Light-modulating pressure sensor with integrated flexible organic light-emitting diode D. Cheneler,1,* M. Vervaeke,2 and H. Thienpont2 1 2
Department of Engineering, University of Lancaster, Lancaster, LA1 4YW, UK
Vrije Universtiteit Brussel, Brussels Photonics Team, Pleinlaan 2, B-1050 Brussels, Belgium *Corresponding author: [email protected] Received 29 October 2013; revised 26 March 2014; accepted 26 March 2014; posted 27 March 2014 (Doc. ID 200369); published 23 April 2014
Organic light-emitting diodes (OLEDs) are used almost exclusively for display purposes. Even when implemented as a sensing component, it is rarely in a manner that exploits the possible compliance of the OLED. Here it is shown that OLEDs can be integrated into compliant mechanical micro-devices making a new range of applications possible. A light-modulating pressure sensor is considered, whereby the OLED is integrated with a silicon membrane. It is shown that such devices have potential and advantages over current measurement techniques. An analytical model has been developed that calculates the response of the device. Ray tracing numerical simulations verify the theory and show that the design can be optimized to maximize the resolution of the sensor. © 2014 Optical Society of America OCIS codes: (080.0080) Geometric optics; (130.3120) Integrated optics devices; (130.6010) Sensors; (160.4890) Organic materials; (250.3680) Light-emitting polymers; (280.5475) Pressure measurement. http://dx.doi.org/10.1364/AO.53.002766
1. Introduction
Organic optoelectronic devices, such as organic lightemitting diodes (OLEDs), field-effect transistors (OFETs), and photovoltaic cells (OPVs) are a maturing technology. Their most significant drawbacks— short lifetime and efficiency [1–4]—have been improved dramatically due to the intense research activity in this area in recent years. Being an innate surface-emitting light source, OLEDs are showing great commercial potential in the lighting industry, in mobile devices, and even in OLED televisions. Indeed OLEDs have been considered for most flat-panel display applications [4–6]. However, OLEDs constructed from multilayer organic thin films [7] can be formed on most substrates such as glass, metals, ceramics, and even flexible surfaces. Therefore, they also have a huge range of heretofore1559-128X/14/132766-07$15.00/0 © 2014 Optical Society of America 2766
APPLIED OPTICS / Vol. 53, No. 13 / 1 May 2014
unexploited potential applications, if they can be integrated into mechanical devices [8]. Flexible organic electronics that can be rolled, bent, and stretched have found a natural application in large-area sensors, particularly useful in robotics and medical monitoring [9–12]. These sensors have largely taken the same form of their rigid counterparts albeit on a polymeric substrate and with more robust circuitry. They rarely include an optoelectronic component, except for a few exceptions. A number of optical chemical and biological sensors have successfully integrated OLEDs [13–19]. The OLED is fixed in these devices and do not exploit their innate compliance, and, as of to date, they have not been implemented as a mechanical sensor, except as a light source in flexible waveguides acting as pressure sensors [20,21]. The cost of manufacturing optoelectronic devices is thus reduced rapidly [6,22], particularly when integrable light detection systems are taken into account [1,23], making them a viable alternative
technology advantages over more classical designs such as silicon-based pressure sensors. Optoelectronic sensors have other advantages [24], such as negating the need for vacuum cavities to maintain high Q-factors needed for resonant devices [25,26]. Also complex electrothermal or electrostatic excitation [27,28], and related compensation/feedback systems [28–30] are not needed, as a simple DC potential will activate the OLED. The aim of this study was to determine the feasibility of using light modulation as a dynamic force transduction mechanism. More specifically, it was to verify whether OLED technology could be integrated into a compliant micro-device as a useful sensing element. The micro-device being considered is a pressure-sensitive membrane, such as that commonly used in piezoresistive and capacitive pressure sensors. The membrane is coated with an OLED layer and thin-film encapsulation. A generic device such as a charged couple device (CCD) or lightdependent resistor (LDR) is assumed to measure the light emitted by the OLED through an aperture. In order to ascertain the effectiveness of this device, a sophisticated analytical model detailing the intensity of photons at the aperture as the OLED membrane deforms under the applied pressure was derived. This was achieved via calculation of the view factor between the aperture and the OLED, with the OLED being treated as a laminated plate. The theory was validated numerically using a commercial nonsequential optical ray-tracing tool. Calculations and parameterized simulations show that the OLED membrane needs to be selectively masked with the maximum change in the intensity with pressure occurring when the diameter of the light-emitting surface matches that of the aperture. 2. Principles of the Device
