High quality factor polymeric Fabry-Perot resonators utilizing a polymer waveguide

Mohammad Amin Tadayon; Martha-Elizabeth Baylor; Shai Ashkenazi

doi:10.1364/OE.22.005904

1. Introduction

An optical resonator is an arrangement of optical components that allows a beam of light to circulate in a closed path [1]. Fabry-Perot interferometers, whispering gallery resonators and photonic crystal microcavities are the most common light confinement mechanisms for this circulation on the micrometer scale [2]. Fabry-Perot resonators are utilized in semiconductor quantum boxes as micropillar resonators to enhance the spontaneous emission rate [3]. Using an ultrahigh quality (Q) factor Fabry-Perot resonator, Hood et al. have demonstrated detection of single atom trajectories [4]. Pruessner et al. have applied the high Q Fabry-Perot resonator to characterize their waveguide microcavity optomechanical system [5]. Adding a Gaussian-shaped defect in a vertical Fabry-Perot microcavity, Ding et al. have predicted higher Q factor and mode confinement using 3D finite difference time domain calculations [6]. Whispering gallery modes, another approach to closed-loop light circulation, are applied in microdisks [7,8], microspheres [9] and microtoroids [10]. Finally, photonic crystals having a defect are another class of optical microresonators that are designed [11] and fabricated [12] in many different ways to achieve the highest Q-factor.

Fabrication of microresonators with polymeric materials is advantageous in mechanical sensing applications. Their flexibility under stress is particularly favorable for fabrication of optical ultrasound receivers. Applying the whispering-gallery concept, polymeric microring resonators have been developed as high-frequency ultrasound receivers [13–15]. Another optical device which has been used for ultrasound detection is the Bragg grating waveguide reflector [16]. Fabry–Perot resonators (e.g., etalons) were one of the first optical resonators used to detect ultrasound [17], either as a single-element detector on the tip of an optical fiber [18] or on a glass substrate [19,20]. The design of these devices offers a seamless approach to integration of laser generation and detection of ultrasound in a single element forming an “all-optical” ultrasound transducer [21].

Although the application of Fabry-Perot resonators in optical micromachined ultrasound transducer (OMUT) technology has shown promising results [19], the demonstrated sensitivity is not high enough for real-time clinical imaging applications such as intravascular ultrasound (IVUS) imaging [21]. Clearly, there is a need to increase the Q-factor without sacrificing the small volume confinement of the optical field.

In this paper, we introduce a simple method to increase the Q-factor of polymeric Fabry-Perot resonators and show that this method can improve the Q-factor by one or more orders of magnitude depending on the reflectivity of the mirrors and the cavity length. This has been done by inducing a small refractive index change in the layer between the mirrors creating a waveguide. To model this structure, we have developed a simple theoretical model for resonators with and without the waveguide. We show that adding a waveguide can significantly improve the device Q-factor in the case of highly reflective mirrors and long cavities. After presenting the modeling results, we describe the fabrication and testing of a device fabricated using a UV-writeable, diffusive photopolymer. Finally, we discuss future directions in developing this technology for miniaturized, fiber optic-based ultrasound imaging devices.

2. Device Principles

The finesse of an optical resonator can be defined as the number of light oscillations between two mirrors at the free space wavelength (λ) before its energy decays by a factor of $e^{- 2 π}$ [22]. Thus the optical path required to achieve this finesse is $2 n L ℱ$ where n, L, $ℱ$ are the refractive index inside the cavity, the length of the cavity, and the finesse respectively. Diffraction, the spreading of light during propagation, causes energy to leak out of the resonator. The farther the light propagates, the greater the amount of energy that is lost from the cavity.

Therefore as one tries to create cavities with higher finesse and Q-factors, the required optical pathlength increases causing a more significant diffraction effect. The effects of diffraction can be eliminated, however, if the light passes through a waveguide between the two mirrors (Fig. 1). This can be done by inducing a small refractive index modulation to create a waveguide in the cavity. In Sections 2.1 and 2.2, we model propagation of the Gaussian beam inside the cavity without and with the waveguide structure, respectively. We compare the results for both cavities and demonstrate that the waveguide can improve the finesse and Q-factor of the cavity by over an order of magnitude depending on the cavity length and reflectivity of the mirrors.

