## Field profiles and spectral properties of chirped Bragg grating Fabry-Perot interferometers

Optics Express, Vol. 13, Issue 6, pp. 1906-1915 (2005)

http://dx.doi.org/10.1364/OPEX.13.001906

Acrobat PDF (207 KB)

### Abstract

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

## 1. Introduction

1. S. Doucet, R. Slavik, and S. LaRochelle, “High-Finesse large band Fabry-Perot fiber filter with superimposed chirped Bragg gratings,” Electron. Lett. **38**, 402–3 (2002). [CrossRef]

2. X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. **15**, 1111–13 (2003). [CrossRef]

*d*, then a given wavelength, λ

_{0}, 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 2

*π*, then the cavity becomes highly transmitting, and hence acts as a FP resonator. Experimental and theoretical [4

4. R. Slavik, S. Doucet, and S. LaRochelle, “High-Performance all-fiber Fabry-Perot filters with superimposed chirped Bragg gratings,” J. Lightwave Technol. **21**, 1059–65 (2003). [CrossRef]

*c*/(2

*nd*), 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 [4

4. R. Slavik, S. Doucet, and S. LaRochelle, “High-Performance all-fiber Fabry-Perot filters with superimposed chirped Bragg gratings,” J. Lightwave Technol. **21**, 1059–65 (2003). [CrossRef]

5. C. Sung-Hak, I Yokota, and M. Obara, “Free spectral range variation of a broadband, high-finesse, multi-channel Fabry-Perot filter using chirped fiber Bragg gratings,” Jpn. J. Appl. Phys. Part 1, **36**, 6383–7 (1997). [CrossRef]

6. G. Town, K. Sugden, J. Williams, I. Bennion, and S. B. Poole, “Wide-Band Fabry-Perot-Like Filters in Optical Fiber,” IEEE Photon. Technol. Letters, **7**, 78–80 (1995). [CrossRef]

7. A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of a Bragg grating and its applications to Fabry-Perot cavities,” J. Opt. Soc. Ame. A **20**, 273–81 (2003). [CrossRef]

*k*

_{eff}, that indicates the spatial regions of the CFBG-FP that are reflective. Using the periodicity of

*k*

_{eff}we explain the FSR of a CFBG-FP. The imaginary portion of

*k*

_{eff}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.

*k*

_{eff}. 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

*n*

_{0}is the original index of refraction of the medium,

*δ*

_{n}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*)=Λ

_{0}+(

*C*

_{h}

*z*/2), where

*C*

_{h}

*z*≪Λ

_{0}across the entire grating length. Using this expression for Λ(

*z*) in Eq. (1) we find that

*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.

*z*

_{i}gives the displacement of the

*i*

^{th}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

*δn*

_{mod}(

*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.

*ω*, as

8. T. Erdogan, “Fiber Grating Spectra,” Journal of Lightwave Technology , **15**, 1277–94 (1997). [CrossRef]

*ω*

_{0}=

*πc/n̄*Λ

_{0}. Since

*z*) is complex we can write

*z*)=

*κ*(

*z*)

*e*

^{iθ}(

*z*), where both

*κ*and

*θ*are real quantities. Making the phase transformation

*A*±(

*z*)=

*Â*±(z)

*e*

^{±iθ(z)/2}we find

*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 [9

9. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E **48**, 4758–67 (1993). [CrossRef]

*κ*(

*z*), and the detuning of light from the local Bragg wavelength. In terms of

*k*

_{eff}, fields acquire phase according to

*e*

^{iζ}, where

*k*

_{eff}is imaginary (|

*δ*(

*z;ω*)|<|

*κ*(

*z*)|), the quantity ζ will also be imaginary, and the ‘phase’ term

*e*

^{iζ}will describe exponential growth or decay. Therefore, the portions of the grating for which

*k*

_{eff}(

*z;ω*) is imaginary can be thought of as reflective, and the portions where

*k*

_{eff}(

*z;ω*) is real as dispersive.

