## Abstract

In this paper the analytic models (AMs) of the spectral responses of fiber-grating-based interferometers are derived from the Fourier mode coupling (FMC) theory proposed recently. The interferometers include Fabry-Perot cavity, Mach-Zehnder and Michelson interferometers, which are constructed by uniform fiber Bragg gratings and long-period fiber gratings, and also by Gaussian-apodized ones. The calculated spectra based on the analytic models are achieved, and compared with the measured cases and those on the transfer matrix (TM) method. The calculations and comparisons have confirmed that the AM-based spectrum is in excellent agreement with the TM-based one and the measured case, of which the efficiency is improved up to ~2990 times that of the TM method for non-uniform-grating-based in-fiber interferometers.

© 2012 OSA

## 1. Introduction

Optical fiber grating has developed into a critical component with wavelength selectivity, for optic fiber communication and sensing systems [1, 2]. According to the propagating direction of coupled mode, fiber gratings are classified to be two categories: fiber Bragg (i.e., short-period) grating (FBG) and long-period fiber grating (LPFG). FBG can cause the mode coupling between forward and backward core-modes, which is regarded as an intrinsic mirror. Thus two cascaded FBGs can form an intrinsic Fabry-Perot (FP) cavity in optical fiber [3]. LPFG can couple core-mode to co-propagating cladding modes, and *vice versa*. Therefore two cascaded LPFGs function as a Mach-Zehnder (MZ) interferometer [4], and one LPFG with a terminal reflector functions as a Michelson interferometer [5]. The core and cladding of optical fiber are taken for the two paths of a LPFG-based interferometer. These in-fiber interferometers are small and completely compatible with optical fiber, and can be employed for comb-like multi-wavelength filters and highly sensitive measurements.

The fiber gratings designed for the interferometers are usually non-uniform (NU), since they can shape their spectra and restrain the undesirable sidelobes prevalent in uniform-gratings. The point-to-point technique that produces NU-gratings requires that the writing parameters can be changed after each point [6]. For timely modifying the writing parameters, especially for the real-time control, the fast analyses of the spectral responses of these interferometers are very significant and necessary in the designs and fabrications. The spectral responses are mainly reflectivity and transmission for FBG-based FP cavity and LPFG-based interferometers, respectively. A grating-based interferometer is usually analyzed by dividing it into three sections, and then multiplying the 2 × 2 transfer matrix (TM) of each section. If the fiber gratings in an interferometer are simple or uniform, the TM of each section can be easily and exactly modeled on the well-known coupled-mode theory. Otherwise, it is necessary to apply some numerical techniques to each section, which include the TM method [7, 8], Rouard’s method [9], Runge-Kutta algorithm [10], direct Fourier transform (FT) [11, 12], the method of single expression [13], the numerical implementations of Bloch wave [14] and electromagnetic scattering [15, 16] etc.. But for analyzing complex gratings, they are substantially time-consuming, and restricted by accumulative computation errors. So it is worthy to look for a fast and exact tool for modeling the spectral responses of the grating-based, especially NU-grating-based interferometers.

The Fourier mode coupling (FMC) theory was proposed recently for the spectra of FBGs and LPFGs [17, 18], which is especially suitable for NU-gratings [19, 20]. For modeling the spectral responses of the aforementioned interferometers, we take an interferometer for a wholly single NU-grating, and then derive the spectral response of the NU-grating from the FMC theory. The spectral response of the NU-grating is regarded as that of the interferometer, which can provide fast and accurate analysis for any grating-based interferometer, and provide analytic solutions for some NU-grating-based interferometers.

## 2. Analytic models

#### 2.1 FMC theory for grating-based interferometers

We assume that the index perturbation in any interferometer is created uniformly across fiber core and nonexistent outside the core. The index perturbation will couple a propagating core mode to backward core mode or forward cladding modes, determined by the case of phase matching. The mode couplings are illustrated in Figs. 1(a) , 1(b) and 1(c), corresponding to the FBG-based FP cavity, LPFG-based MZ and Michelson interferometers.

According to the FMC theory [17, 18], the coupled-mode equation for counter- and co-directional couplings can be constructed to be

*z*is the wave propagation direction,

*Δn*(

*z*) is the profile of the index perturbation in an interferometer,

*m*and

*s*are the orders of propagating mode and coupled mode, respectively,

*B*(

_{m}*z*) and

*B*(

_{s}*z*) are amplitude coefficients,

*β*and

_{m}*β*are propagation constants; sign “±” is replaced with “+” for counter-directional coupling, and with “-” for co-directional coupling;

_{s}*k*is the coupling coefficient between the propagating and coupled modes, defined as

_{s}*ε*

_{0}is the permittivity of vacuum,

*ω*is angular frequency,

*n*

_{0}is the refractive index of fiber core,

*r*and

*ϕ*are the axes of cylindrical coordinate system,

*A*is cross-section area,

*A∞*is whole cross-section area,

*E*and

_{m}*E*are the electric fields of the propagating and coupled modes, respectively.

