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Optics Express

  • Editor: J. H. Eberly
  • Vol. 1, Iss. 9 — Oct. 27, 1997
  • pp: 242–249
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Dynamics of ultrashort pulse propagation through fiber gratings

Lawrence R. Chen, J. E Sipe, Seldon D. Benjamin, Howard Jung, and Peter W. E. Smith  »View Author Affiliations


Optics Express, Vol. 1, Issue 9, pp. 242-249 (1997)
http://dx.doi.org/10.1364/OE.1.000242


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Abstract

By directly integrating the time-domain coupled-mode equations, we can explicitly obtain and examine the backward and forward propagating waves as a function of position and time within fiber grating structures. We apply this numerical procedure to calculate the temporal reflection and transmission response of fiber gratings subject to ultrashort pulse inputs. This allows us to study the dynamics of the ultrashort pulse-grating interaction.

© Optical Society of America

1. Introduction

In-fiber Bragg gratings have been successfully implemented to satisfy a variety of optical communications needs such as wavelength-division-multiplexing and dispersion compensation, and in numerous sensing applications. [1

1. See, for example, W. W. Morey, G. A. Ball, and G. Meltz, “Photoinduced Bragg gratings in optical fibers,” Opt. and Photon. News 5, 8–14 (1994). [CrossRef]

] Typically, fiber gratings are used with incoherent broadband sources or cw and quasi-cw (pulsed) sources whose spectral bandwidth is narrower than that of the grating response. The characteristics of and pulse propagation through gratings in this regime are well understood. [2

2. See, for example, J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994). [CrossRef]

], [3

3. D. Taverner, D. J. Richardson, J.-L. Archambault, L. Reekie, P. St. J. Russell, and D. N. Payne, “Experimental investigation of picosecond pulse reflection from fiber gratings,” Opt. Lett. 20, 282–284 (1995). [CrossRef] [PubMed]

]

Recently, there have been several investigations involving the use of fiber gratings under situations other than those already listed. [4

4. See, for example, B.J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996). [CrossRef] [PubMed]

] – [8

8. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse propagation through fiber gratings: theory and experiment,” presented at Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals Topical Meeting 1997, Williamsburg, Virginia, paper BMB2.

] In particular, our research has focused on the linear propagation of ultrashort pulses through fiber gratings (i.e. the ultrashort pulse response of fiber gratings). Here the spectral bandwidth of the incident pulse is considerably broader than that of the grating response. Furthermore, the physical length of the pulse is significantly shorter than the grating length. We have shown, both theoretically and experimentally, that under such excitation, the temporal reflection and transmission responses of fiber gratings are complex and would not be expected based on the cw and quasi-cw responses. [7

7. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” IEEE/OSA J. Lightwave Technol. 15, 1503–1512 (1997). [CrossRef]

], [8

8. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse propagation through fiber gratings: theory and experiment,” presented at Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals Topical Meeting 1997, Williamsburg, Virginia, paper BMB2.

] We have also qualitatively explained the features of the ultrashort pulse response, correlating them with various grating characteristics, in order to gain physical insight into the corresponding interaction. [7

7. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” IEEE/OSA J. Lightwave Technol. 15, 1503–1512 (1997). [CrossRef]

]

Our prior numerical simulations were based on a linear systems approach: the temporal reflection or transmission responses were obtained by taking the inverse Fourier Transform of the product between the appropriate grating frequency response and input pulse spectrum. With this approach, only the resultant reflection and transmission responses (signals) were available. In this paper, we use an alternative approach based on the time-domain coupled-mode equations (TDCMEs) to simulate the interaction between ultrashort pulses and fiber gratings. The TDCMEs have previously been used to study the switching dynamics of finite periodic nonlinear media. [9

9. C. M. de Sterke, K. R. Jackson, and B. D. Robert, Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am B 8, 403–412 (1991). [CrossRef]

] Here, we extend this approach to our regime of interest. Specifically, directly integrating the TDCMEs allows us to explicitly obtain, at any desired time, the backward and forward propagating waves as a function of position within different grating structures. As the input pulse is stepped through a grating, the backward and forward propagating waves are computed, plotted, and sequentially stored in order to generate a “movie” that illustrates the dynamics of the ultrashort pulse-grating interaction. These “movies” confirm our earlier calculations and qualitative explanations.