## 3. Results and discussion

*k*

_{eff}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

*k*

_{eff}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.

*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=533

*nm*and

*C*

_{h}=2.5

*nm/cm*. The agreement between the two is excellent.

*k*

_{eff}is significantly more complicated. We define

*γ=κ*

_{2}/

*κ*

_{1}, the ratio between the strengths of the two superimposed gratings. Then,

*κ*

_{1}=

*κ*

_{2}≡

*κ*

_{0}) the formulae for

*κ*(

*z*) and

*dθ/dz*take on a particularly simple form:

*k*

_{eff}(

*z;ω*). In Fig. 4 (thick dotted line) we plot

*Im*[

*k*

_{eff}] for the wavelength indicated by the black dot in Fig. 3

*a*, normalized so that its peak value is one.

*Im*[

*k*

_{eff}] has two large side-lobes, and it vanishes at the vanishing point of

*κ*(

*z*). After these side-lobes,

*Im*[

*k*

_{eff}] 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*[

*k*

_{eff}]. Consequently, instead of the decaying exponential associated with a DFB, the field |

*A*+(

*z*)| for the

*N*

^{th}resonance, centred at

*zN*, is roughly

*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*[

*k*

_{eff}]=0 and

*Im*[

*k*

_{eff}]≠0. Note that in the vicinity of its vanishing point

*Im*[

*k*

_{eff}] is roughly linear with respect to

*z*, so that ∫

*Im*[

*k*

_{eff}(

*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.

*b*we note that they occur when

*Im*[

*k*

_{eff}] 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

*N*

^{th}resonance is

*zN*in the expression (13) for

*δ*(

*z*), the resonance frequencies occur at

_{0}/

*d*. This means that a small change in the value of

*d*will have a large effect on in

*d*of about 0.25

*µm*will change the position of the resonances (but not their spacing) by about 1

*mm*. Furthermore, since the FSR is independent of Λ0/

*d*, the resonances are not necessarily integer multiples of the FSR.

*Im*[

*k*

_{eff}(

*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*[

*k*

_{eff}(

*z*)](dotted line) for a transmission peak using the same parameters as used for Fig. 3, but with a cavity spacing

*d*=3.2

*mm*. We also plot (solid line) |

*A*+(

*z;ω*

_{N})|

^{2}. In the accompanying movie clip for Fig. 5 we plot the field profile for cavity lengths varying from

*d*=1

*mm*to

*d*=10

*mm*. For smaller values of

*d*, the field profile is roughly Gaussian, but eventually side lobes appear and the approximate expression (23) for |

*A*+(

*z;ω*

_{N})| is no longer valid. Nevertheless, the oscillations in |

*A*+(

*z*)|

^{2}correspond to the regions where

*Im*[

*k*

_{eff}(

*z*)]=0 and

*Im*[

*k*

_{eff}(

*z*)]≠0. In principle, one can use

*k*

_{eff}(

*z*) to get a quantitative approximation for the field profile |

*A*(

*z*)|

^{2}for a long cavity CFBG-FP. In regions where

*k*

_{eff}(

*z*) is imaginary (real), both A± (

*z*) are evanescent (oscillatory). One need only match the solutions at the points where

*k*

_{eff}(

*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 [9

9. L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E **48**, 4758–67 (1993). [CrossRef]

*k*

_{eff}(

*z*) is complicated, as in Fig. 5, one needs to match a large number of matrices to generate the solution.

## 4. Conclusion

*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.