As shown in Fig. 1, the design considered is a micromachined flexible circular membrane upon which is deposited multiple layers of organic polymers and pertinent electrodes forming a generic organic light emitting diode (see [3–5] for examples). The organic layers are covered with a thin-film encapsulation
layer for protection to extend the life of the device. This aspect is the most challenging because OLEDs require the excellent protection from moisture and oxygen penetration in order for the device to have a sufficiently long lifespan to be useful [5]. This layer also adds to the stiffness of the membrane, decreasing the resolution and necessitating optimization of the design. The entire surface of the OLED is assumed to emit photons isotropically at a constant intensity over the entire surface. It will be shown later that it is necessary to cover the OLED with an opaque mask that only permits photons to pass through a small circular opening at the center of the membrane. A small distance above the OLED is an aperture, which permits a portion of the emitted photons to reach a detection system, such as an OPV or LDR. A pressure difference across the membrane causes it to deflect and change the amount of photons that reach the aperture and, in turn, the detection system. The output of the detection system is therefore a direct measure of the applied pressure. 3. Theory A. Membrane Deflection
The light intensity at the aperture changes with pressure due to the deflection of the membrane. This deflection has two effects: it moves the light-emitting surface closer to the aperture while changing the relative gradient of that surface. The deflection of the membrane in the light modulator is derived using classical laminated plate theory (CLPT) [31,32]. CLPT is essentially based on the Kirchhoff’s plate theory wherein a multilayered heterogeneous plate structure is reduced to a kinematically equivalent single layer, thereby simplifying a 3D elasticity problem to an equivalent 2D problem. In this case, the membrane takes the form of a circular plate, clamped all around the edges and with constant pressure acting across it. The deflection, wx; y, can be shown to be wx; y
P R2 − x2 y2 2 ; 64D i
(1)
where P is the applied pressure, Ri is the membrane radius, x and y are the Cartesian coordinates, and D is the flexural rigidity defined as D
Fig. 1. Schematic of the light modulating pressure sensor. The arrows in the center denote the coordinate system used in the derivation.
A11 D11 − B211 : A11
(2)
A, B, and D are matrices defined in Appendix A. Later it shall be shown that issues arise when the whole surface of the membrane is allowed to emit light. Here the problem has been negated by adding an extra mask layer, which limits the area from which photons are emitted. An alternative solution would be to have a smaller OLED specifically deposited to form a small circular light-emitting area in 1 May 2014 / Vol. 53, No. 13 / APPLIED OPTICS
2767
the center of the membrane or indeed off center, which will also provide interesting results. This case is not considered here, but the consequence of this alternative design would be the need to include the effects of partial lamination. How to do this is discussed in [33].
The aperture, surface j, is planar and horizontal and so has a simple vertical unit normal given by
B.
The aperture is a distance h from the un-deformed light-emitting surface. Therefore the position vector of a point on the aperture with coordinates xj ; yj ; h and a point on the membrane with coordinates xi ; yi ; wxi ; yi is given by
Intensity as a Function of Pressure
Given that the OLED layer emits light diffusely [7], i.e., it behaves like a Lambertian surface in that the surface luminance is isotropic; the proportion of emitted photons that reaches the aperture is given by the view factor. The view factor between two infinitesimal elements on the light-emitting surface, dAi , and, at the aperture, dAj , is a function of the relative orientation of the two areas and the distance between them. In general, the view factor, dF, is defined as [34] dF dAi →dAj
nˆ i · sij nˆ j · sji dAj ; πS4
(3)
where nˆ is the unit normal of the infinitesimal areas located at ri and rj , respectively. S is the distance between the areas, defined as the magnitude of sij ri − rj . The radiative intensity, I, is defined as the energy emitted as photons per unit solid angle, per unit time and unit area normal to the direction of emittance. The flux of photons, J, emitted by a surface per unit area can therefore be defined as [34]
nˆ j 0; 0; 1:
(9)
sij xj − xi ; yj − yi ; h − wxi ; yi ; with magnitude S xj − xi 2 yj − yi 2 h − wxi ; yi 2 1∕2 :
2π
Ir; sˆ nˆ · sˆ dΩ;
(4)
where Ω is the solid angle of the area of the projected photons and sˆ is the direction vector of the photons. The total energy leaving dAi toward surface Aj is Z EdAi →Aj Iri dAi
nˆ i · sij nˆ j · sji dAj : πS4 Aj
(5)
0s1 ∂wxi ; yi 2 ∂wxi ; yi 2 1Adxi dyi : dAi @ ∂xi ∂yi (12) For the aperture, it is simply dAj dxj dyj :
q Iri I 0 Φ Rm − x2i y2i :
Z Iri dAi :
Z Z Ai
Aj
Iri
(6)
nˆ i · sij nˆ j · sji dAj dAi : πS4
Φx is the Heaviside step function, therefore Eq. (14) equals I 0 when the distance from the center is less than the radius of the mask, Rm , and zero everywhere else. Equation (7) becomes
