, r =

√x2 + y2.

If r is chosen to be zero at the center of the beam, then thebeam profile for the top-hat circular beam is the same as(25), except with x replaced by r =

√x2 + y2. By trans-

forming from the Cartesian coordinates (x, y) to the cylin-drical polar coordinates (r, ζ), the corresponding directiv-ity factor becomes:

Ψ =1

πa2

∫ a0

∫ 2π0

eikxr cos ζ rdζdr (28)

=2a2

∫ a0

J0(kxr)rdr =2J1(kxa)

kxa. (29)

So, using (2) and (3), the overall directivity D(f, θ) nowcan be written:

D = ikz2Ψ(C+(eikz2d − 1) − C−(e−ikz2d − 1)

)+ ikxΨ

(B+(eiktz2d − 1) + B−(e−iktz2d − 1)

)− d. (30)

Eq. (30) can be used to calculate the complete, frequency-dependent, complex directional response, i.e., both mag-nitude and phase.

IV. Directivity Measurements

As a first test of the model described in Section III,it was compared to a previous, experimentally validated


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

kz1A+

00

λ1k21A

+

000

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

kz1 kx kx kz2 -kz2 0 00 kxe1 kx/e1 kz2e2 -kz2/e2 -kxe3 -kz3e40 -ktz2e1 ktz2/e1 kxe2 kx/e2 ktz3e3 -kxe4

−λ1k21 -λ̄2kxktz2 λ̄2kxktz2 γ2 γ2 0 00 λ̄2kxktz2e1 -λ̄2kxktz2/e1 -γ2e2 -γ2/e2 -λ̄3kxktz3e3 γ3e40 µ2(k2tz2 − k2x) µ2(k2tz2 − k2x) -2µ2kxkz2 2µ2kxkz2 0 00 µ2(k2tz2 − k2x)e1 µ2(k2tz2 − k2x)/e1 -2µ2kxkz2e2 2µ2kxkz2/e2 −µ3(k2tz3 − k2x)e3 2µ3kz3kxe4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

A−

B+

B−

C+

C−

D+

E+

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(22)

Fig. 3. Experimental arrangement for the plane-wave directivity mea-surements.

model [6], which describes the sensor’s normal-incidencefrequency response. This other model has been shown toagree with normal-incidence, plane-wave, pressure mea-surements for a variety of different substrates and poly-mer layers. Both models give the same normal-incidenceresponse; the model described here, however, can also ac-count for waves with oblique angles of incidence. An ex-periment with which to compare the model output forobliquely incident waves was therefore devised.

Fig. 3 shows the experimental arrangement used tomeasure the directional response of the detector, i.e., theresponse to plane waves incident at oblique angles. APanametrics V312-SU ultrasound transducer (6-mm di-ameter, 10 MHz center frequency, −6 dB points at 6.3and 12.9 MHz) driven with a Panametrics 5052 PR pulser-receiver was mounted on an arm capable of being rotatedsuch that the sound beam from the transducer always wasincident on the same point on the sensor, from the samedistance away. To ensure that the wavefronts arriving atthe sensor were plane, the transducer was placed in the farfield, 140 mm from the sensor face. This is in the far fieldfor frequencies up to about 20 MHz. The maximum outputof the transducer was 130 kPa, well within the linear rangeof the sensor [12]. The sensor used in these measurements,described in detail in [12], consisted of a 3.8-mm thick sub-strate of borosilicate glass (cl ≈ 5640 m/s, cs ≈ 3280 m/s,ρ = 2240 kg/m3) and a 40-µm thick Parylene C (Spe-cialty Coatings Systems, Indianapolis, IN) polymer layer(cl ≈ 2200 m/s, cs ≈ 1100 m/s, ρ = 1180 kg/m3). Thereflective coatings were aluminium in this case and with a