2. Time-domain coupled-mode equations

The refractive index in the fiber core is modeled as [2

2. See, for example, J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994). [CrossRef]

,7

7. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” IEEE/OSA J. Lightwave Technol. 15, 1503–1512 (1997). [CrossRef]

,10

10. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E , 54, 2963–2975 (1996). [CrossRef]

]

n(z)=n0+σ(z)+2κ(z)cos[2k0z+ϕ(z)]
(1)

where k0 is the wavenumber at the design resonant wavelength (with corresponding resonance frequency ω0 = (ck0)/n0 and c is the speed of light in vacuum), n0 is the effective mode index of the unmodified core, and σ, κ, ϕ are slowly varying functions of z. The function σ characterizes the space averaged background refractive index increase due to the envelope of the induced index change (DC background envelope), κ is the position dependent coupling coefficient, and ϕ describes the position dependent phase (chirp) of the grating. Writing the electric field within the grating structure as a sum of forward and backward propagating waves

Ezt=a+ztej(ω0t+δtk0z+ϕ(z)2)+aztej(ω0t+δt+k0zϕ(z)2)
(2)

and substituting Eq. (2) in the scalar wave equation, we find that the envelope functions a ± (z, t) satisfy the following time-domain coupled-mode equations [10

10. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E , 54, 2963–2975 (1996). [CrossRef]

]:

j(a+ζtζ+n0ck0a+ζtζ)+(σ(ζ)2n02+n0δck012ϕ(ζ))a+(ζ,t)+κ(ζ)2n02aζt=0
j(aζtζn0ck0aζtt)+(σ(ζ)2n02+n0δck012ϕ(ζ))aζt+κ(ζ)2n02a+ζt=0
(3)

subject to the boundary conditions a +(0,t) = A(t) and a -(L,t) = 0 and initial conditions a ±(ζ,0) = 0 where A(t) is the envelope of the input pulse, L is the length of the grating, and ζ = k0z (further details regarding the derivation or the significance of the terms in Eq. (3) can be found in [7

7. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” IEEE/OSA J. Lightwave Technol. 15, 1503–1512 (1997). [CrossRef]

], [10

10. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E , 54, 2963–2975 (1996). [CrossRef]

]). The above equations are solved using a modified predictor-corrector method that is both stable and fourth order. [9

9. C. M. de Sterke, K. R. Jackson, and B. D. Robert, Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am B 8, 403–412 (1991). [CrossRef]

] This particular method can easily be implemented on any personal computer and an average run takes several tens of minutes (on a 100 MHz Pentium processor). Although this process is computationally more involved than those previously used (see, for example [7

7. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” IEEE/OSA J. Lightwave Technol. 15, 1503–1512 (1997). [CrossRef]

]), it provides additional insight into the dynamics of the ultrashort pulse-grating interaction as shown below.

3. Simulations and discussion

In this section, we present simulations of ultrashort pulse propagation through three different gratings whose parameters are listed below in Table 1. A transform-limited 1-ps Gaussian pulse at 1.55 μm with a peak intensity of 100 (arbitrary units) is assumed as the ultrashort input. Notice that (i) the corresponding FWHM bandwidth of the input pulse is ≈ 3.5 nm which is considerably broader than those of the gratings considered and (ii) the physical length of the pulse is ≈ 0.2 mm which is considerably shorter than the grating lengths. In view of this latter remark, the input pulse will interact, at any given time, only with a fraction of the grating over which it overlaps. This is important in understanding the dynamics of the ultrashort pulse response for these gratings.