(7) EAi →Aj
The unit vector of the light-emitting surface can be defined, given Eq. (1), as: ∂wx;y − ∂wx;y ∂x ; − ∂y ; 1 : nˆ i r ∂wx;y 2 ∂wx;y 2 1 ∂x ∂y APPLIED OPTICS / Vol. 53, No. 13 / 1 May 2014
I 0 π
Z
Ri −Ri
Z p Z R2 −y2 i
−
i
p 2 2 Ri −yi
Rj Rj
Z −p R2 −y2 j
p 2 2
−
q × Φ Rm − x2i y2i …
2768
(14)
Ai
Therefore the total energy leaving Ai toward surface Aj is EAi →Aj
(13)
The masked area prevents light from being emitted from everywhere except for an area of constant intensity, I 0 , and radius Rm, in the center of the membrane. The intensity is therefore
The total amount of energy leaving Ai is EAi π
(11)
The differential surface area of the light-emitting surface is
Z Jr
(10)
Rj −y
× f xj ; yj ; xi ; yi dxj dyj dxi dyi ;
(8) where
(15)
h f xj ; yj ; xi ; yi
i i ;yi i ;yi − ∂wx fxj − xi g − ∂wx fyj − yi g fh − wxi ; yi g h − wxi ; yi ∂xi ∂yi xj − xi 2 yj − yi 2 h − wxi ; yi 2 2
(16)
Upon substitution of Eq. (1), Eq. (16) becomes h f xj ; yj ; xi ; yi
i Pyi − x2i y2i fxj − xi g 16D R2i − x2i y2i fyj − yi g z z i2 2 h P xj − xi 2 yj − yi 2 h − 64D R2i − x2i y2i 2
Pxi 2 16D Ri
(17)
where z h − P∕64DR2i − x2i y2i 2 . Converting to polar coordinates and setting θ θi − θj and integrating over θi gives EAi →Aj
−I 0 32D
Z
Rj
rj 0
Z
Rm ri 0
Z
2π θ0
gri ; rj ; θdθdri drj
(18)
with 4Pri rj cos θR2i − r2i PR4i − 3r4i 2PR2i r2i − 64hDri rj P 2 2 2 h i2 2 R − ri : gri ; rj ; θ h− P 64D i R2i − r2i 2 r2i r2j − 2ri rj cos θ h − 64D
This is the fraction of energy that reaches an aperture given an OLED membrane light source. Note that when P 0, Eq. (19) becomes identical to the expression between two flat discs given in [35]. 4. Numerical Simulations
Numerical simulations were carried out using a commercial nonsequential optical ray-tracing tool, Zemax. Nonsequential ray tracing treats optical rays as straight paths from one surface to another. At each surface, the ray can reflect, refract, scatter, or be absorbed. Ray directions and interactions are calculated in 3D throughout the complete system. Each run simulates 108 rays. The source was modeled as a deformable circular disc emitting rays in a Lambertian emission pattern. The deformation of the disc was governed by Eq. (1), and the so simulation was directly a function of the applied pressure. It was assumed that the rays traveling toward the top cover, but not through the aperture, are fully absorbed. In doing so, the simulation will yield an underestimation of the optical power going to the detection plane. 5. Results and Discussion
To simulate the performance of the pressure sensor, it was assumed the membrane had a radius of 4.2 mm with an effective flexural rigidity of 1.72 × 10−3 N∕m. The initial gap between the lightemitting surface and the aperture was set at 425 μm. The maximum pressure difference across the membrane was 150 kPa, which resulted in a maximum deflection of 424 μm. Therefore, the minimum distance between the light-emitting surface and the aperture was just less than 1 μm. In Fig. 2,
(19)
the relative change in the output signal amplitude calculated from Eq. (18), in decibels, for various aperture and mask dimensions is given. It can be seen that if the membrane is not masked so that the entire surface can emit light, then the relative change in signal amplitude for any given deflection is negligible. This is a consequence of the Lambertian nature of the OLED, which emits light equally in all directions. As it is assumed that light is not attenuated in the sensor, changing the distance photons need to travel by deflecting the surface has no effect. This is only true when the area of the light-emitting surface is much larger than the aperture and so is effectively infinite. Any photons that are directed away from the aperture in this case due to the deflection of the membrane are compensated by photons that are consequently directed toward the aperture from elsewhere on the light-emitting surface. When the radius of the mask is comparable to the radius of the aperture, the solid angle and, hence, the view factor can increase significantly when the membrane is deflected and the reduced light-emitting surface is brought closer to the aperture. It should be noted, however, that the mask reduces the overall signal amplitude by the same ratio as the area of the exposed light-emitting surface to the area of the membrane. In order to determine the signal amplitude between a pressurized and nonpressurized membrane, we have performed numerous automated parameter sweeps to determine the ideal configuration of the sensor for a large signal swing between an unpressurized and pressurized membrane. The maximum signal swing is found with the largest membrane 1 May 2014 / Vol. 53, No. 13 / APPLIED OPTICS
2769
Fig. 3. Signal amplitude ratio (dB) between applied pressures of 0 and 150 kPa. The red line is the predicted response from Eq. (18), and the black circles denote that predicted from the numerical analysis. Membrane radius is 4.2 mm, the OLED active area radius is 0.5 mm, the gap between membrane and aperture is 425 μm, and the top cover thickness is 100 μm. The depicted sweep of the camera aperture radius shows an optimum at 0.5 mm, i.e., the same radius as the active area.