Fig. 1 Schematic of the Fabry-Perot Resonator with a waveguide having a core refractive index of n₁ and a cladding refractive index of n₂. The layers of dielecteric mirrors are presented by high (n_h) and low n_l refractive index.

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2.1 Modeling of the cavity without waveguide

The unfolded equivalent model of wave propagation inside both cavities can be seen in Fig. 2(a) and (b). To model the propagation of the wave inside the cavity without the waveguide, we assume that the cavity does not affect the propagation parameters of the beam and only decreases the beam amplitude by a factor of $\sqrt{R_{1} R_{2}}$ during each oscillation where R₁ and R₂ are the mirrors reflectivities. We consider the incident light as a Gaussian wave with the electric field phasor [23]

E_{0} (r, z) = A_{0} \frac{w_{0}}{w (z)} \exp [- i (k z - η (z)) - r^{2} (\frac{1}{w {(z)}^{2}} + i \frac{k}{2 R (z)})],

where r is the radial distance from the center of the beam, z is the axial distance from the beam waist, k is the wavenumber (

2 π n / λ

),

A_{0}

is the electric field amplitude at its waist,

w_{0}

is the waist size,

w (z) = z_{0} \sqrt{1 + z^{2} / z_{0}^{2}}

,

R (z) = z (1 + z^{2} / z_{0}^{2})

,

η (z) = \tan^{- 1} z / z_{0}

and

z_{0}

is the Rayleigh range determined by

π n w_{0}^{2} / λ

. The electric field amplitude of the s-th reflection (

B_{s}

) can be calculated:

B_{0} (r) = - \sqrt{R_{1}} E_{0} (r, 0), B_{s} (r) = T_{1} \sqrt{R_{2}} {(\sqrt{R_{1} R_{2}})}^{s - 1} E_{0} (r, 2 s L), f o r s > 0,

where

T_{1}

is the transmittance through the first mirror. The total reflected field at each point is

B_{t} (r) = \sum_{s = 0}^{\infty} B_{s} (r) = - \sqrt{R_{1}} E_{0} (r, 0) + T_{1} \sqrt{R_{2}} \sum_{s = 1}^{\infty} {(\sqrt{R_{1} R_{2}})}^{s - 1} E_{0} (r, 2 s L),

and the total intensity is

I_{t} (r) = B_{t} {(r)B}_{t}^{*} (r)

. Then, the total power of the reflected light (

P_{t}

) is,

P_{t} = 2 π \int_{0}^{r_{domain}} I_{t} (r) r d r,

the value of r_domain, which indicates the domain of integration. This value varies depending on the method which is used to the reflected light. Figure 3(a) shows the reflected light intensity as a function of the incident wavelength for a plane wave and a Gaussian wave. The graph corresponds to a beam waist radius of 9.4 μm, which is the approximate value for our actual beam size; the reflectivity of the mirrors was set to be 0.995 and the cavity length is 100 μm. As we can see, the case of plane wave illumination has a much sharper dip than that of Gaussian wave and a 39 times larger finesse.

Fig. 2 Gaussian beam propagation in the cavity (a) without and (b) with waveguide embedded in the Fabry-Perot etalon layer (t:transmission, r:reflection).

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Fig. 3 (a) Resonance reflection spectrum of a 100 μm Fabry-Perot optical resonator excited with a Gaussian wave and a plane wave. (b) Finesse variation versus reflectivity for different cavity length with 10 μm steps .

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In Fig. 3(b) we show finesse versus reflectivity for different cavity lengths for the case of n = 1. For the Gaussian beam resonator, increasing both the cavity length and mirror reflectivity causes the cavity finesse to drastically deviate from the plane wave predicted value. The main reason for the lower finesse for the Gaussian wave in this case is the variation of the wave-front amplitude and phase profile during propagation resulting in a phase mismatch of successive reflections.

2.2 Modeling of the cavity with waveguide

In this section, we show that the waveguide optical cavity can behave as a plane wave cavity. Therefore, it is possible to design a very high Q-factor and finesse Fabry-Perot resonator, limited only by absorption in the mirrors and bulk polymer layer. To model wave propagation inside the cavity with the waveguide for photon confinement, we assume that the light travels inside the cavity with the propagation parameters and profile of a circular dielectric waveguide. The beam amplitude is assumed to decrease by a factor of $\sqrt{R_{1} R_{2}}$ during each oscillation.