Table 1. Characteristics of gratings used in the numerical simulations.

table-icon
View This Table

In each of Figs. 1 – 3, a single frame from the movies illustrating the propagation of the 1-ps Gaussian pulse through the three different gratings is displayed. Once all of the interactions have taken place, the resultant backward and forward propagating waves correspond precisely to those calculated using the linear systems approach (see, for example, [7

7. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” IEEE/OSA J. Lightwave Technol. 15, 1503–1512 (1997). [CrossRef]

]). Thus, the movies confirm our qualitative explanations for the features of the ultrashort pulse-grating interaction. They also provide new insight into the dynamics of the responses, not previously available by simply considering the resultant signals. In the following subsections, we qualitatively discuss the ultrashort pulse response of the three gratings.

3.1 Weak uniform grating

The reflected signal from the weak uniform grating consists of two components: a main reflection peak whose duration equals the round-trip propagation time through the grating immediately followed by a transient subpulse. The movie (see Fig. 1) clearly illustrates the dynamics of the interaction and the origin of these two components. The main reflection peak is a coherent sum of the individual reflections generated as the input pulse (or the forward propagating wave) propagates through the grating. Specifically, the continual interaction between the input pulse and the grating results in reflections that contribute to the backward propagating wave which is seen to grow longer. The last reflected component that contributes to the main reflection peak arises when the input pulse leaves the grating (at this point, there are no more contributions to the main reflection peak). This last component must then travel the length of the grating; thus, the total duration of the main reflection peak is precisely the round-trip propagation time through the grating.

Fig. 1. A single frame from the movie illustrating the propagation of a 1-ps pulse through a weak uniform grating. The dotted lines indicate the grating boundaries. The backward propagating wave appears in red and the forward propagating wave in blue. The peak of the forward propagating pulse is off the scale since the scale has been expanded to retain as much detail as possible. To run the movie, click on the above figure. [Media 1]

Also note that it is when the input pulse leaves the grating that the transient subpulse appears and starts to grow. The subpulse was previously explained in terms of an effective-medium picture [7

7. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” IEEE/OSA J. Lightwave Technol. 15, 1503–1512 (1997). [CrossRef]

]: frequencies in the input pulse corresponding to those in the sidelobes of the cw grating response and which propagate in the grating are reflected only once they reach the discontinuity between the end of the grating and the unmodified fiber, giving rise to this transient subpulse. Since these frequencies must then travel back through the grating they will only appear after the round-trip propagation time through the grating. On the other hand, if we examine the forward propagating wave on an expanded scale, we actually see the growth of a trailing component (that lags the main pulse). The interaction of trailing component in the forward propagating wave with the grating creates the transient subpulse. Although at first these two explanations for the appearance of the transient are seemingly different, they are in fact equivalent. The effective medium picture is an alternate means of representing the dispersive nature of the grating (the grating is replaced by an effective medium that contains no physical grating, but has a refractive index that is frequency dependent). [2

2. See, for example, J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994). [CrossRef]

] The dispersive nature of the grating gives rise to the trailing component in the forward propagating wave and hence, the transient subpulse. More concretely, frequencies in the sidelobes of the cw grating response travel slower than those farther detuned from the grating resonance condition and the trailing component in the forward propagating wave that lags the main pulse consists of these frequencies (the main pulse travels faster since it consists of frequencies having larger group velocities, being farther detuned from resonance). Thus, their subsequent interactions with the grating will result in the reflection components that appear after those between the main pulse and the grating. Furthermore, the continual interaction between the input pulse and the grating results in a depletion of energy from the same frequency spectrum of the input (the input pulse effectively sees the same grating segments due to the uniformity of the grating). This depletion of energy results in a gradual decrease in the intensity of the main reflection peak and more importantly, a “hole” in the input pulse spectrum. This hole is equivalent to the cw grating response carved into the input pulse spectrum.

3.2 Weak grating with linear chirp

The above features can be explained as follows. As the input pulse propagates through the grating, each grating fraction that it interacts with is slightly different due to the changing grating period. Frequencies are reflected at various positions in the grating so that energy is not constantly reflected from the same portion of the input pulse spectrum. Thus, the backward propagating wave does not experience as much of a decrease in intensity as in the uniform case. When examining the movie in Fig. 2, we see this effect manifest through the creation of an inflection point where the intensity of backward propagating wave stops to decrease.