Fig. 2. Change in signal amplitude calculated from Eq. (18) due to change in the applied pressure relative to the signal amplitude when there is no deflection for different mask radii (Rm). (a) Aperture radius is 50 μm. (b) Aperture radius is 500 μm.
In Fig. 5 we determine that the smallest possible top cover thickness will yield the best performance with regards to signal variation and, hence, sensor resolution in the 0 to 150 kPa pressure range. A thicker top cover leads to a larger aspect ratio of the aperture and, hence, a smaller acceptance cone of the aperture with respect to the emissive surface. For the given pressure range this leads to smaller signal swings. It has been shown that for a light-modulating pressure sensor with the dimensions and rigidity discussed at the beginning of this section, the pressure range detectable is 150 kPa. It was also explained
displacement. This means we need to implement the largest possible membrane diameter (4.2 mm) and the largest possible gap (425 μm) between the OLED and the top cover of the stack. Figure 3 suggests that for a given geometry the highest difference in signal strength between an un-deformed and deformed membrane can be found when the OLED active area radius and the camera aperture radius match. Figure 4 shows the evolution of the signal amplitude with varying camera aperture radius while the OLED active area radius is matched to it. The smaller the aperture and matched OLED, the larger the signal amplitude. This behavior can be explained by observing that a small camera aperture radius limits the cone of light that can reach the detection surface. Any deformation or shift in position of the OLED on the membrane will result in a different match between the emission light cone and acceptance light cone. Matching the aperture of the camera and the active area of the OLED yields a match between the etendue of the emissive surface and the detector and, hence, a large signal swing between the deformed and un-deformed membrane.
Fig. 4. Signal amplitude ratio (dB) between applied pressures of 0 and 150 kPa. The red line is the predicted response from Eq. (18), and the black circles denote that predicted from the numerical analysis. Membrane radius is 4.2 mm, the gap between membrane and aperture is 425 μm, and the top cover thickness is 100 μm. OLED active area radius equals the camera aperture radius. Smallest aperture in the simulation is 50 μm radius.
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pressure sensor and was validated using a commercial ray-tracing tool. It has also been shown that in order to be effective, the compliant diode cannot emit light over the entire surface of the diode and that the sensor will offer the best resolution when the lightemitting surface has a similar radius to the aperture. Appendix A:Derivation of the Membrane Deflection Equation
A, B, and D are the extensional stiffness matrix, extensional-bending coupling stiffness matrix and bending stiffness matrix, respectively, and are expressed as A Fig. 5. Normalized signal amplitude of sensor between applied pressures of 0 and 150 kPa for different aperture thicknesses. Membrane radius is 4.2 mm, and the gap between membrane and aperture is 425 um. The OLED active area radius equals the camera aperture radius (0.5 mm). A large thickness limits the amount of angles that are admitted through the aperture because of the increased aspect ratio.