_{s}The two fiber gratings in an interferometer are usually identical. The expression of the perturbation of Michelson interferometer will be similar to that of MZ interferometer, if the terminal reflector is with the reflectivity of 1. Every interferometer aforementioned can be regarded as only two cascaded fiber gratings. Thus all the interferometers are with the similar expression of perturbation distribution, which just have the different values of grating period. If the origin of axis *z* is located at the center of any interferometer, the index perturbation *Δn*(*z*) of the interferometer consisting of two identical fiber gratings, can be modeled as

*L*is the length of each fiber grating,

*P*is the interval between the two cascaded gratings, Δ

*n*

_{0}(

*z*) is the perturbation distribution of one fiber grating shifted mathematically to the central position. In Michelson interferometer,

*P*equals 2 times the space between the LPFG and the terminal reflector.

We know *β _{m}* = 2

*π*(

*n*+

_{m}*δ*

_{n0})/

*λ*and

*β*= 2

_{s}*πn*/

_{s}*λ*, where

*δ*

_{n0}is the effective-index change caused by the index perturbation,

*λ*is the wavelength in vacuum,

*n*and

_{m}*n*are the effective indices of propagating and coupled modes, respectively. Note that

_{s}*n*and

_{s}*k*in backward coupling are different from those in forward coupling, which are related to the order

_{s}*s*of coupled mode. By considering two-mode coupling and incorporating Eq. (3) into Eq. (1), the integration of Eq. (1) is performed along the axis

*z*within the limits of the start position

*z*

_{0}= -(

*L*+

*P*/2) and the end position

*z*

_{1}=

*L*+

*P*/2 of an interferometer, which directly leads to

*ν*= (

_{s}*n*+

_{m}*δ*

_{n0}±

*n*)/

_{s}*λ*. The boundary conditions are

*B*(

_{m}*z*

_{0}) = 1,

*B*(

_{s}*z*

_{1}) = 0 for counter-directional couplings, and are

*B*(

_{m}*z*

_{0}) = 1,

*B*(

_{s}*z*

_{0}) = 0 for co-directional couplings. According to the boundary conditions and the law of flux conservation [21], the relations between the amplitude coefficients

*B*(

_{m}*z*) and

*B*(

_{s}*z*) can be described by Eqs. (5) and (6) for counter- and co-directional couplings, respectively

The right part of Eq. (4) is the Fourier transform of the index perturbation *Δn*_{0}(*z*) in one fiber grating. We define an equation as

*ν*is independent variable as defined previously,

_{s}*γ*(

_{s}*ν*) and

_{s}*η*(

_{s}*ν*) represent the real and imaginary, respectively, components due to the Fourier transform. Equation (7) can be implemented by discrete Fourier transform suitable for all types of index perturbations, also by analytic solution if there exists closed-form Fourier integration suitable for some perturbations. By substituting Eqs. (5), (6) and (7) into Eq. (4), we can solve the integral Eq. (4) for

_{s}*B*(

_{m}*z*) and

*B*(

_{s}*z*), and then obtain the reflectivity

*R*= |

*B*

_{s}(

*z*

_{0})|

^{2}/|

*B*

_{m}(

*z*

_{0})|

^{2}and the

*bar*-transmission

*T*= |

*B*

_{m}(

*z*

_{1})|

^{2}/|

*B*

_{m}(

*z*

_{0})|

^{2}of FBG- and LPFG-based interferometers, respectively, given by

*ν*= (

_{B}*n*+

_{m}*δ*

_{n0}+

*n*)/

_{s}*λ*,

*ν*= (

_{L}*n*+

_{m}*δ*

_{n0}-

*n*)/

_{s}*λ*. Equations (8) and (9) are the general solutions for the spectra responses of the aforementioned interferometers which may be constructed by uniform or non-uniform gratings. Once the perturbation Δ

*n*

_{0}(

*z*) of one fiber grating is determined, the spectral responses of the corresponding interferometer can be modeled and calculated on Eqs. (7), (8) and (9), which provide analytic solutions for the interferometers formed by uniform-gratings and some NU-gratings. The followings are some examples of the analytic models.

