## Abstract

We report the design and fabrication of long-period waveguide gratings (LPWGs) in benzocyclobutene (BCB) ridge waveguides. We apply an accurate perturbation theory to analyze the LPWGs. In particular, the phase-matching condition, the coupling coefficients, the temperature dependence of the resonance wavelength, the bandwidth, and the polarization dependence of the resonance wavelength are discussed. Several LPWGs in BCB ridge waveguides are fabricated by a UV-writing technique using a KrF excimer laser. The transmission spectra of the gratings are measured and discussed. An LPWG with a polarization-insensitive resonance wavelength at a specific temperature is demonstrated. Experimental results agree well with the theory. Our results are useful for the design of LPWG-based devices for various applications.

©2005 Optical Society of America

## 1. Introduction

Much research effort has been devoted to the study of long-period fiber gratings (LPFGs), which allow the coupling of light between the guided mode and the cladding modes of an optical fiber at specific wavelengths. LPFGs have become increasingly popular for their many applications as gain flatteners of erbium-doped fiber amplifiers [1], band-rejection filters [2], add-drop couplers [3], dispersion controllers [4], and various kinds of sensors [5,6]. As an ordinary fiber has a round shape and is made of a highly stable material (silica), it is in general difficult to realize tunable devices with LPFGs formed in ordinary fibers. To remove the geometry and material constraints of a fiber, long-period waveguide gratings (LPWGs), namely, long-period gratings fabricated in planar waveguides, have been proposed [7,8].

A large number of materials are available for the fabrication of optical waveguides, such as glass, lithium niobate, semiconductor, and polymer. Among these materials, polymer is relatively easy to process and also offers many favorable optical and electrical properties. Polymer can be spin-coated on glass, silica, silicon, etc., and therefore is compatible with most optical materials. In addition, it can provide a high degree of ruggedness and allow mass production of components at a low cost. A distinct feature of polymer, compared with other optical materials, is the strong temperature dependence of its refractive index. The large thermo-optic coefficient of polymer (about 25 times larger than that of glass) can be leveraged to produce efficient thermo-optically tunable components. For example, an LPWG filter, which can be tuned linearly over the entire C+L band (~90 nm) with a temperature control of ~10 °C, has been demonstrated with a polymer-clad ion-exchanged BK7 glass waveguide recently [9]. Both the wavelength-tuning range and the temperature sensitivity of this filter exceed those achieved with LPFGs [10,11].

Among different polymers, benzocyclobutene (BCB), which is available from Dow Chemical, offers a number of advantages, such as good thermal stability (*T*
_{g}>350°C), low moisture uptake, good adhesion properties, and relatively low cost. It has been used widely as an optical waveguide material [12,13]. Recently, LPWGs in a BCB channel waveguide [14] and a BCB rib waveguide [15] were demonstrated. The conventional technique of making an LPWG [14] is by first corrugating the surface of the core of a waveguide with a prescribed pitch by photolithography and reactive ion etching (RIE), and then coating the core with a cladding with prescribed characteristics. By changing the stress in the waveguide through etching of the cladding width, a polarization-insensitive resonance wavelength thermally tunable from 1520 nm to 1610 nm with a temperature range of only 8 °C was achieved [14].

In this paper, we report a comprehensive study of LPWGs in BCB ridge waveguides. Thanks to the ridge geometry, we are able to apply an accurate perturbation theory to the calculation of the phase-matching condition and the coupling coefficient of an LPWG. The theory simplifies tremendously the analysis and design of an LPWG and provides much insight into the transmission characteristics of the LPWG, including the temperature dependence of the resonance wavelength, the bandwidth, and the polarization independence of the resonance wavelength. We also describe the fabrication of a ridge waveguide and the writing of an LPWG on the cladding of the waveguide using the UV technique we recently proposed [16]. The transmission characteristics of several fabricated LPWGs are discussed and compared with theoretical results. In particular, we demonstrate an LPWG that exhibits a polarization-independent resonance wavelength by controlling the waveguide dimensions.