A12
A12 A11 " Pn Ek
k1 1−υ2 zk k
− zk−1
υk E k k1 1−υ2 zk k
− zk−1
Pn
Pn
υk E k k1 1−υ2 zk k
− zk−1
Ek k1 1−υ2 zk k
− zk−1
Pn
# ;
(A1)
earlier that the light intensity variations due to changes in pressure are detectable using a wide range of common components such as LDRs. Such components are easily interrogated using simple circuitry such as potential divider circuits and data acquisition systems (DAQs). Modern DAQs can routinely measure voltages with a resolution of 18 bits. When combined with a LDR circuit designed to operate over the full input range of the DAQ, it can be seen that the sensor described here has a theoretical resolution of 0.57 Pa. This resolution and range is better than the 0.9 Pa resolution claimed in [36], which states their Au-coated polydimethylsiloxane strain-gauge based pressure sensor exhibits the best results reported so far for any flexible membranebased sensor over the pressure range tested (
Department of Engineering, University of Lancaster, Lancaster, LA1 4YW, UK
Vrije Universtiteit Brussel, Brussels Photonics Team, Pleinlaan 2, B-1050 Brussels, Belgium *Corresponding author: [email protected] Received 29 October 2013; revised 26 March 2014; accepted 26 March 2014; posted 27 March 2014 (Doc. ID 200369); published 23 April 2014
Organic light-emitting diodes (OLEDs) are used almost exclusively for display purposes. Even when implemented as a sensing component, it is rarely in a manner that exploits the possible compliance of the OLED. Here it is shown that OLEDs can be integrated into compliant mechanical micro-devices making a new range of applications possible. A light-modulating pressure sensor is considered, whereby the OLED is integrated with a silicon membrane. It is shown that such devices have potential and advantages over current measurement techniques. An analytical model has been developed that calculates the response of the device. Ray tracing numerical simulations verify the theory and show that the design can be optimized to maximize the resolution of the sensor. © 2014 Optical Society of America OCIS codes: (080.0080) Geometric optics; (130.3120) Integrated optics devices; (130.6010) Sensors; (160.4890) Organic materials; (250.3680) Light-emitting polymers; (280.5475) Pressure measurement. http://dx.doi.org/10.1364/AO.53.002766
1. Introduction
Organic optoelectronic devices, such as organic lightemitting diodes (OLEDs), field-effect transistors (OFETs), and photovoltaic cells (OPVs) are a maturing technology. Their most significant drawbacks— short lifetime and efficiency [1–4]—have been improved dramatically due to the intense research activity in this area in recent years. Being an innate surface-emitting light source, OLEDs are showing great commercial potential in the lighting industry, in mobile devices, and even in OLED televisions. Indeed OLEDs have been considered for most flat-panel display applications [4–6]. However, OLEDs constructed from multilayer organic thin films [7] can be formed on most substrates such as glass, metals, ceramics, and even flexible surfaces. Therefore, they also have a huge range of heretofore1559-128X/14/132766-07$15.00/0 © 2014 Optical Society of America 2766
APPLIED OPTICS / Vol. 53, No. 13 / 1 May 2014
unexploited potential applications, if they can be integrated into mechanical devices [8]. Flexible organic electronics that can be rolled, bent, and stretched have found a natural application in large-area sensors, particularly useful in robotics and medical monitoring [9–12]. These sensors have largely taken the same form of their rigid counterparts albeit on a polymeric substrate and with more robust circuitry. They rarely include an optoelectronic component, except for a few exceptions. A number of optical chemical and biological sensors have successfully integrated OLEDs [13–19]. The OLED is fixed in these devices and do not exploit their innate compliance, and, as of to date, they have not been implemented as a mechanical sensor, except as a light source in flexible waveguides acting as pressure sensors [20,21]. The cost of manufacturing optoelectronic devices is thus reduced rapidly [6,22], particularly when integrable light detection systems are taken into account [1,23], making them a viable alternative
technology advantages over more classical designs such as silicon-based pressure sensors. Optoelectronic sensors have other advantages [24], such as negating the need for vacuum cavities to maintain high Q-factors needed for resonant devices [25,26]. Also complex electrothermal or electrostatic excitation [27,28], and related compensation/feedback systems [28–30] are not needed, as a simple DC potential will activate the OLED. The aim of this study was to determine the feasibility of using light modulation as a dynamic force transduction mechanism. More specifically, it was to verify whether OLED technology could be integrated into a compliant micro-device as a useful sensing element. The micro-device being considered is a pressure-sensitive membrane, such as that commonly used in piezoresistive and capacitive pressure sensors. The membrane is coated with an OLED layer and thin-film encapsulation. A generic device such as a charged couple device (CCD) or lightdependent resistor (LDR) is assumed to measure the light emitted by the OLED through an aperture. In order to ascertain the effectiveness of this device, a sophisticated analytical model detailing the intensity of photons at the aperture as the OLED membrane deforms under the applied pressure was derived. This was achieved via calculation of the view factor between the aperture and the OLED, with the OLED being treated as a laminated plate. The theory was validated numerically using a commercial nonsequential optical ray-tracing tool. Calculations and parameterized simulations show that the OLED membrane needs to be selectively masked with the maximum change in the intensity with pressure occurring when the diameter of the light-emitting surface matches that of the aperture. 2. Principles of the Device