The optical cavity consists of a core with a refractive index of n₁ and a radius of $a$ surrounded by the cladding with a refractive index of n₂. The Cartesian components of an electric field for approximately y-polarized light with a small z-component in cylindrical coordinates are [23]:

\begin{array}{l} E_{x_{l m}} (r, ϕ, z) = 0; E_{y_{l m}} (r < a, ϕ, z) = A_{l m} J_{l} (h_{l m} r) e^{i l ϕ} e^{- i β_{l m} z}, \\ E_{y_{l m}} (r > a, ϕ, z) = B_{l m} K_{l} (q_{l m} r) e^{i l ϕ} e^{- i β_{l m} z}; \\ E_{z_{l m}} (r < a, ϕ, z) = \frac{h_{l m}}{β_{l m}} \frac{A_{l m}}{2} [J_{l + 1} (h_{l m} r) e^{i (l + 1) ϕ} + J_{l - 1} (h_{l m} r) e^{i (l - 1) ϕ}] e^{- i β_{l m} z}, \\ E_{z_{l m}} (r > a, ϕ, z) = \frac{q_{l m}}{β_{l m}} \frac{B_{l m}}{2} [K_{l + 1} (q_{l m} r) e^{i (l + 1) ϕ} - K_{l - 1} (q_{l m} r) e^{i (l - 1) ϕ}] e^{- i β_{l m} z}; \end{array}

where lm indicates the mode number for different modes (

{LP}_{l m}

),

A_{l m}

is the mode factor and

B_{l m} =

A_{l m} J_{l} (h_{l m} a) / K_{l} (q_{l m} a)

. The mode factor is dependent on the mode shape and the incident light profile.

β_{l m}

is the propagation constant,

q_{l m} = \sqrt{β_{l m}^{2} - n_{2}^{2} k_{0}^{2}}

,

h_{l m} =

\sqrt{n_{1}^{2} k_{0}^{2} - β_{l m}^{2}}

. For

{LP}_{l m}

mode, the propagation parameter can be calculated using

h_{l m} a J_{l + 1} (h_{l m} a) / J_{l} (h_{l m} a)

= q_{l m} a K_{l + 1} (q_{l m} a) / K_{l} (q_{l m} a)

, where m is defined as the m-th root of this equation. The electric field for the specific mode can be written as

E_{l m z} (r, ϕ, z) = E_{x_{l m}} e_{x}

+ E_{y_{l m}} e_{y} + E_{z_{l m}} e_{z} =

E_{l m} (r, ϕ) e^{- i β_{l m} z}

. Following the same method for the derivation of Eq. (3), the reflected electric field for the lm mode can be calculated:

B_{t_{l m}} (r, ϕ) = E_{l m} (r, ϕ) [- \sqrt{R_{1}} + T_{1} \sqrt{R_{2}} \sum_{s = 1}^{\infty} {(\sqrt{R_{1} R_{2}})}^{s - 1} e^{- i β_{l m} 2 s L}],

assuming

β_{l m} = n_{e f f_{l m}} k_{0}

. The effective index,

n_{e f f_{l m}}

, can be considered a constant close to the resonance frequency. For each mode in Eq. (6), the electric field distribution (

E_{l m} (r, ϕ)

) and its integral are slowly varying with respect to variation of the wavelength and can be treated as a constant. Therefore, for each mode,

E_{l m} (r, ϕ)