## 2. Method of analysis

Figure 1 shows the cross section of a ridge waveguide, which consists of a substrate of refractive index *n*_{s}
, a guiding layer of refractive index *n*
_{f} and thickness *d*
_{f}, a cladding layer of refractive index *n*
_{cl} and thickness *d*
_{cl}, and an external medium of refractive index *n*
_{ex} that extends to infinity, where *n*
_{f}>*n*
_{cl}>*n*
_{s}, *n*
_{ex}. The guiding layer and the cladding have the same width *w*. We assume that only the fundamental quasi-TE and quasi-TM modes are guided, which are referred to as the ${E}_{11}^{x}$ and ${E}_{11}^{y}$ modes with effective indices *N*
_{11,i}(*i*=*x*, *y*) where *n*
_{cl}<*N*
_{11,i}<*n*
_{f}. A long-period grating with pitch Λ is introduced on the surface of the guiding layer or the cladding layer. The grating allows light coupling from the fundamental mode (${E}_{11}^{x}$ or ${E}_{11}^{y}$) to the cladding modes of the same polarization (${E}_{\mathit{\text{mn}}}^{x}$
or ${E}_{\mathit{\text{mn}}}^{y}$
), whose effective indices *N*_{mn}
,_{i} (*m*, *n* are integers except *m*=*n*=1) are smaller than *n*
_{cl}, i.e., *n*
_{s}<*N*
_{mn}
,_{i}<*n*
_{cl}. The resonance wavelength *λ*
_{0}, at which the coupling between the two modes is strongest, is determined by the phase-matching condition:

The effective indices of the guided mode and the cladding modes of the waveguide can be calculated accurately from the following perturbation formula [17]:

where *S*_{x}
=1 and *S*_{y}
=0. In Eq. (2), *V*=(1/2)*d*
_{f}
*k*
_{0}(${n}_{\mathrm{f}}^{2}$-${n}_{\mathrm{s}}^{2}$)^{1/2} is the normalized frequency (with *k*
_{0}=2*π*/*λ* the free-space wavenumber and *λ* the free-space wavelength), Δ_{1}=(${n}_{\mathrm{f}}^{2}$-${n}_{\mathrm{s}}^{2}$)/2${n}_{\mathrm{f}}^{2}$ and Δ_{2}=(${n}_{\mathrm{f}}^{2}$-*n _{ex}*

^{2})/2${n}_{\mathrm{f}}^{2}$ are the relative index steps, and

*N̑*

_{n-1,i}is the effective index of the TE

_{n-1}(for

*i*=

*x*) or TM

_{n-1}(for

*i*=

*y*) mode of the four-layer slab waveguide formed by extending the width of the ridge waveguide to infinity. The perturbation formula is accurate, as long as the external index is sufficiently low, i.e.,

*n*

_{f}-

*n*

_{s}≪

*n*

_{f}-

*n*

_{ex}and the mode of concern is not close to cutoff [17].

For all the numerical results given in subsequent sections, unless stated otherwise, the following waveguide parameters are used: *n*
_{s}=1.444, *n*
_{f}=1.54, *n*
_{cl}=1.50, *n*
_{ex}=1.0 (air), and *d*
_{f}=2.0 µm. These values are typical of a polymer waveguide fabricated on a silica substrate. The cladding thickness *d*
_{cl} and the waveguide width *w*, which are important design parameters, are allowed to vary. Material dispersion and thermally induced stress birefringence are ignored in the calculation. We only consider the coupling to the first cladding mode. The coupling to a higher-order mode can be analyzed by the same approach.

## 2.1 Phase-matching curves

The phase-matching condition given by Eq. (1) governs the dependence of the resonance wavelength *λ*
_{0} on the grating pitch Λ and, therefore, plays a central role in the study of long-period gratings. The variation of *λ*
_{0} with Λ (the phase-matching curve) is shown in Fig. 2 for different values of cladding thickness and waveguide width. As shown in Fig. 2(a), when the cladding is relatively thick (*d*
_{cl}≥4 µm), *λ*
_{0} increases linearly with Λ. When the cladding is thin, however, the relationship between *λ*
_{0} and Λ is no longer linear. It is also noted in Fig. 2(a) that the phase-matching curves for the quasi-TE and quasi-TM modes can become close as the cladding thickness changes and, in fact, cross each other at a particular value of cladding thickness (*d*
_{cl}=3 µm). This implies the presence of a specific LPWG design where the couplings for both polarizations occur at the same wavelength, as in the case of an LPWG in a slab waveguide [8]. The phase-matching curves for the LPWGs with different waveguide widths are shown in Fig. 2(b). Only the phase-matching curves for the quasi-TE polarization are given as the results for the quasi-TM polarization are similar (and therefore not shown). It is clear from Fig. 2(b) that the phase-matching condition depends only weakly on the waveguide width provided that the cladding mode of concern remains above cut-off.