As shown in Fig. 1, the design considered is a micromachined flexible circular membrane upon which is deposited multiple layers of organic polymers and pertinent electrodes forming a generic organic light emitting diode (see [3–5] for examples). The organic layers are covered with a thin-film encapsulation
layer for protection to extend the life of the device. This aspect is the most challenging because OLEDs require the excellent protection from moisture and oxygen penetration in order for the device to have a sufficiently long lifespan to be useful [5]. This layer also adds to the stiffness of the membrane, decreasing the resolution and necessitating optimization of the design. The entire surface of the OLED is assumed to emit photons isotropically at a constant intensity over the entire surface. It will be shown later that it is necessary to cover the OLED with an opaque mask that only permits photons to pass through a small circular opening at the center of the membrane. A small distance above the OLED is an aperture, which permits a portion of the emitted photons to reach a detection system, such as an OPV or LDR. A pressure difference across the membrane causes it to deflect and change the amount of photons that reach the aperture and, in turn, the detection system. The output of the detection system is therefore a direct measure of the applied pressure. 3. Theory A. Membrane Deflection
The light intensity at the aperture changes with pressure due to the deflection of the membrane. This deflection has two effects: it moves the light-emitting surface closer to the aperture while changing the relative gradient of that surface. The deflection of the membrane in the light modulator is derived using classical laminated plate theory (CLPT) [31,32]. CLPT is essentially based on the Kirchhoff’s plate theory wherein a multilayered heterogeneous plate structure is reduced to a kinematically equivalent single layer, thereby simplifying a 3D elasticity problem to an equivalent 2D problem. In this case, the membrane takes the form of a circular plate, clamped all around the edges and with constant pressure acting across it. The deflection, wx; y, can be shown to be wx; y
P R2 − x2 y2 2 ; 64D i
(1)
where P is the applied pressure, Ri is the membrane radius, x and y are the Cartesian coordinates, and D is the flexural rigidity defined as D
Fig. 1. Schematic of the light modulating pressure sensor. The arrows in the center denote the coordinate system used in the derivation.
A11 D11 − B211 : A11
(2)
A, B, and D are matrices defined in Appendix A. Later it shall be shown that issues arise when the whole surface of the membrane is allowed to emit light. Here the problem has been negated by adding an extra mask layer, which limits the area from which photons are emitted. An alternative solution would be to have a smaller OLED specifically deposited to form a small circular light-emitting area in 1 May 2014 / Vol. 53, No. 13 / APPLIED OPTICS
2767
the center of the membrane or indeed off center, which will also provide interesting results. This case is not considered here, but the consequence of this alternative design would be the need to include the effects of partial lamination. How to do this is discussed in [33].
The aperture, surface j, is planar and horizontal and so has a simple vertical unit normal given by
B.
The aperture is a distance h from the un-deformed light-emitting surface. Therefore the position vector of a point on the aperture with coordinates xj ; yj ; h and a point on the membrane with coordinates xi ; yi ; wxi ; yi is given by
Intensity as a Function of Pressure
Given that the OLED layer emits light diffusely [7], i.e., it behaves like a Lambertian surface in that the surface luminance is isotropic; the proportion of emitted photons that reaches the aperture is given by the view factor. The view factor between two infinitesimal elements on the light-emitting surface, dAi , and, at the aperture, dAj , is a function of the relative orientation of the two areas and the distance between them. In general, the view factor, dF, is defined as [34] dF dAi →dAj
nˆ i · sij nˆ j · sji dAj ; πS4
(3)
where nˆ is the unit normal of the infinitesimal areas located at ri and rj , respectively. S is the distance between the areas, defined as the magnitude of sij ri − rj . The radiative intensity, I, is defined as the energy emitted as photons per unit solid angle, per unit time and unit area normal to the direction of emittance. The flux of photons, J, emitted by a surface per unit area can therefore be defined as [34]
nˆ j 0; 0; 1:
(9)
sij xj − xi ; yj − yi ; h − wxi ; yi ; with magnitude S xj − xi 2 yj − yi 2 h − wxi ; yi 2 1∕2 :
2π
Ir; sˆ nˆ · sˆ dΩ;
(4)
where Ω is the solid angle of the area of the projected photons and sˆ is the direction vector of the photons. The total energy leaving dAi toward surface Aj is Z EdAi →Aj Iri dAi
nˆ i · sij nˆ j · sji dAj : πS4 Aj
(5)
0s1 ∂wxi ; yi 2 ∂wxi ; yi 2 1Adxi dyi : dAi @ ∂xi ∂yi (12) For the aperture, it is simply dAj dxj dyj :