We have also examined the resultant reflection response for gratings with the same index modulation but varying in chirp from 0.0 nm/cm (uniform grating) to 0.4 nm/cm. As the chirp parameter increases, the transient subpulse pulse decreases in size, almost disappearing for a chirp of 0.3 nm/cm, while the trailing edge of the backward propagating wave grows. These features are attributed to the dispersive nature of the linearly chirped grating that is considerably different from its uniform counterpart. In the case of a uniform grating, the transient subpulse was due to frequencies in the sidelobes of the cw grating response. For the linearly chirped grating, the corresponding frequencies travel faster, and hence are reflected sooner. Thus, the transient subpulse moves into the main reflection peak and adds to the trailing edge of the backward propagating wave. Also notice that since these frequencies travel faster, there is not as much temporal spreading of the trailing component (i.e. the trailing component has a shorter duration) in the forward propagating wave.

Fig. 2. A single frame from the movie illustrating the propagation of a 1-ps pulse through a weak uniform grating. The dotted lines indicate the grating boundaries. The backward propagating wave appears in red and the forward propagating wave in blue. The peak of the forward propagating pulse is off the scale since the scale has been expanded to retain as much detail as possible. To run the movie, click on the above figure. [Media 2]

3.3 Very strong uniform grating

As in the case for a weak uniform grating, the reflected signal also comprises a main reflection peak and transient subpulses. By examining the spectrum in each of the reflected components, we find that the main reflection peak consists primarily of frequencies in the stop band of the cw grating response while the transients are composed of frequencies in the sidelobes. Since the index modulation is strong and the grating stop band forms a considerable fraction of the input pulse spectrum, a significant amount of energy is reflected within the first few grating periods, resulting in the relatively short duration of the main reflection peak. This is in contrast to the weak grating where only a small amount of energy is reflected by each grating period so that the entire input pulse can propagate well into the grating.

Fig. 3. A single frame from the movie illustrating the propagation of a 1-ps pulse through a very strong uniform grating. The dotted lines indicate the grating boundaries. The backward propagating wave appears in red and the forward propagating wave in blue. The peak of the forward propagating pulse is initially off the scale since the scale has been expanded to retain as much detail as possible. To run the movie, click on the above figure. [Media 3]

An equivalent alternative interpretation of the oscillations in the propagating fields is the formation of a quasi-standing wave within the grating structure. [11

11. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrafast pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996). [CrossRef]

] The frequencies in the forward propagating wave that enter the grating scatter energy into the quasi-standing wave which in turn scatters energy back into the forward and backward propagating waves. Thus, there is a transient storage of energy within the grating culminating in the transient subpulses.

4. Summary

Acknowledgment

This research was supported in part by Photonics Research Ontario and the Natural Sciences and Engineering Research Council (Canada).

Footnotes

J. E. Sipe is with the Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada M5S 1A7.

References and links

1.

See, for example, W. W. Morey, G. A. Ball, and G. Meltz, “Photoinduced Bragg gratings in optical fibers,” Opt. and Photon. News 5, 8–14 (1994). [CrossRef]

2.

See, for example, J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994). [CrossRef]

3.

D. Taverner, D. J. Richardson, J.-L. Archambault, L. Reekie, P. St. J. Russell, and D. N. Payne, “Experimental investigation of picosecond pulse reflection from fiber gratings,” Opt. Lett. 20, 282–284 (1995). [CrossRef] [PubMed]

4.

See, for example, B.J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996). [CrossRef] [PubMed]

5.

K. Rottwitt, M. J. Guy, A. Boskovic, D. U. Noske, J. R. Taylor, and R. Kashyap, “Interaction of uniform phase picosecond pulses with chirped and unchirped photosensitive fiber Bragg gratings,” Electron. Lett. 30, 995–996 (1994). [CrossRef]

6.

L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, and S. Juma, “Ultrashort pulse propagation in multiple-grating fiber structures,” Opt. Lett. 22, 402–404 (1997). [CrossRef] [PubMed]

7.