## Acknowledgements

## References and links

1. | S. Doucet, R. Slavik, and S. LaRochelle, “High-Finesse large band Fabry-Perot fiber filter with superimposed chirped Bragg gratings,” Electron. Lett. |

2. | X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev, and I. Bennion, “Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,” IEEE Photon. Technol. Lett. |

3. | G. Brochu, R. Slavik, and S. LaRochelle, “Ultra-Compact 52 mW 50-GHz spaced 16 channels narrow-line and single polarization fiber laser,” in Optical Fiber Communication Conference (The Optical Society of America, Washington, DC, 2004), postdeadline paper PDP22. |

4. | R. Slavik, S. Doucet, and S. LaRochelle, “High-Performance all-fiber Fabry-Perot filters with superimposed chirped Bragg gratings,” J. Lightwave Technol. |

5. | C. Sung-Hak, I Yokota, and M. Obara, “Free spectral range variation of a broadband, high-finesse, multi-channel Fabry-Perot filter using chirped fiber Bragg gratings,” Jpn. J. Appl. Phys. Part 1, |

6. | G. Town, K. Sugden, J. Williams, I. Bennion, and S. B. Poole, “Wide-Band Fabry-Perot-Like Filters in Optical Fiber,” IEEE Photon. Technol. Letters, |

7. | A. Melloni, M. Floridi, F. Morichetti, and M. Martinelli, “Equivalent circuit of a Bragg grating and its applications to Fabry-Perot cavities,” J. Opt. Soc. Ame. A |

8. | T. Erdogan, “Fiber Grating Spectra,” Journal of Lightwave Technology , |

9. | L. Poladian, “Graphical and WKB analysis of nonuniform Bragg gratings,” Phys. Rev. E |

**OCIS Codes**

(050.2230) Diffraction and gratings : Fabry-Perot

(050.2770) Diffraction and gratings : Gratings

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 8, 2005

Revised Manuscript: March 3, 2005

Published: March 21, 2005

**Citation**

Suresh Pereira and Sophie LaRochelle, "Field profiles and spectral properties of chirped Bragg grating Fabry-Perot interferometers," Opt. Express **13**, 1906-1915 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-6-1906

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### References

- S. Doucet, R. Slavik and S. LaRochelle, �??High-Finesse large band Fabry-Perot fiber filter with superimposed chirped Bragg gratings,�?? Electron. Lett. 38, 402-3 (2002). [CrossRef]
- X. Shu, K. Sugden, P. Rhead, J. Mitchell, I. Felmeri, G. Lloyd, K. Byron, Z. Huang, Igor Khrushchev and I. Bennion, �??Tunable Dispersion Compensator Based on Distributed Gires-Tournois Etalons,�?? IEEE Photon. Technol. Lett. 15, 1111-13 (2003). [CrossRef]
- G. Brochu, R. Slavik and S. LaRochelle, �??Ultra-Compact 52 mW 50-GHz spaced 16 channels narrow-line and single polarization fiber laser,�?? in Optical Fiber Communication Conference (The Optical Society of America, Washington, DC, 2004), postdeadline paper PDP22.
- R. Slavik, S. Doucet and S. LaRochelle, �??High-Performance all-fiber Fabry-Perot filters with superimposed chirped Bragg gratings,�?? J. Lightwave Technol. 21, 1059-65 (2003). [CrossRef]
- C. Sung-Hak, I Yokota and M. Obara, �??Free spectral range variation of a broadband, high-finesse, multi-channel Fabry-Perot filter using chirped fiber Bragg gratings,�?? Jpn. J. Appl. Phys. Part 1, 36, 6383-7 (1997). [CrossRef]
- G. Town, K. Sugden, J. Williams, I. Bennion and S. B. Poole, �??Wide-Band Fabry-Perot-Like Filters in Optical Fiber,�?? IEEE Photon. Technol. Letters, 7, 78-80 (1995). [CrossRef]
- A. Melloni, M. Floridi, F. Morichetti and M. Martinelli, �??Equivalent circuit of a Bragg grating and its applications to Fabry-Perot cavities,�?? J. Opt. Soc. Ame. A 20, 273-81 (2003). [CrossRef]
- T. Erdogan, �??Fiber Grating Spectra,�?? Journal of Lightwave Technology, 15, 1277-94 (1997). [CrossRef]
- L. Poladian, �??Graphical and WKB analysis of nonuniform Bragg gratings,�?? Phys. Rev. E 48, 4758-67 (1993). [CrossRef]

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