does not affect the characteristic curve. The total reflected electric field is,

B_{t} = \sum_{l = 0}^{\infty} \sum_{m = 1}^{\infty} B_{t_{l m}}

. Using the same parameters as the cavity without the waveguide, we used a 100 µm cavity length with mirror reflectivities of 0.995 and a Gaussian beam of radius 9.4 µm. and are 1.53 and 1.50 respectively. The core radius is 10 μm. In the first fourteen modes of the waveguide, only four of them have nonzero contributions in response to the Gaussian beam. Among these four modes 99% of the light is coupled to the L₀₁. Therefore, as we see in Fig. 4(a) with Gaussian beam illumination, the finesse of the waveguide optical cavity will be exactly the same as the plane wave cavity with approximately the same effective refractive index. This shows that the only limitation for the waveguide cavity to achieve a very high Q-factor or finesse is the material absorption. Material absorption in the cavity medium or in the mirror material would reduce the quality of the resonance. We have not explicitly considered the diffraction of light inside the layers of the dielectric Bragg reflector. However, this diffraction might slightly decrease the finesse of the cavity and might shift its resonance frequency as well [24]. From Fig. 4(a), we can identify the excitation of some other modes, however, these modes are not as significant as the first mode. The normalized amplitude of the first mode shown in Fig. 4(b) is about 0.99. Although the higher modes do not affect the finesse of the system they can be removed by having the normalized frequency (

V = a k_{0} \sqrt{n_{1}^{2} - n_{2}^{2}}

) below 2.4. In this case, the cavity only supports the LP₀₁ mode which is known as the fundamental mode. Therefore, with a refractive index change of about 0.003 in a 4 µm radius cavity the higher modes will vanish. The other way to remove the higher modes in the cavity without a waveguide is to use an aperture to create an effective Fresnel number (e.g.,

a^{2} / 2 L λ

) below 3 [25]. However, in this case it is unlikely that high finesse can be achieved since there is a change in the phase profile of the propagating light.

Fig. 4 (a) Comparison of plane-wave resonance in an un-guided cavity with multi-mode resonance in the waveguide cavity, (b) cross-sectional plot of the modes inside the waveguide cavity.

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3. Fabrication

The fabricated device consists of two dielectric Bragg reflectors with a 100 μm thick polymer layer between them. A 4 µm dielectric Brag reflector is designed to have high UV transmittance and with more than 99% C-Band NIR reflectance (Evaporated Coatings, Inc., Willow Grove, PA). The holographic photopolymer utilized to fabricate this device is a thiol-ene/methacrylate photopolymer whose optical index can be modified utilizing standard photo-lithography processes [26]. Holographic photopolymers, also known as diffusive photopolymers, are a class of polymers that self-develop refractive index structures in response to optical exposure. In these materials, optical exposure locally consumes monomer, driving in-diffusion of replacement monomer which increases the index of refraction [26]. Figure 5(a) shows the steps required to make the device. The chromium mask is built on one of these mirror-coated wafers (step 1). The mask has been made on the uncoated side of the borofloat wafer, 500 μm away from the miror surface, to avoid any interference between the chromium film and the optical cavity. Then the mirrors are separated with a 100 μm spacer and the gap between them is filled with the photopolymer (step 2). The polymer is exposed from the backside for 54 s using the 13.5 mW/cm² UV mercury arc lamp to bring the polymer to its gel point [26]. An optical filter has been applied to 365 nm with the narrow line-width lamp . Then the fabricated mask, consisting of circle sizes of 5-15 μm, is exposed for 110 s from the mask side to expose the cavity area (step 3). In this step, the refractive index of the exposed area is around 1.521 which is 0.03 higher than the unexposed area [26]. Since the mask is 500 μm away from the polymer, using the Fresnel diffraction theorem, the radius of the generated cavities is calculated to be 5-20 μm. Finally the device is placed in the dark for two days for the monomer to diffuse to the center of the waveguide region creating a roughly uniform index across the center of the waveguide. Figure 5(b) shows an array of resonators. The distance between them is about 500 µm.

Fig. 5 (a) Fabrication steps of waveguide-Fabry-Perot device by permanent refractive index modification in a photopolymer, (b) microscope image of the fabricated array of the devices.

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4. Experimental testing of the resonator

To test an optical cavity with a 20 μm diameter, we use the experimental set-up shown in Fig. 6(a). Light from an infrared laser source is coupled to the cavity using a collimator, polarized beam splitter, a λ/4 waveplate, and a lens. Linearly polarized light from a continuous near infrared (NIR) tunable laser source (8168F, Agilent HP) is transmitted to the collimator through a polarization-maintaining single-mode fiber with mode field diameter of 10.5 μm. The linearly polarized collimated beam is reflected by 90° by the polarizing beam splitter. The λ/4 waveplate changes the linear polarization to circular polarization. An IR camera (1/2” CCD, Cat#56-567, Edmund Optics) is used to visualize the cavity from the top to align and couple the beam to a specific cavity element in the wafer that typically holds a large number of elements of different cavity radii. The circularly polarized light is coupled to the cavity using a lens. With our arrangement of lens and collimator, the spot size of the focused beam is approximately 18.8 μm. Upon reflecting from the device and passing through the λ/4 waveplate a second time, the resulting polarization is linear and perpendicular to the incident polarization. Because of the rotated polarization, the beam passes through the beamsplitter and is detected by a power meter.