The difference in the effective index squared between the guided mode (*m*=*n*=1) and the first cladding mode (*m*=1, *n*=2) is obtained from Eq. (2) as

which is independent of the waveguide width. Note that *N̑*_{0,i} and *N̑*_{1,i} are the effective indices of the first two modes of the four-layer slab waveguide. From Eq. (1) and Eq. (3), the grating pitch required for a given resonance wavelength λ_{0} is given by

The grating pitch for a ridge waveguide is insensitive to the waveguide width and, therefore, can be approximated by that for the four-layer slab waveguide obtained by extending the width of the ridge waveguide to infinity. This result is very much of practical significance as the slab waveguide is an intermediate step in the fabrication of the ridge waveguide and the effective indices of the slab waveguide can be measured accurately by the well-known prism-coupler technique. This special property of the ridge waveguide helps the realization of LPWGs with more predictable characteristics.

## 2.2 Coupling coefficients

The strength of the rejection band of an LPWG is governed by the coupling coefficients *κ*_{i}
, where *i*=*x* and *y* for the quasi-TE and quasi-TM polarizations, respectively [7,8]. For a ridge waveguide, the mode field can be expressed accurately as a product of the fields in the *x* and *y* directions [18]. The coupling coefficients of the LPWG in the ridge waveguide can thus be expressed in terms of those of the LPWG in the appropriate slab waveguide:

$${\kappa}_{y}={\Gamma}_{\mathrm{TE}}^{x}{\stackrel{\u2322}{\kappa}}_{\mathrm{TM}}^{y},$$

where ${\mathrm{\Gamma}}_{\text{TE}}^{x}$ and ${\mathrm{\Gamma}}_{\text{TM}}^{x}$ are the confinement factors for the TE_{0} and TM_{0} modes of the slab waveguide formed by extending the thickness of the rectangular core to infinity, and ${\stackrel{\u2322}{\kappa}}_{\text{TE}}^{y}$ and ${\stackrel{\u2322}{\kappa}}_{\text{TM}}^{y}$ are the coupling coefficients for the LPWG in the slab waveguide formed by extending the width of the rectangular core to infinity. The dependence of the confinement factors on the waveguide width is shown in Fig. 3. Because the index difference between the waveguide material and air is large, the mode field is well confined in the *x* direction. Therefore, both ${\mathrm{\Gamma}}_{\text{TE}}^{x}$ and ${\mathrm{\Gamma}}_{\text{TM}}^{x}$ are close to 1 and Eq. (5) can be simplified to

$${\kappa}_{y}\cong {\stackrel{\u2322}{\kappa}}_{\mathrm{TM}}^{y}$$

which are insensitive to the waveguide width. The expressions for ${\stackrel{\u2322}{\kappa}}_{\text{TE}}^{y}$ and ${\stackrel{\u2322}{\kappa}}_{\text{TM}}^{y}$ can be found in [7] for a phase grating and in [8] for a corrugation grating.

## 2.3 Temperature sensitivity

The temperature sensitivity of the resonance wavelength of an LPWG, *dλ*
_{0}/*dT*, can also be derived from a perturbation theory, as in the case of an LPFG [19]. The result is

with

where *C*
_{co} and *C*
_{cl} are the thermal-optic coefficients of the core and the cladding, respectively. *η*
_{11f} and *η*
_{11cl} denote the fractional powers of the guided mode in the core and the cladding, respectively, and *η*
_{mnf} and *η*
_{mncl} denote the fractional powers of the cladding mode in the core and the cladding, respectively. The factor *γ*, evaluated at *λ*
_{0}, describes the effect of modal dispersion. When the cladding is thick enough, the mode fields in the substrate become negligible. In that case, we have *η*
_{11f}+*η*
_{11cl}≅1 and *η*
_{mnf}
+*η*_{mncl}
≅1, and Eq. (7) reduces to the formula for LPFGs [19]:

It is clear from Eq. (7) and Eq. (9) that the change of λ_{0} with the temperature *T* at a given grating pitch Λ is governed by the modal dispersion factor *γ*. Figure 4(a) shows the variation of *γ* with the cladding thickness at λ_{0}=1550 nm and the corresponding required grating pitch for the coupling between the guided mode (*m*=*n*=1) and the first cladding mode (*m*=1, *n*=2) for the quasi-TE polarization. It can be seen from Fig. 4(a) that the value of *γ* flips from negative to positive at a particular value of cladding thickness, which is due to the presence of a singularity, i.e., when the denominator in Eq. (8) goes to zero. This behavior has been observed with LPFGs when *γ* is plotted as a function of the mode order [20]. In fact, the factor *γ* is proportional to *dλ*
_{0}/*d*Λ, the slope of the phase-matching curve. The singularity point, |*γ*|=∞, corresponds to the turning point on the phase-matching curve, which gives rise to the generation of dual resonance wavelengths from coupling to the same cladding mode. Such dual resonance wavelengths have been observed with an LPFG operating at a high cladding mode order [20]. Because the fiber cladding is thick (62.5 µm in radius), dual resonance occurs only at a high cladding mode order [20]. As shown by our results, when the cladding is thin, dual resonance can actually be obtained with even the first-order cladding mode. It can also be seen that when the cladding is thick enough, the magnitude of *γ* is small and depends weakly on the cladding thickness. The grating pitch required increases with the cladding thickness, which is consistent with the results shown in Fig. 2(a). This can be explained by the fact that the effective index of the cladding mode increases much faster than that of the guided mode as the cladding thickness increases, so a longer grating pitch is required to keep the resonance wavelength unchanged according to Eq. (1).

The temperature sensitivity of an LPWG depends on the thermal-optic coefficients of the materials and the waveguide structure, especially the cladding thickness. The variation of *dλ*
_{0}/*dT* with the cladding thickness is shown in Fig. 4(b) for different values of *C*
_{cl} (assuming *C*
_{f}=-1.0×10^{-4}/°C and *λ*
_{0}=1550 nm). In practice, the value of *C*
_{cl} can be changed by changing the polymer material for the cladding. It can be seen from Fig. 4(b) that the temperature sensitivity of an LPWG can be controlled over a wide range by simply controlling the cladding thickness or by carefully tuning the difference between *C*
_{f} and *C*
_{cl}. These curves also exhibit the flip-flop behavior due to the factor *γ*. In the case of a thick cladding, the cladding mode is confined mainly in the cladding and the guided mode is confined mainly in the guiding layer. The fractional power difference, *η*
_{11f}-*η*
_{mnf}≅*η*
_{11f}, is not much affected by the cladding thickness. Therefore, according to Eq. (9), the magnitude of *dλ*
_{0}/*dT* is insensitive to the cladding thickness and the sign of *dλ*
_{0}/*dT* is determined by the sign of *C*
_{f}-*C*
_{cl}. The condition of zero temperature sensitivity is thus given by *C*
_{cl}=*C*
_{f}, which is the same as that for an LPFG. As the cladding becomes thinner, the cladding mode is pushed deeper into the guiding layer/substrate and, at the same time, the guided mode penetrates more into the cladding. As the cladding becomes thin enough, the fractional powers of the modes in the guiding layer and the cladding become comparable and highly sensitive to the cladding thickness. The condition of zero temperature sensitivity is given by *C*
_{f} (*η*
_{11f}-*η*
_{mnf})+*C*
_{cl} (*η*
_{11cl}-*η*
_{mncl}
)=0.

The effects of the waveguide width on the modal dispersion factor *γ* and the temperature sensitivity of the resonance wavelength are shown in Fig. 5(a) and (b), respectively. As expected from the phase-matching curves in Fig. 2(b), the waveguide width has only a weak effect on the factor *γ* and, hence, on the temperature sensitivity of the resonance wavelength.

