We analyze, theoretically, a Fabry-Perot interferometer constructed from superimposed, chirped fiber Bragg gratings. Interference effects between the superimposed gratings play a large role in determining the exact positions of the FP resonances. We give formulae to determine the spatial position of the resonances of the system, and, in certain cases, the profile of their field intensity.
©2005 Optical Society of America
In recent years, Fabry-Perot (FP) resonators have been constructed using superimposed, chirped fiber Bragg gratings (CFBGs) for use as filters , or as building blocks for more complicated structures such as Gires-Tournois interferometers  or multi-wavelength fiber lasers . The transmissive and reflective properties of a CFBG-FP can be understood via the following simple model. In a CFBG, the local period of the grating varies with position, so different wavelengths are reflected at different spatial positions. When two CFBGs are superimposed, but separated by a distance d, then a given wavelength, λ0, is reflected at two different positions, also separated by d. This is shown schematically in Fig. 1: Grating 1 reflects wavelength λ 0 at position z 0, while grating 2 reflects the same wavelength at z 0+d. If the phase accumulation in one round trip between the two reflection points is a multiple of 2π, then the cavity becomes highly transmitting, and hence acts as a FP resonator. Experimental and theoretical  analyses have shown that the structure possesses a free spectral range (FSR) equal to c/(2nd), and that the resonance peaks have a Lorentzian lineshape, both of which are exactly what one would expect in a FP filter. However, this model gives no information on the nature of the field distribution in the CFBG-FPs. Furthermore, the implicit assumption that the two CFBGs can be treated independently is incorrect, because the superposition of the CFBGs induces a beating between them that must be accounted for. Previous analyses of CFBG-FPs have either been numerical  or have concentrated on limiting cases where the gratings were not superimposed . The work of Town et al.  includes some discussion of superimposed gratings, but it was assumed that the effects of the superimposition were negligible. Melloni et al.  considered FP interferometers constructed by non-chirped gratings. In all of these works, little attention has been paid to the nature of the field profiles within the structures at resonance. An understanding of these profiles is essential for active devices, such as erbium-doped fiber lasers , constructed using CFBG-FPs, because the width of the field profile will impact gain saturation and spatial hole burning.
In this paper, we present a more complete model to describe both the transmissive properties of a CFBG-FP, and the field profiles. We first show that the beating between the individual CF-BGs plays a large role in determining the resonances of the structure. We introduce an effective wavenumber, keff , that indicates the spatial regions of the CFBG-FP that are reflective. Using the periodicity of keff we explain the FSR of a CFBG-FP. The imaginary portion of keff gives qualitative insight into the field profiles associated with transmission maxima of the FP. In the special case where the superimposed CFBGs that compose the FP have the same strength and chirp, and are displaced by a relatively small distance, we present an analytical formula for the field profile. We also show that the spatial positions of the resonances in the structure are very sensitive to the displacement, d, between the superimposed CFBGs. When the chirped gratings have unequal strengths, we show that the wavelengths associated with the peaks shift linearly with respect to the index modulation of the grating, while maintaining the same FSR.
The outline of this paper is as follows. In section 2, we present the basic physical model, and the coupled mode equations used for the numerical simulations. We also introduce the effective wavenumber, keff . In section 3, we discuss the transmission spectra and field profiles for several CFBG-FP configurations. In section 4 we conclude.
2. Coupled mode theory for superimposed, chirped FBGs
We start with the following model for the effective index profile of a single ‘raised-cosine’ CFBG fabricated via UV illumination:
where n 0 is the original index of refraction of the medium, δn is the effective index modulation and Λ(z) is the local Bragg period of the grating. In this paper we consider a grating with a weak linear chirp, so that the local period is Λ(z)=Λ0+(Chz/2), where Chz≪Λ0 across the entire grating length. Using this expression for Λ(z) in Eq. (1) we find that
where the quadratic phase
We assume that the grating starts at z=0, and ends at z=L. The effects of apodization can be easily included in Eq. (1). However, for our purposes apodization would complicate the presentation of the paper without affecting the basic physics.