thickness of ≈ 50 nm were considered acoustically negligi-ble. The lateral dimensions of the sensor were 5×3 cm. Thelaser beam used to interrogate the sensor was a 70 mW,850 nm, distributed Bragg reflector laser diode. Its outputwas collimated and expanded to provide a large area el-liptical beam of dimensions 16 × 12 mm. The bias pointto ensure a linear response was chosen by angle tuningthe incident beam [12], [20]. The light reflected from thesensor was directed through an aperture, 400-µm radius,to give a top-hat profile, to a photodiode. The high-passfiltered (> 300 kHz) output of which was recorded usinga 500 MHz digitizing oscilloscope. The arm was manuallyrotated in steps of 1◦, and a measurement of the acousticsignal from the transducer was made at each angle (av-eraged over 100 pulses). These measurements were thenFourier transformed to give the frequency-angle sensor re-sponse.

Fig. 4 shows a measurement of the frequency-angle re-sponse for angles from 0◦ (normal incidence) to 80◦ for afrequency range from 1 to 15 MHz. These measurementswere normalized to the frequency response at normal inci-dence predicted by the model. The prediction of the model,using the estimated material parameters stated above, isshown in Fig. 5. The minima that form the series of con-cave curves are due to spatial averaging over the illumi-nated spot. This occurs when the horizontal componentof the wavelength is an integer multiple of the spot diam-eter, i.e., at the zeros of Ψ (29), which is approximatelykxa = π(n + 1/4) for n = 1, 2, . . . . The two vertical fea-tures, labeled A and B, are caused by critical angle ef-fects. At low frequency, these minima occur close to thecritical angles for transmission of compressional (15◦) andshear waves (27◦) from water to glass, a consequence ofthe fact that, at these low frequencies, the polymer layeris much thinner than a wavelength and thus loses its abil-ity to influence the waves’ behavior. At these critical an-gles, the wavefronts are traveling horizontally in the glassand, hence, almost horizontally in the polymer. Therefore,there is little displacement in the vertical direction and thesensor response falls to a minimum. At higher frequencies,more complex interference effects affect the response, asthe polymer layer becomes a significant proportion of awavelength.

The comparison between the measurement and themodel can be seen more clearly in the horizontal profilesthrough Figs. 4 and 5 for frequencies from 1 to 13 MHz,


Fig. 4. The measured, plane-wave, frequency-angle response in deci-bels, for a frequency range of 1–15 MHz and incidence angles of0–80◦, for a glass-backed, Parylene film FP ultrasound sensor withaluminium reflective coatings and a 40-µm Parylene C layer. Theresponse has been normalized to show the same normal incidenceresponse as the model, shown in Fig. 5.

Fig. 5. The plane-wave, frequency-angle response of a glass-backedsensor predicted by the model of Section III, and corresponding to themeasurements in Fig. 4. The parameters used were: 40-µm ParyleneC layer (cl = 2200 m/s, cs = 1100 m/s, ρ = 1180 kg/m3), glasssubstrate (cl = 5640 m/s, cs = 3280 m/s, ρ = 2240 kg/m3), and400-µm radius top-hat interrogation beam.

shown in Fig. 6. From these directivity plots and the full,frequency-angle plots, it is clear that there is good qualita-tive agreement between the model and the measurementsoverall and good quantitative agreement at lower frequen-cies and smaller incidence angles. However, there are dis-crepancies in the amplitude if not the shape of the responseat higher angles of incidence and higher frequencies, largelydue to signal-to-noise limitations. Three factors contributeto the low signal-to-noise ratio (SNR): the reduced out-put of the transducer at higher frequencies (at 15 MHzit is 10 dB below its maximum output at 10 MHz), theincreased attenuation at higher frequencies and the large(140 mm) transducer-sensor distance leads to at least afurther 7 dB of attenuation, and the sensor itself is less

Fig. 6. Directivity plots at frequencies of 1–13 MHz obtained bytaking horizontal profiles through Figs. 4 and 5 and normalizing tonormal incidence. ‘-’ model, ‘o’ measurements.