## 2.4 3-dB bandwidth

Assuming complete power transfer, the full width at half maximum (FWHM) of the resonance peak, Δ*λ*
_{3dB}, is given by [20]

where Δ*n*
_{eff}=*N*
_{11,i}-*N*_{mn}
,
_{i}
, *L* is the grating length. It can be seen from Eq. (10) that the 3-dB bandwidth of the resonance peak is proportional to the modal dispersion factor *γ* and the grating pitch, while it is inversely proportional to the grating length.

## 2.5 Polarization dependence

The resonance wavelengths for the quasi-TE and quasi-TM polarizations are in general different. To facilitate the discussion of this property, we use a waveguide parameter *D*_{mn}
, which is defined as [8]

According to Eq. (1), *D*_{mn}
is a measure of the difference between the resonance wavelengths of the quasi-TE and quasi-TM polarizations. According to Eq. (4), we obtain

Therefore, *D*
_{12} can be estimated from *D̑*_{12} for the slab waveguide, which is a measurable parameter. The condition required for achieving a polarization-insensitive resonance wavelength is *D*_{mn}
=0. The factor *D*_{mn}
depends on both the geometry and the material of the waveguide. The cladding thickness required for *D*
_{12}=0 is shown in Fig. 6(a) as a function of the resonance wavelength. The results are in fact close to those for a slab waveguide [8], as implied by Eq. (12). The waveguide width can also affect the polarization dependence of the grating, though to a much less extent, as shown in Fig. 6(b).

## 3. Waveguide and grating fabrication

We first prepared an epoxy-clad BCB ridge waveguide. A BCB film was spin-coated on a 20-mm-long silica-on-silicon substrate. It was cured in N_{2} atmosphere with a temperature ramp up to 250 °C for ~60 minutes. An epoxy (UV 11-3) film was next spin-coated on the BCB film. The epoxy film was exposed to a 350 W UV lamp for 10 minutes and post-cured at 120 °C for 10 minutes. The BCB and epoxy films were then etched into a ridge waveguide by photolithography and RIE. In order to study the dependence of the resonance wavelength on the waveguide width, we fabricated several ridge waveguides with different waveguide widths on the same substrate by using a properly designed mask. The width of the ridge waveguide varied from 4 µm to 7 µm, and the thicknesses of the BCB and epoxy films were fixed at 1.9 µm and 2.6 µm, respectively. An SEM image of one of the waveguides is shown in Fig. 7. The propagation loss of the ridge waveguide, measured by the cut-back method, was ~2 dB/cm, which is larger than the material loss of BCB. The additional loss is believed to be due to the roughness of the etched sidewalls of the waveguide. The epoxy cladding was made shorter than the BCB core by about 3 mm at both ends to facilitate the excitation and detection of the guided mode. Before the BCB and epoxy films had been etched, their refractive indices were measured at 1550 nm by a prism coupler (Metricon 2010). The refractive indices of the BCB film were measured to be 1.5491 and 1.5480 for the TE and TM polarizations, respectively, and those of the epoxy film were 1.5052 (TE) and 1.5046 (TM). The birefringence observed was believed to be due to the stress induced in the polymer thin films [21]. We chose a pitch of 39 µm, which, according to the calculation, should correspond to the coupling between the guided mode (${E}_{11}^{x}$ or ${E}_{11}^{y}$) and the first cladding mode (${E}_{12}^{x}$ or ${E}_{12}^{y}$) and produce resonance wavelengths around 1550 nm for both polarizations.

We formed a long-period grating on the surface of the epoxy cladding by the UV technique we proposed recently [16]. The UV source we used was a KrF excimer laser, which generated UV pulses at 248 nm with a full width at half maximum (FWHM) of 23 ns. The energy density of the UV pulses was set at 10 mJ/cm^{2}. We found that the UV light changed both the refractive index and the thickness of the epoxy film [16], in agreement with the findings for a different polymer material, PMMA [22]. These results suggest the possibility of inducing a grating in the epoxy cladding of the ridge waveguide.

We next exposed the ridge waveguide to the UV pulses through an amplitude mask, which was a thin quartz plate consisting of a periodic chromium pattern with a pitch of 39 µm and a length of 10 mm. We monitored the transmission spectrum of the waveguide by using a C+L band ASE source covering a wavelength range from 1500 nm to 1620 nm and an optical spectrum analyzer (OSA).