For two superimposed gratings written with the same phase mask, but with potentially different grating strengths,
where zi gives the displacement of the ith grating. We have introduced n̄=n 0+δn 1+δn 2 to describe the average grating index including the contributions from the UV illumination, and δnmod (z) to describe the oscillatory part of n(z). To simplify the presentation we set z 1=0 and z 2=d for the remainder of the paper.
To accurately determine the optical properties of a CFBG-FP we first write the electric field for a given frequency, ω, as
where Â± are the slowly-varying amplitudes of fields propagating in the forward and backward directions respectively. Then, using this definition of the electric field in Maxwell’s equations in one dimension, and applying the usual rotating-wave approximation we arrive at the following coupled mode equations (CME) 
as the strength of the individual gratings, we have introduced
a complex coupling coefficient, and
which accounts for the detuning from the fundamental Bragg frequency, ω 0=πc/n̄Λ0. Since (z) is complex we can write (z)=κ(z)eiθ (z), where both κ and θ are real quantities. Making the phase transformation A±(z)=Â±(z)e ±iθ(z)/2 we find
Equation (12) has the form of the familiar CME: δ (z;ω) can be interpreted as the detuning from the local Bragg frequency, and κ(z) as the local grating strength. If δ and κ are independent of z, then the solutions to (12) will be oscillatory when |δ|>|κ|, and exponentially growing or decaying when |δ|<|κ|. When δ and κ depend weakly on z, one can still identify oscillatory and exponential solutions for the regions where |δ(z;ω)|>|κ(z)| and |δ(z;ω)|<|κ(z)| respectively, but it is difficult to match the solutions when |δ(z;ω)|=|κ(z)| (the turning points). To a very good approximation, the matching can be performed using WKB theory from quantum mechanics . However, for our purposes it is sufficient to introduce an effective wavenumber,
which incorporates information about the beat frequency of κ(z), and the detuning of light from the local Bragg wavelength. In terms of keff , fields acquire phase according to eiζ , where
If keff is imaginary (|δ(z;ω)|<|κ(z)|), the quantity ζ will also be imaginary, and the ‘phase’ term eiζ will describe exponential growth or decay. Therefore, the portions of the grating for which keff (z;ω) is imaginary can be thought of as reflective, and the portions where keff (z;ω) is real as dispersive.
3. Results and discussion
In this section we analyze several CFBG configurations. We start by using keff to generate a simple formula for the transmission of a single CFBG. We then present formulae for κ(z) and δ(z) for a CFBG-FP which we use to explain the FSR of the structure. Then, concentrating on the special case where the two superimposed gratings have the same strength, we present analytical formulae for the wavelengths and spatial positions of the FP resonances. We use the value of keff to determine the spatial profile of the resonance. We then investigate two more complicated situations: symmetric CFBG-FPs with a long cavity length; and asymmetric CFBG-FPs.
For a single CFBG with grating strength k 0 the value of θ is directly related to the phase, ϕ(z), of the chirped grating, and keff is particularly easy to evaluate:
The transmission of the grating is
|t| is independent of frequency because we have implicitly assumed that the grating is infinitely long, so all frequencies experience the same transmission. In practice, Eq. (17) is valid for frequencies well within the stop gap of a finite grating. In Fig. 2 we plot |t|2 as a function of k 0 using using the CME and Eq. (17). In the simulations we use Λ=533nm and Ch =2.5nm/cm. The agreement between the two is excellent.
For a CFBG-FP, keff is significantly more complicated. We define γ=κ 2/κ 1, the ratio between the strengths of the two superimposed gratings. Then,
is the difference between the quadratic phases of the displaced gratings.