L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse reflection from fiber gratings: a numerical investigation,” IEEE/OSA J. Lightwave Technol. 15, 1503–1512 (1997). [CrossRef]

8.

L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse propagation through fiber gratings: theory and experiment,” presented at Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals Topical Meeting 1997, Williamsburg, Virginia, paper BMB2.

9.

C. M. de Sterke, K. R. Jackson, and B. D. Robert, Nonlinear coupled-mode equations on a finite interval: a numerical procedure,” J. Opt. Soc. Am B 8, 403–412 (1991). [CrossRef]

10.

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E , 54, 2963–2975 (1996). [CrossRef]

11.

M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, “Ultrafast pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss,” Phys. Rev. E 54, R1078–R1081 (1996). [CrossRef]

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(060.5530) Fiber optics and optical communications : Pulse propagation and temporal solitons
(320.7140) Ultrafast optics : Ultrafast processes in fibers

ToC Category:
Research Papers

History
Original Manuscript: September 9, 1997
Revised Manuscript: August 21, 1997
Published: October 27, 1997

Citation
Lawrence Chen, Seldon Benjamin, Howard Jung, Peter W. E. Smith, and John Sipe, "Dynamics of Ultrashort Pulse Propagation Through Fiber Gratings," Opt. Express 1, 242-249 (1997)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-1-9-242


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References

  1. See, for example, W. W. Morey, G. A. Ball, and G. Meltz, "Photoinduced Bragg gratings in optical fibers," Opt. and Photon. News 5, 8-14 (1994). [CrossRef]
  2. See, for example, J. E. Sipe, L. Poladian, and C. M. de Sterke, "Propagation through nonuniform grating structures," J. Opt. Soc. Am. A 11, 1307-1320 (1994). [CrossRef]
  3. D. Taverner, D. J. Richardson, J.-L. Archambault, L. Reekie, P. St. J. Russell, and D. N. Payne, "Experimental investigation of picosecond pulse reflection from fiber gratings," Opt. Lett. 20, 282-284 (1995). [CrossRef] [PubMed]
  4. See, for example, B.J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, "Bragg grating solitons," Phys. Rev. Lett. 76, 1627-1630 (1996). [CrossRef] [PubMed]
  5. K. Rottwitt, M. J. Guy, A. Boskovic, D. U. Noske, J. R. Taylor, and R. Kashyap, "Interaction of uniform phase picosecond pulses with chirped and unchirped photosensitive fiber Bragg gratings," Electron. Lett. 30, 995-996 (1994). [CrossRef]
  6. L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, and S. Juma, "Ultrashort pulse propagation in multiple-grating fiber structures," Opt. Lett. 22, 402-404 (1997). [CrossRef] [PubMed]
  7. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, "Ultrashort pulse reflection from fiber gratings: a numerical investigation," IEEE/OSA J. Lightwave Technol. 15, 1503-1512 (1997). [CrossRef]
  8. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, "Ultrashort pulse propagation through fiber gratings: theory and experiment," presented at Bragg Gratings, Photosensitivity, and Poling in Glass Fibers and Waveguides: Applications and Fundamentals Topical Meeting 1997, Williamsburg, Virginia, paper BMB2.
  9. C. M. de Sterke, K. R. Jackson, and B. D. Robert, Nonlinear coupled-mode equations on a finite interval: a numerical procedure," J. Opt. Soc. Am B 8, 403-412 (1991). [CrossRef]
  10. L. Poladian, "Resonance mode expansions and exact solutions for nonuniform gratings," Phys. Rev. E, 54, 2963-2975 (1996). [CrossRef]
  11. M. Scalora, R. J. Flynn, S. B. Reinhardt, R. L. Fork, M. J. Bloemer, M. D. Tocci, C. M Bowden, H. S. Ledbetter, J. M. Bendickson, J. P. Dowling, and R. P. Leavitt, "Ultrafast pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss," Phys. Rev. E 54, R1078-R1081 (1996). [CrossRef]

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