## 4. Experimental results and discussion

The effects of the cladding thickness on the resonance wavelength of an LPWG and its temperature sensitivity have been studied experimentally with slab waveguides [23]. Similar results are expected for ridge waveguides. In the present experimental work, we focus on the effects arising from a change in the width of the ridge waveguide.

Figure 8 shows the measured resonance wavelengths at 20.8 °C for several LPWGs in ridge waveguides with different waveguide widths. As shown in Fig. 7, the sidewalls of the fabricated waveguide were not absolutely straight, so we used the average of the maximum and minimum widths measured from the SEM image. The error was around 0.1 µm. The calculated results are also shown in the figure for comparison. The measured results agree quite well with the theoretical values. As expected, the resonance wavelength varies slowly with the waveguide width. Furthermore, the difference between the resonance wavelengths for the quasi-TE and quasi-TM polarizations is insensitive to the waveguide width, in agreement with the prediction in Section 2.5.

An LPWG was written in another ridge waveguide, which had a width of 5.9 µm. The thicknesses of the BCB core and the epoxy cladding were 2.0 µm and 1.8 µm, respectively. The refractive indices of the BCB film were measured to be 1.5380 and 1.5360 for the quasi-TE and quasi-TM polarizations, respectively, and those of the epoxy film were 1.5054 (TE) and 1.5052 (TM). The grating pitch was 39 µm and the same UV dosage was used for writing the grating. The measured spectra at 22.2 °C for the quasi-TE and quasi-TM polarizations are shown in Fig. 9(a). The resonance wavelength is almost the same for both polarizations at this temperature. The temperature dependences of the resonance wavelengths for both polarizations are presented in Fig. 9(b), which shows that the resonance wavelengths shift to the shorter wavelength as the temperature increases. By linear fitting of the measurement data in Fig. 9(b), the temperature sensitivities for the quasi-TE and quasi-TM polarizations are -7.0 nm/°C and -1.3 nm/°C, respectively. The thermal-optic coefficients for BCB and epoxy films were measured separately by the prism coupler technique and the values obtained were -1.0×10^{-4}/°C and -1.3×10^{-4}/°C, respectively, which were practically the same for both polarizations. From Eq. (7), the temperature sensitivities for the quasi-TE and quasi-TM polarizations are estimated to be -8.4 nm/°C and -1.8 nm/°C, respectively, which agree well with the experimental data. The corresponding modal dispersion factors *γ* are calculated to be -22.5 and -6.9. The large temperature-sensitivity difference between the two polarizations is due to the presence of material birefringence and the use of a thin cladding, as discussed in Section 2.3, which result in a large difference in the modal dispersion factors between the two polarizations. Although the temperature sensitivity of the resonance wavelength of the present grating is smaller than those reported previously of LPWGs produced by the RIE process [9,14], it is still much higher than that of a typical LPFG [19]. Furthermore, as shown in Fig. 9(a), the bandwidth of the resonance peak for the quasi-TE polarization is larger than that for the quasi-TM polarization, which is consistent with Eq. (10), i.e., the bandwidth of the resonance peak is proportional to the magnitude of the factor *γ*.

## 5. Conclusion

With the help of an accurate perturbation theory, we carry out a detailed analysis of an LPWG in a ridge waveguide. Analytical expressions are presented to highlight the characteristics of the LPWG, including the phase-matching condition, the coupling coefficients, the 3-dB bandwidth, the temperature sensitivity of the resonance wavelength, and the polarization dependence of the resonance wavelength. We also present the transmission characteristics of several experimental LPWGs that were formed in BCB/epoxy ridge waveguides by the UV-writing technique. The dependence of the resonance wavelength on the width of the waveguide is discussed and an LPWG showing a polarization-insensitive resonance wavelength at a specific temperature is demonstrated. The experimental results are found to agree well with the theoretical calculations. Our results provide a better understanding of the properties of LPWGs and should facilitate the design of LPWG-based devices.

## Acknowledgments

The authors wish to thank C. K. Chow and H. P. Chan for their technical assistance and useful discussions. This work was supported by the grants from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 1160/01E and CityU 1255/03E].

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