Fig. 7. The plane-wave, frequency-angle response of a polycarbonate-backed sensor predicted by the model of Section III. The parametersused were: 40-µm Parylene C layer (cl = 2200 m/s, cs = 1100 m/s,ρ = 1180 kg/m3), polycarbonate substrate (cl = 2180 m/s, cs =960 m/s, ρ = 1180 kg/m3), 400-µm radius top-hat interrogationbeam.


sensitive at large angles of incidence and high frequencies.The large transducer-sensor separation was a necessary re-quirement to ensure that the sensor was in the transducer’sfar field so that the wavefronts were essentially plane whenthey reached it. As well as the noise introduced by the lowSNR, there was some experimental error due to the diffi-culty of aligning the experimental system to ensure thatthe illuminated, sensitive point on the detector remainedat the same point in the field of the ultrasound transducerfor every measurement angle.

V. Applications

A. Sensor Design

Perhaps the most direct application of the model is itsuse as a predictive tool to improve the sensor design itselfthrough careful investigation of the effects of the mate-rial and geometric parameters. For instance, by replacingthe glass substrate with polycarbonate (cl = 2180 m/s,cs = 960 m/s, ρ = 1180 kg/m3), which has a lower shearspeed than the sound speed in the water, the vertical fea-ture, B, can be eliminated, Fig. 7. The minimum in Fig. 7at 42◦ corresponds to the compressional wave critical an-gle, feature A in Fig. 4, although clearly shifted from 15◦

because of the slower wave speed. At the critical angle thecompressional wave is traveling perpendicular to the sen-sor inside the polymer layer and so causes no motion in thevertical direction. The minimum remains at a constant an-gle as the frequency increases because the acoustic prop-erties of the polycarbonate and Parylene are similar (sothere is no significant acoustic boundary between them);therefore, even when the polymer layer is a significant pro-portion of a wavelength, interference effects do not dom-inate (d is no longer a characteristic length scale for theproblem).

B. Effective Element Radius

It is often assumed that the size of the acoustically sen-sitive measuring “element” is given by the size of the regionilluminated with the interrogation light beam. In fact, asignal arriving at the sensor outside the illuminated areamay propagate as a surface wave to the illuminated regionand be detected. This phenomenon has also been notedin piezoelectric PVDF detectors such as membrane hy-drophones, in which Lamb waves propagate to the sensi-tive region from outside it. These guided surface waves andother modes cause the directivity of the sensor to deviatefrom that of a purely spatially-averaging pressure detector.The “effective” element radius is a useful but approximatemeasure of a sensor’s directional characteristics [21], [22],obtained by comparing the directivity of the real sensorto the directivity of a rigid, circular, pressure detector ofradius aeff, whose directivity is due purely to spatial aver-aging and is given by:

Deff =2J1(kxaeff)

kxaeff, (31)

Fig. 8. Directivity plots at frequencies of 1–13 MHz obtained bytaking horizontal profiles through Fig. 7 and normalizing to normalincidence. Also shown is the directivity function J1(kxaeff)/(kxaeff),for an ideal, rigid, circular, pressure detector of radius a = 400 µm(dashed line) indicating that, in this case, the effective element radiusequals the illuminated spot radius.

The value of aeff for which Deff(aeff) most closelymatches the measured directivity is called the effective el-ement radius, irrespective of the radius of the illuminatedspot. The effective and optically defined element radii dif-fer when the sensor does not behave like a rigid, circular,pressure detector. In this section the directivity model isused to answer two, related, questions:

• Under what conditions is the effective element radiusthe same as the illumination spot radius?

• Is there a minimum, attainable, effective element ra-dius for a given sensor thickness d? Is there a limit be-yond which a smaller illumination spot radius makesno difference to the directivity pattern?