Fig. 6 (a) The experimental set-up for testing reflection spectrum of the device (b) characteristic reflection spectrum curve of the tested device.

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To find the resonance wavelength of the optical cavity, the NIR wavelength is scanned from 1530 nm to 1550 nm. The reflected light intensity is recorded using an optical power meter (PM100 optical power meter with S122B Germanium Sensor, Thorlabs) and the results are presented in Fig. 6(b). The finesse of the optical cavity is about 682 and the Q-factor is about 55000 with a full width half maximum (FWHM) of about 0.031 nm and an average free spectral range of 8.45 nm. The noise in the characteristic curve shown in Fig. 6(b) is mainly due to laser and photodetector noise. As we can see in Fig. 6(b), there are two different modes involved in the characteristic curve. One of them is the first mode (LP₀₁) and since the device is not a single mode cavity the second dip can result from coupling of light to other modes. Based on our theoretical calculation, this second dip can be due to the LP₆₁ mode that has a resonance wavelength of 0.086 nm smaller than the first mode. Although it appears that this mode participation amplitude should be zero with fully Gaussian beam imperfect illumination might result in enhancement of this mode. This additional mode can be eliminated either by decreasing the refractive index difference between the core and cladding or decreasing the size of the cavity. Due to the low tuning resolution of our laser and the noises from the photodetector, the dip does not have a completely Lorentzian shape. The measured characteristic curve was repeatable over different cavities in the wafer and was independent of alignment accuracy. We measured the characteristic curve for cavities without a waveguide structure and did not find any resonance for these cavities. This is likely due to the fact that the signal was small and unresolved within our current signal to noise ratio limits.

5. Conclusion

We present a method that significantly improves the Q-factor and finesse of polymeric Fabry-Perot resonators. By creating a waveguide inside the cavity, a confined mode propagates in the cavity having a phase-front perfectly matched to the flat mirror geometry. This improves the sharpness of the resonance peak by one or more orders of magnitude. This method also keeps the coupled light energy confined to a smaller volume, which is extremely important in applying the technology to miniaturized ultrasound imaging devices. Further improvement of the characteristic curve of the resonator is expected by integrating the cavity onto the tip of an optical fiber. In this way, we expect better coupling due to higher overlap between the fiber modes and the cavity modes. Additionally, higher modes can be eliminated by using a single-mode cavity design.

To achieve single-mode operation, we can either decrease the radius of the cavity or induce a smaller refractive index difference between the core and cladding of the waveguide. To decrease the refractive index difference between the core and cladding, the device is flood-cured under UV light for 270 s from the back-side to fully cure the polymer and use up any remaining monomer chemistry after the final diffusion step. In this case, the expected refractive index change is about 0.006 instead of the current value which is 0.03 [26].

One of the major advantages of this device is its compatibility with imaging fiber bundles. Large arrays of several thousands of high Q-factor resonators can be fabricated on the tip of a mirror-coated optical fiber bundle without any need for alignment or a complicated fabrication process. Our final goal is to apply this device to high-resolution, miniaturized 3D ultrasound imaging probes in medical imaging applications.

Acknowledgments

This work is supported by the National Science Foundation under grant no. CMMI-1266270. Parts of this work were carried out in the Minnesota Nano Center which receives partial support from NSF through the NNIN program. The first author gratefully thanks the University of Minnesota for awarding him a Doctoral Dissertation Fellowship and Benjamin Cerjan for his assistance with the photopolymer.

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High quality factor polymeric Fabry-Perot resonators utilizing a polymer waveguide

Abstract

1. Introduction

2. Device Principles

2.1 Modeling of the cavity without waveguide

2.2 Modeling of the cavity with waveguide

3. Fabrication

4. Experimental testing of the resonator

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (6)

Optics Express