#### 2.2 For uniform-grating-based interferometers

For any interferometer formed by uniform FBGs or LPFGs, the perturbation Δ*n*_{0u}(*z*) of one uniform grating can be written to be

*Λ*and

*δ*

_{n}are the period and amplitude, respectively, of the perturbation in one fiber grating. In Eq. (10), the difference between FBG and LPFG is only the value of period

*Λ*. By substituting Eq. (10) into Eq. (7), we can derive the closed-form of the Fourier transform result, and then get

*γ*= 0 and

_{su}*η*as

_{su}*R*of the uniform-FBG-based FP cavity, and for the bar-transmission

_{Fu}*T*of the uniform-LPFG-based MZ or Michelson interferometer, governed by Eqs. (12) and (13), respectively

_{Mu}*σ*=

_{Β}*v*1/Λ and

_{B}−*σ*=

_{L}*v*1/Λ are the detuning parameters of counter- and co-directional couplings, respectively.

_{L}−#### 2.3 For GA-grating-based interferometers

For any interferometer constructed by GA-FBGs or -LPFGs, the perturbation Δ*n*_{0G}(*z*) of one GA-grating can be modeled as

*α*is GA-coefficient,

*Λ*and

*δ*

_{n}are the nominal period and amplitude, respectively, of the perturbation. By incorporating Eq. (14) with Eqs. (7), (8) and (9), we can derive the closed-form of the Fourier transform of the perturbation Δ

*n*

_{0G}(

*z*), and then obtain the analytic solutions for the reflectivity

*R*of the GA-FBG-based FP cavity, and for the bar-transmission

_{FG}*T*of the GA-LPFG-based MZ or Michelson interferometer, described by Eqs. (15) and (16), respectively

_{MG}Equations (12) and (15) are the analytic models (AMs) of the reflectivities of the FP cavities constructed by uniform- and GA-FBGs, respectively. Equations (13) and (16) are the analytic models of the transmissions of the MZ or Michelson interferometers formed by uniform- and GA-LPFGs, respectively. They all are analytic solutions for the spectral responses.

## 3. Simulations and comparisons

To confirm the accuracy and efficiency of the analytic models, we now apply Eqs. (12), (13), (15) and (16) to the calculations of the spectral responses of the interferometers, and compare them with the measured and TM-based spectra, since the TM method is preferred and most often employed.

#### 3.1 Uniform-FBG-based FP cavity

Two uniform FBGs with sinusoidal index perturbation are induced in a single mode fiber with the effective index *n _{m}* = 1.4635 of core mode LP

_{01}. The two FBGs are identical, and cascaded to form a FP cavity, which are described by the following parameters: period

*Λ*= 0.532

*μm*, grating length

*L*= 1000.16

*μm*(integer times the period), perturbation amplitude

*δ*= 4 × 10

_{n}^{−4}, normalized coupling coefficient

*k*= 2531πΝ/s and interval

_{s}*P*= 2.5

*mm*. The transmission

*T*of the FP cavity can be measured easily by an optical spectrum analyzer, where

_{Fu}*T*= 1-

_{Fu}*R*. Figures 2(a) and 2(b) plot the calculated reflectivities and transmissions, respectively, of the FP cavity, according to Eq. (12) (solid lines) and the TM method (dotted lines). Figure 2(c) shows the measured transmission of a FBG-based FP cavity with the similar parameters abovementioned. Figures 2(a), 2(b) and 2(c) indicate that the Eq. (12)-based spectrum is very close to the TM-based one and the measured case, and that the spectral response of uniform-FBG-based FP cavity is with some sidelobes.

_{Fu}#### 3.2 Uniform-LPFG-based MZ and Michelson interferometers

Two uniform LPFGs with quasi-sinusoidal index perturbation are inscribed in a single-mode fiber that possesses the effective index *n _{m}* = 1.46533 and

*n*= 1.462 of core mode and cladding mode, respectively. The two LPFGs are also identical, and cascaded to construct a MZ interferometer with the following parameters:

_{s}*Λ*= 400

*μm*,

*L*= 60

*mm*,

*P*= 550

*mm*,

*δ*= 1.8 × 10

_{n}^{−4}and

*k*= 69.8πΝ/s. Figure 3(a) illustrates the calculated transmissions of the MZ interferometer, depending on Eq. (13) (solid lines) and the TM method (dotted lines). Figure 3(b) exhibits the measured transmission of a MZ interferometer with the similar parameters abovementioned. Figures 3(a) and 3(b) show that the Eq. (13)-based spectrum is in good agreement with the TM-based one, and is also very close to the measured case. A Michelson interferometer can be formed by a terminal reflector and one LPFG with the same parameters abovementioned. If the value of the interval between the LPFG and the terminal reflector without attenuation is equal to 275

_{s}*mm*, the spectral response of the Michelson interferometer is the same as the MZ interferometer.