Since κ(z) is related to cos (Δϕ(z)), it has a spatial period Z=2/(Chd), which arises from the beat period between the two exponentials that comprise κ̂ (z) (10). The quantity dθ/dz is not periodic, because the first term in the parentheses of the expression (19) is linearly dependent on z. In moving from a position z to z+Z while keeping the frequency constant, κ (z) is unchanged while δ (z;ω) increases by π/d. To keep δ (z;ω) unchanged, one must reduce the frequency by a value δω=πc/n̄d. Thus, for an infinite structure, any feature found at (ω,Z) will also be found at (ω-πc/n̄,z+Z). Note that δω=πc/n̄d corresponds to δf=c/2n̄d, which is the usual FSR associated with a FP. Therefore, in the CFBG-FP, the FSR is associated with the spatial beat period between the displaced CFBGs.
When both gratings of a CFBG-FP have the same strength (κ 1=κ 2≡κ 0) the formulae for κ (z) and dθ/dz take on a particularly simple form:
The formula for dθ/dz is valid everywhere except at the vanishing points of κ (z). At these vanishing points θ experiences a π phase-shift, so its derivative diverges. In Fig. 3 a we plot κ (z) using n 0=1.45, δn 1=δn 2=7.38×10-4, Ch=2.5nm/cm, d=1mm and Λ0=533nm which leads to a spatial beat period Z=2.27mm. We use a total length L=24mm. In Fig. 3 b, we plot the transmission spectrum of the grating, simulated using the CME. The transmission peaks have a FSR of 0.824nm, which is the expected value for a FP resonator with index n̄ and cavity spacing d. Each transmission peak can be identified with a vanishing point of κ (z) in the following sense. In Fig. 3 b, we identify the transmission peak at 1.5503µm with a black dot. When light of this frequency traverses the structure, there is a large build-up of field intensity centred about the vanishing point of κ (z) indicated by a black dot in Fig. 3 a. Transmission through adjacent FP peaks is correlated with a build-up of field intensity at adjacent vanishing points.
The field build-up at a resonance can be understood both qualitatively and quantitatively using keff (z;ω). In Fig. 4 (thick dotted line) we plot Im [keff ] for the wavelength indicated by the black dot in Fig. 3 a, normalized so that its peak value is one. Im [keff ] has two large side-lobes, and it vanishes at the vanishing point of κ (z). After these side-lobes, Im [keff ] vanishes for the rest of the grating, because the local Bragg frequency is no longer sufficiently close to the incident wavelength to make the grating reflective. For the symmetric situation (κ 1=κ 2≡κ 0) depicted in Figs. 3 and 4, θ (z) experiences a π-shift at the vanishing points of κ (z), so the structure is directly analagous to a DFB, except that the mirrors on either side of the vanishing point have a reflectivity profile given by Im [keff ]. Consequently, instead of the decaying exponential associated with a DFB, the field |A+(z)| for the Nth resonance, centred at zN, is roughly
In Fig. 4 (dotted line) we plot |A+(z)|2 using (23). We also plot |A+(z)|2 as determined directly from the CME (solid line). The main differences between these solutions occur in the wings of the field, because the approximate solution does not account for the boundary conditions at the interface between the regions where Im [keff ]=0 and Im [keff ]≠0. Note that in the vicinity of its vanishing point Im [keff ] is roughly linear with respect to z, so that ∫ Im [keff (z)]∝ (z-zN)2, which gives a Gaussian profile for |A+(z)|2. Eventually this approximation is no longer valid, and the profile deviates from a Gaussian. However, if the grating is sufficiently strong, then the field will decay almost to zero before the approximation is invalid.
To determine the exact positions and the wavelengths of the resonances of Fig. 3 b we note that they occur when Im [keff ] is symmetric about its vanishing point. This occurs for wavelengths where δ (z)=0 at the nodes of κ (z). Using (21) and (20) we find that the position zN of the Nth resonance is
Using zN in the expression (13) for δ (z), the resonance frequencies occur at
It is evident from Eq. (24) that the exact position of the resonances in the CFBG-FP is dependent on the ratio Λ0/d. This means that a small change in the value of d will have a large effect on in d of about 0.25µm will change the position of the resonances (but not their spacing) by about 1mm. Furthermore, since the FSR is independent of Λ0/d, the resonances are not necessarily integer multiples of the FSR.