In Fig. 8 the solid lines show the directivity of a 40-µm thick, polycarbonate-backed sensor with a circular il-luminated spot of 400-µm radius (i.e., horizontal profilesthrough Fig. 7), and the dashed lines are the function Deff,also with radius aeff = 400 µm. The good agreement be-tween the two directivity functions suggests that, in thiscase, with an illuminated spot radius much larger than the


Fig. 9. Directivity plots from the model (solid line) of a 40-µm thick,polycarbonate-backed sensor at a frequency of 25 MHz for illumina-tion spot radii of 160, 80, 40, 20, 10, and 5 µm shown on a linear scale.Also shown (dashed-dotted line) is the function J1(kxaeff)/(kxaeff)at which the radius aeff has been found through a best-fit over anglesfrom 0 to 40◦ (the dashed part of the line).

sensor layer thickness, the effective element radius corre-sponds closely to the illumination spot radius, at least forfrequencies above 3 MHz, confirming our intuitive expec-tation.

Fig. 9 shows an example in which the approximationaeff ≈ a begins to break down as the illuminating spot ra-dius is reduced. The solid lines show the directivity, on alinear scale, at 25 MHz for the 40-µm sensor for decreas-ing illumination spot radii of 160, 80, 40, 20, 10, and 5 µm(from 4 to 1/8 times the FP layer thickness). Eq. (31)was fitted to these directivity patterns in a least-squaressense for angles from 0 to 40◦, as at larger angles the rigiddisc approximation differs significantly from the sensormodel, and the comparison would have been meaningless.Similarly, approximate but useful small-angle comparisonshave been made in the past for hydrophone directivities,by comparing the widths of the first lobe (beamwidths)and ignoring higher-angle effects [21], [22]. In Fig. 9 wesee that, for large a, the spatial-averaging term, Ψ from(29), dominates the behavior, and we see good agreementbetween aeff and a. As a is reduced in size, Ψ approachesunity and interference effects within the sensor affect the

Fig. 10. The effective-element radius, aeff, as a function of the illumi-nated spot radius, a, for a sensor thicknesses d = 40 and 25 µm at afrequency of 25 MHz (polycarbonate-backed sensor). As the illumi-nated spot radius a is reduced, the effective-element radius eventuallyreaches a minimum value below which further reductions in a haveno effect.

response more noticeably, so that, when a has been re-duced to half the sensor thickness d/2, the effective radiusaeff is still twice this size. By reducing a still further, wesee that a minimum aeff = 36 µm is reached at a = d/4,10 µm, so that any further reductions in a lead to no fur-ther reduction in aeff.

In Fig. 10, which shows the effective radius, aeff, as afunction of the illuminated spot radius, a, for sensor thick-nesses of 40 and 25 µm and a frequency of 25 MHz, thiseffect can be seen clearly. The effective radius reduces to aminimum below which any further reductions in the illu-minated spot radius leave the effective radius unchanged.Fig. 11, which shows the normalized effective-element ra-dius as a function of the normalized wavenumber ka fora number of fixed spot radii, gives a more general result.From Fig. 11, it is clear that, to achieve the absolute mini-mum, requires both an illuminated spot size less than d/4and a sufficiently high frequency. At lower frequencies, tothe left of the graph, a cosine-type dependence charac-teristic of a gradient or difference detector dominates theresponse, and the effective element radius becomes muchlarger than the illuminated spot radius. We can now seethat the examples in Figs. 9 and 10 were all chosen sothat the frequency was sufficiently high to ensure the min-imum effective radius reached was the absolute minimumachievable.

Three conclusions may be drawn from these graphs:

• for sufficiently large spot sizes and frequencies (a > 2dand ka � 1) the effective element radius equals thespot radius aeff = a,

• it is not possible to achieve an effective element ra-dius smaller than ∼ 0.9d however small the illumi-nated spot radius, and

• to achieve this minimum, effective-element radius, theilluminated spot radius must be less than d/4, and thefrequency sufficiently high, kd � 4.


Fig. 11. The effective-element radius, aeff, normalized by the FP layerthickness d, as a function of ka, the wavenumber normalized by theillumination spot radius, showing that, for a > 2d and ka � 1, theeffective element radius aeff = a; however small the illuminated spotradius, it is not possible to achieve an effective element radius smallerthan ∼ 0.9d; and to achieve this minimum effective-element radiusthe actual spot radius must be less than d/4 and the frequency suchthat kd � 4 (polycarbonate-backed sensor).