#### 3.3 GA-grating-based interferometers

Like analogous the uniform-grating-based interferometers, we can also calculate the spectral responses of GA-grating-based interferometers. A FP cavity consists of two identical GA-FBGs in a single-mode fiber with the effective index *n _{m}* = 1.49 of core mode LP

_{01}. The FP cavity is described by the following parameters: GA-coefficient

*α*= 20,

*Λ*= 0.52

*μm*,

*L*= 3

*mm*,

*P*= 4

*mm*,

*δ*= 3.5 × 10

_{n}^{−4}and

*k*= 2531πΝ/s. Figure 4(a) plots the calculated reflectivities of the GA-FBG-based FP cavity, according to Eq. (15) (solid lines) and the TM method (dotted lines). In the TM method, each GA-FBG is divided into 150 piecewise-uniform segments.

_{s}Two identical GA-LPFGs are written into a single-mode fiber with the effective index *n _{m}* = 1.4654 and

*n*= 1.4619 of core mode and cladding mode, respectively, which results in a MZ interferometer. The MZ interferometer is described by the parameters:

_{s}*α*= 20,

*Λ*= 400

*μm*,

*L*= 40

*mm*,

*P*= 150

*mm*,

*δ*= 4 × 10

_{n}^{−4}and

*k*= 115.7πΝ/s. Figure 4(b) demonstrates the calculated transmissions of the GA-LPFG-based MZ or Michelson interferometers, by using Eq. (16) (solid lines) and the TM method (dotted lines). In the TM method, each GA-LPFG is also divided into 150 piecewise-uniform segments. Figures 4(a) and 4(b) indicate that the calculated spectra based on Eqs. (15) and (16) are in excellent agreements with those on the TM method, for GA-grating-based interferometers.

_{s}#### 3.4 Efficiency of analytic models

In respect of calculation efficiency, the AM-based efficiency is much higher than the TM-based one related to the number of segments. All the calculations aforementioned were achieved in a PC as Founder S360R with the Vista OS and Matlab5.3. In the cases of 2-pm wavelength resolution and 4-nm span, the TM-based calculation requires the average time of about 0.2628*s* and 11.824*s* for the FP cavities formed by uniform FBGs and GA-FBGs with the segment number of 150, respectively, whereas the AM-based calculation needs correspondingly about 2.73*ms* and 4.19*ms*. In the cases of 20-pm wavelength resolution and 120-nm span, the TM-based calculation requires the average time of about 0.6136*s* and 29.157*s* for the MZ or Michelson interferometers formed by uniform LPFGs and GA-LPFGs with the segment number of 150, respectively, whereas the AM-based calculation needs correspondingly about 8.132*ms* and 9.75*ms*.

The calculation times as listed above imply that the AM-based efficiencies are improved up to 96 and 2990 times the TM-based ones for uniform- and GA-grating-based, respectively, interferometers, which are the highest as the coupled-mode theory of single uniform-grating. This means that the fast analyses of the spectral responses of some NU-grating-based interferometers can be achieved on the base of the FMC theory. If two NU-gratings in an interferometer are different, the AM-based efficiency will be increased up to about 5980 times the TM-based one.

## 4. Conclusion

In conclusion we have presented the analytic models of the spectral responses of some grating-based interferometers on the FMC theory. The interferometers include FP cavity, MZ and Michelson interferometers, which are constructed by uniform- and GA-FBGs, or uniform- and GA-LPFGs. The analytic models of GA-grating-based interferometers illustrate that the closed-form solutions for the spectral responses of some NU-grating-based interferometers can be derived from the FMC theory, which have overturned the traditional viewpoint that is the absence of analytic solutions for the spectra of NU-grating-based devices. The AM-based calculations have been achieved, and compared with the TM-based ones and the measured cases. The comparisons illustrate that the efficiency of the AM-based calculation is increased up to several thousands times that of the TM-based one, and also demonstrate that the AM-based spectrum is in excellent agreement with practical case and the TM-based one, in the aspects of shape, central wavelength, bandwidth, sidelobes and coupled intensity etc. The calculations and comparisons have confirmed that the analytic models on the FMC theory are extremely accurate and highly efficient for the interferometers formed by uniform and non-uniform fiber gratings.

The FMC-based modeling can be exploited for the fast and accurate analyses of the spectral responses of grating-based interferometers, especially for that of complex-grating-based interferometers. Like analogous the GA-grating-based interferometers, we can also model the spectral responses of the interferometers constructed by other complex gratings, such as Moire, superstructure, raised-cosine-apodized, linearly chirped and phase-shifted gratings etc.

## Acknowledgments

This work was supported partially by the Key Laboratory of Optical Fiber Sensing and Communications (UESTC), Ministry of Education, P. R. China, under grant Open Fund Project 2010, and also by the Key Laboratory of Optoelectronic Technology and Systems, Ministry of Education, P. R. China, under grant Visiting Scholar Fund 2011.

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