Up to this point we have considered a CFBG-FP with a small cavity length. If we increase the cavity length then the beat period, Z, decreases, which has two direct consequences. First, the width of the lobes where Im [keff (z)]≠0 is much narrower. Second, the region over which the detuning is relatively small spans many more lobes than in Fig. 3. In Fig. 5 we plot Im [keff (z)](dotted line) for a transmission peak using the same parameters as used for Fig. 3, but with a cavity spacing d=3.2mm. We also plot (solid line) |A+(z;ωN )|2. In the accompanying movie clip for Fig. 5 we plot the field profile for cavity lengths varying from d=1mm to d=10mm. For smaller values of d, the field profile is roughly Gaussian, but eventually side lobes appear and the approximate expression (23) for |A+(z;ωN )| is no longer valid. Nevertheless, the oscillations in |A+(z)|2 correspond to the regions where Im [keff (z)]=0 and Im [keff (z)]≠0. In principle, one can use keff (z) to get a quantitative approximation for the field profile |A(z)|2 for a long cavity CFBG-FP. In regions where keff (z) is imaginary (real), both A± (z) are evanescent (oscillatory). One need only match the solutions at the points where keff (z)=0, and then impose the chosen boundary conditions at z=0 and z=L to determine the entire field pattern of the system. However, the usual method for performing this field matching, WKB theory, introduces small errors into the solution . Furthermore, when the distribution of keff (z) is complicated, as in Fig. 5, one needs to match a large number of matrices to generate the solution.
The situation becomes increasingly complicated when κ 1≠κ 2. If we vary the value of κ 2 we find two results. First, as expected for a FP constructed with unequal mirrors, the maximum transmission of the peaks in the reflection spectrum no longer reaches 100%. The variation in maximum transmission with γ (=κ 2/κ 1) can be quite accurately modelled by the familiar Airy function for FP resonators using the reflectivities of the individual gratings. The second change is that the wavelength of the resonance peaks varies linearly with γ, as shown in Fig. 6 (solid line). In varying γ we vary both the value of n̄ and δnmod . Since the FSR is related to the value of n̄, it is perhaps not too surprising that the positions of the resonance wavelengths change as we augment γ. However, these positions vary, albeit less sharply, even when we hold n̄ constant (dashed line) and vary only δnmod . Inset into Fig. 6 is Im [keff (z)] associated with γ=1.166. The (normalized) lobes are of unequal amplitude, which reflects the fact that the displaced grating (k 2) is stronger than the first grating (k 1). Also, the region between the lobes is now much larger than a single point, because the variation in dθ/dz significantly modifies the local Bragg frequency in this region. By solving the CME starting at a point in between the two lobes, we have confirmed that the transmission peaks correspond to the situation where one round trip in the cavity accumulates a multiple of 2π worth of phase. Furthermore, we have confirmed that the attenuation of the first (second) lobe corresponds to r 1 (r 2), the reflectivites of the individual gratings in the absence of each other. The field profile in the grating (not shown) is asymmetric, because the reflectivity to the right of the cavity is much larger than the reflectivity to the left of the cavity.
We have presented a theoretical analysis of a Fabry-Perot interferometer constructed using chirped, superimposed Bragg gratings. By introducing an effective wavenumber to describe the structure, we have shown that the spatial beat period associated with the superimposed gratings is responsible for the FSR of the cavity. We have identified the positions of the resonances with vanishing points of the grating coefficient, κ(z). We have also shown that the field patterns within the structure are remarkably more complicated than would be expected for a normal FP resonator. In the simplest case, where the field profile is roughly Gaussian, we presented an analytical formula to describe the profile. However, for larger cavities, the profile exhibits several oscillations that correspond to the field entering and leaving reflective portions of the grating. The work in this paper can be helpful for designing lasing structures, where, due to spatial hole burning, the field distribution inside the structure is of prime importance.
This research is part of a program funded by the Québec Government and TeraXion. Suresh Pereira acknowledges funding from the Natural Sciences and Engineering Research Council of Canada (NSERC).
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