q Iri I 0 Φ Rm − x2i y2i :
Z Iri dAi :
Z Z Ai
Aj
Iri
(6)
nˆ i · sij nˆ j · sji dAj dAi : πS4
Φx is the Heaviside step function, therefore Eq. (14) equals I 0 when the distance from the center is less than the radius of the mask, Rm , and zero everywhere else. Equation (7) becomes
(7) EAi →Aj
The unit vector of the light-emitting surface can be defined, given Eq. (1), as: ∂wx;y − ∂wx;y ∂x ; − ∂y ; 1 : nˆ i r ∂wx;y 2 ∂wx;y 2 1 ∂x ∂y APPLIED OPTICS / Vol. 53, No. 13 / 1 May 2014
I 0 π
Z
Ri −Ri
Z p Z R2 −y2 i
−
i
p 2 2 Ri −yi
Rj Rj
Z −p R2 −y2 j
p 2 2
−
q × Φ Rm − x2i y2i …
2768
(14)
Ai
Therefore the total energy leaving Ai toward surface Aj is EAi →Aj
(13)
The masked area prevents light from being emitted from everywhere except for an area of constant intensity, I 0 , and radius Rm, in the center of the membrane. The intensity is therefore
The total amount of energy leaving Ai is EAi π
(11)
The differential surface area of the light-emitting surface is
Z Jr
(10)
Rj −y
× f xj ; yj ; xi ; yi dxj dyj dxi dyi ;
(8) where
(15)
h f xj ; yj ; xi ; yi
i i ;yi i ;yi − ∂wx fxj − xi g − ∂wx fyj − yi g fh − wxi ; yi g h − wxi ; yi ∂xi ∂yi xj − xi 2 yj − yi 2 h − wxi ; yi 2 2
(16)
Upon substitution of Eq. (1), Eq. (16) becomes h f xj ; yj ; xi ; yi
i Pyi − x2i y2i fxj − xi g 16D R2i − x2i y2i fyj − yi g z z i2 2 h P xj − xi 2 yj − yi 2 h − 64D R2i − x2i y2i 2
Pxi 2 16D Ri
(17)
where z h − P∕64DR2i − x2i y2i 2 . Converting to polar coordinates and setting θ θi − θj and integrating over θi gives EAi →Aj
−I 0 32D
Z
Rj
rj 0
Z
Rm ri 0
Z
2π θ0
gri ; rj ; θdθdri drj
(18)
with 4Pri rj cos θR2i − r2i PR4i − 3r4i 2PR2i r2i − 64hDri rj P 2 2 2 h i2 2 R − ri : gri ; rj ; θ h− P 64D i R2i − r2i 2 r2i r2j − 2ri rj cos θ h − 64D
This is the fraction of energy that reaches an aperture given an OLED membrane light source. Note that when P 0, Eq. (19) becomes identical to the expression between two flat discs given in [35]. 4. Numerical Simulations
Numerical simulations were carried out using a commercial nonsequential optical ray-tracing tool, Zemax. Nonsequential ray tracing treats optical rays as straight paths from one surface to another. At each surface, the ray can reflect, refract, scatter, or be absorbed. Ray directions and interactions are calculated in 3D throughout the complete system. Each run simulates 108 rays. The source was modeled as a deformable circular disc emitting rays in a Lambertian emission pattern. The deformation of the disc was governed by Eq. (1), and the so simulation was directly a function of the applied pressure. It was assumed that the rays traveling toward the top cover, but not through the aperture, are fully absorbed. In doing so, the simulation will yield an underestimation of the optical power going to the detection plane. 5. Results and Discussion
To simulate the performance of the pressure sensor, it was assumed the membrane had a radius of 4.2 mm with an effective flexural rigidity of 1.72 × 10−3 N∕m. The initial gap between the lightemitting surface and the aperture was set at 425 μm. The maximum pressure difference across the membrane was 150 kPa, which resulted in a maximum deflection of 424 μm. Therefore, the minimum distance between the light-emitting surface and the aperture was just less than 1 μm. In Fig. 2,
(19)
the relative change in the output signal amplitude calculated from Eq. (18), in decibels, for various aperture and mask dimensions is given. It can be seen that if the membrane is not masked so that the entire surface can emit light, then the relative change in signal amplitude for any given deflection is negligible. This is a consequence of the Lambertian nature of the OLED, which emits light equally in all directions. As it is assumed that light is not attenuated in the sensor, changing the distance photons need to travel by deflecting the surface has no effect. This is only true when the area of the light-emitting surface is much larger than the aperture and so is effectively infinite. Any photons that are directed away from the aperture in this case due to the deflection of the membrane are compensated by photons that are consequently directed toward the aperture from elsewhere on the light-emitting surface. When the radius of the mask is comparable to the radius of the aperture, the solid angle and, hence, the view factor can increase significantly when the membrane is deflected and the reduced light-emitting surface is brought closer to the aperture. It should be noted, however, that the mask reduces the overall signal amplitude by the same ratio as the area of the exposed light-emitting surface to the area of the membrane. In order to determine the signal amplitude between a pressurized and nonpressurized membrane, we have performed numerous automated parameter sweeps to determine the ideal configuration of the sensor for a large signal swing between an unpressurized and pressurized membrane. The maximum signal swing is found with the largest membrane 1 May 2014 / Vol. 53, No. 13 / APPLIED OPTICS
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Fig. 3. Signal amplitude ratio (dB) between applied pressures of 0 and 150 kPa. The red line is the predicted response from Eq. (18), and the black circles denote that predicted from the numerical analysis. Membrane radius is 4.2 mm, the OLED active area radius is 0.5 mm, the gap between membrane and aperture is 425 μm, and the top cover thickness is 100 μm. The depicted sweep of the camera aperture radius shows an optimum at 0.5 mm, i.e., the same radius as the active area.