So, for the 40-µm thick polymer layer studied here,the minimum attainable, effective-element radius is about36 µm, and this can be achieved only with an illuminationspot radius of about 10 µm and at frequencies above about20 MHz.

VI. Conclusions

A model of the directivity and frequency response of aplanar, FP ultrasound sensor has been described and ex-perimentally validated. Although a three-layer geometrywas used here, the model may be straightforwardly ex-tended to sensors with an arbitrary number of layers [19].

Such sensor directivity models have many possible uses.Examples of two applications were given here: the modelwas used as a design tool to improve the sensor’s direc-tional characteristics, and the relationship between ac-tual and effective element radius was investigated. It wasfound that, for a given sensor thickness, d, the mini-mum achievable, high-frequency, effective-element radiusis about 0.9d. There are many other potential applications.To improve accuracy in high frequency ultrasonic calibra-tion and metrology, it is becoming increasingly common totake into account the normal-incidence frequency responseof the sensor [23], [24]. For sound field characterization, inwhich the sound waves are not necessarily normally in-cident, an improvement in accuracy may be achieved byusing the model to deconvolve the directional response ofthe sensor too. In a similar way, deconvolution of the sen-sor response could reduce artefacts and blurring in imag-ing applications, in which images are formed by spatiallymapping the output of the sensor. As well as removing theeffects of the detector response from measurements, the

model could be included in a forward model of ultrasonicpropagation to predict more accurately real measurementsof an ultrasound field [25].

Appendix A

As stated in Section III-A, a change in the optical phasecould be due to a physical change in the thickness of theFP layer or by a change in its refractive index. This secondmechanism, whereby a strain causes a change in refractiveindex, is called the photoelastic effect. In Section III thismechanism was assumed to be negligible, and the opticalphase change was modeled as purely due to a change inthe thickness. The reasons for making this assumption arediscussed in this appendix.

If an acoustic wave traveling through the sensor causesa time-varying stress birefringence in the sensor, the direc-tivity measurements would be expected to depend on thepolarization of the interrogating laser beam. Experimentswith beams of different polarization showed no significantdifferences in the directivity measurements. However, al-though this rules out an anisotropic change in refractiveindex due to stress birefringence, it does not eliminate thepossibility of an isotropic change in the refractive indexdue to the density changes induced by the wave. This pos-sible transduction mechanism is discussed below, and italso is found to be insignificant for the examples studiedhere.

In the absence of an acoustic wave, the difference inoptical phase between light reflected from the two sidesof the interferometer is φ = 4πnd/λ0 radians. The changein phase due to the two mechanisms, therefore, may bewritten as:

∆φ =∂φ

∂d∆d +

∂φ

∂n∆n =

4πλ0

(n0∆d + d∆n) .(32)

The second term—due to the photoelastic effect—maybe neglected if:

∆nn0

� ∆dd

. (33)

Following Pitts and Greenleaf [26], a change in the re-fractive index of an isotropic, dielectric medium may berelated to a small change in the density, ∆ρ = ρ − ρ0, by:

∆n =(

n20 − 12n0

)∆ρρ0

, (34)

where ρ and ρ0 are the densities in the presence and ab-sence of a wave, respectively. As the dilatation ∇ · u, thesum of the volumetric strains is related to the density by:

∇ · u = ρ0ρ

− 1, (35)

the change in refractive index can be written:

∆n =(

n20 − 12n0

)(−∇ · u

1 + ∇ · u

). (36)


This change can be calculated using the wave modeldescribed in Section III-B. Calculations of the ratiod∆n/n0∆d showed it be to below 0.001 for the range offrequencies, layer thicknesses, and types of materials dis-cussed here. In addition, this isotropic photoelastic mech-anism predicts a frequency response quite different fromthat measured experimentally [6] and exhibits peaks andtroughs in the directional response that disagree with ex-periment results (Fig. 4).

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The Frequency-Dependent Directivity of a Planar Fabry-Perot … · 2014. 8. 15. · Perot (FP) polymer ﬁlm sensor has been described [6]. However, this model provided the sensor