Fig. 2. Change in signal amplitude calculated from Eq. (18) due to change in the applied pressure relative to the signal amplitude when there is no deflection for different mask radii (Rm). (a) Aperture radius is 50 μm. (b) Aperture radius is 500 μm.
In Fig. 5 we determine that the smallest possible top cover thickness will yield the best performance with regards to signal variation and, hence, sensor resolution in the 0 to 150 kPa pressure range. A thicker top cover leads to a larger aspect ratio of the aperture and, hence, a smaller acceptance cone of the aperture with respect to the emissive surface. For the given pressure range this leads to smaller signal swings. It has been shown that for a light-modulating pressure sensor with the dimensions and rigidity discussed at the beginning of this section, the pressure range detectable is 150 kPa. It was also explained
displacement. This means we need to implement the largest possible membrane diameter (4.2 mm) and the largest possible gap (425 μm) between the OLED and the top cover of the stack. Figure 3 suggests that for a given geometry the highest difference in signal strength between an un-deformed and deformed membrane can be found when the OLED active area radius and the camera aperture radius match. Figure 4 shows the evolution of the signal amplitude with varying camera aperture radius while the OLED active area radius is matched to it. The smaller the aperture and matched OLED, the larger the signal amplitude. This behavior can be explained by observing that a small camera aperture radius limits the cone of light that can reach the detection surface. Any deformation or shift in position of the OLED on the membrane will result in a different match between the emission light cone and acceptance light cone. Matching the aperture of the camera and the active area of the OLED yields a match between the etendue of the emissive surface and the detector and, hence, a large signal swing between the deformed and un-deformed membrane.
Fig. 4. Signal amplitude ratio (dB) between applied pressures of 0 and 150 kPa. The red line is the predicted response from Eq. (18), and the black circles denote that predicted from the numerical analysis. Membrane radius is 4.2 mm, the gap between membrane and aperture is 425 μm, and the top cover thickness is 100 μm. OLED active area radius equals the camera aperture radius. Smallest aperture in the simulation is 50 μm radius.
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pressure sensor and was validated using a commercial ray-tracing tool. It has also been shown that in order to be effective, the compliant diode cannot emit light over the entire surface of the diode and that the sensor will offer the best resolution when the lightemitting surface has a similar radius to the aperture. Appendix A:Derivation of the Membrane Deflection Equation
A, B, and D are the extensional stiffness matrix, extensional-bending coupling stiffness matrix and bending stiffness matrix, respectively, and are expressed as A Fig. 5. Normalized signal amplitude of sensor between applied pressures of 0 and 150 kPa for different aperture thicknesses. Membrane radius is 4.2 mm, and the gap between membrane and aperture is 425 um. The OLED active area radius equals the camera aperture radius (0.5 mm). A large thickness limits the amount of angles that are admitted through the aperture because of the increased aspect ratio.
A12
A12 A11 " Pn Ek
k1 1−υ2 zk k
− zk−1
υk E k k1 1−υ2 zk k
− zk−1
Pn
Pn
υk E k k1 1−υ2 zk k
− zk−1
Ek k1 1−υ2 zk k
− zk−1
Pn
# ;
(A1)
earlier that the light intensity variations due to changes in pressure are detectable using a wide range of common components such as LDRs. Such components are easily interrogated using simple circuitry such as potential divider circuits and data acquisition systems (DAQs). Modern DAQs can routinely measure voltages with a resolution of 18 bits. When combined with a LDR circuit designed to operate over the full input range of the DAQ, it can be seen that the sensor described here has a theoretical resolution of 0.57 Pa. This resolution and range is better than the 0.9 Pa resolution claimed in [36], which states their Au-coated polydimethylsiloxane strain-gauge based pressure sensor exhibits the best results reported so far for any flexible membranebased sensor over the pressure range tested (
