## Abstract

We demonstrate numerically that acoustooptic interaction between two co-propagating modes in an optical fiber can be utilized to obtain optical delays. Both positive and negative delays of several pulse lengths can be obtained. Based on the simulations we consider relevant experimental parameters.

© 2009 Optical Society of America

## 1. Introduction

There has recently been much interest in studying technologies for optical buffers [1–4], which may find applications within telecommunications, optical processing, and quantum computing.

Several approaches for optical buffers exist. The simplest approach is a free-space delay line, where the delay is due to propagation over a certain length. Another approach is to use a slow light device, where the group velocity of light is significantly reduced, compared to the speed of light in vacuum. Examples of such systems include electromagnetically induced transparency (EIT) [5, 6], coherent population oscillations [7], and coupled resonator optical waveguides [8]. Recently, a delay of several tens of pulse lengths has been demonstrated using a double absorption resonance in cesium [9].

It has also been shown that light can be delayed and advanced in optical fibers [10]. Several exciting results have been obtained using stimulated Brillouin and Raman scattering [11–14]. Light pulses have even been stored for several nanoseconds by converting the data pulses to an acoustic excitation [15].

It is well known that both positive and negative delays can be obtained in fiber Bragg gratings [16]. This has recently been demonstrated experimentally for linear and nonlinear gratings [17–19].

We here show numerically that pulse delays and advancements can be obtained in optical fibers using acoustooptic coupling between two co-propagating optical modes. Coupling between the two optical modes can be achieved using flexural or torsional acoustic waves [20–22]. It is shown that pulse advancements and delays can be several pulse lengths. This result is not in conflict with relativistic causality, but is a consequence of the different group velocities of the two co-propagating optical modes.

The paper is structured as follows: In Sec. 2, we review the application of coupled-mode theory to acoustooptic coupling between two co-propagating modes for a uniform grating. It is also pointed out that the group delay of the grating can be either positive or negative, depending on which of the two coupled modes has the lowest group velocity. In Sec. 3, we review the propagation of a Gaussian pulse through a uniform acoustooptic interaction region. The application of the Kramers-Kronig relations to the acoustooptic transmission coefficient is also discussed. The main results of the paper are contained in Sec. 4. It concerns propagation of a Gaussian pulse through two acoustooptic interaction regions, separated by a section of unperturbed fiber. It is pointed out that delays and advancements of several pulse lengths can be obtained. We also give a numerical example with relevant experimental parameters, and discuss practical limitations on the maximum achievable delay.

## 2. Theory

A schematic overview of a setup for acoustooptic coupling is shown in Fig. 1. The acoustic wave can be excited using a piezoelectric transducer and an acoustic horn [20]. It propagates in the stripped fiber section, and is absorbed in the protective plastic jacket of the fiber. Thus, the stripped section defines the acoustooptic interaction region. Several types of acoustic waves can be excited, such as the lowest order flexural mode or the lowest order torsional mode. Expressions for the mode pattern and dispersion relation for the various acoustic modes are given in Ref. [21]. The frequency and amplitude of the acoustic wave can be dynamically adjusted by varying the voltage applied to the piezoelectric transducer.

The acoustic wave acts as a travelling long-period grating. Coupling between two optical modes due to the acoustic wave can be analyzed with coupled-mode theory [16,23]. The spatial and temporal evolution of a mode, denoted mode *l*, can be written

where Ψ_{l}(**r**
_{⊥}) is the transverse mode profile, β_{l} is the propagation constant, *ω*
_{l} is the frequency, and *a _{l}*(

*z*) is assumed to be a slowly varying amplitude due to acoustooptic coupling.

The group velocity of mode *l* is given by

In general, the group velocity is different for different modes.

The amplitudes *a _{l}* evolve according to the coupled-mode equations [23],

where

is the phase-mismatch, and *K* is the propagation constant of the acoustic wave. Mode 1 and mode 2 could for example be the fundamental (LP_{01}) and the second order (LP_{11}) mode when using a flexural acoustic wave, or the *x*- and *y*-polarization of the fundamental mode when using a torsional acoustic wave. κ is the acoustooptic coupling coefficient.

The simplest grating structure is a uniform grating. By solving Eqs. (3)–(4) for a uniform acoustooptic interaction region of length *L*[23], the amplitude *a*
_{1} at the end of the interaction region is

where we have assumed that all light is contained in mode 1 at *z* = 0. The quantity γ is given by

Assuming perfect phase-match (∆*β* = 0) is obtained at the frequency *ω*
_{0}, we can Taylor-expand ∆*β* to first order around *ω*
_{0}:

where

is the difference in group delay between mode 2 and mode 1 along the acoustooptic interaction region. We note that ∆*t _{g}* can be positive or negative, depending on which of the two coupled modes has highest group velocity. One can simply switch the labelling of the modes to obtain the desired sign of ∆

_{tg}.

It is useful to introduce the normalized coupling coefficient

and the normalized frequency offset

The transmission coefficient *T* = |*T*|exp(*i*ϕ) can now be written as

Experimentally, the amplitude of the acoustic wave can be varied to obtain the desired coupling coefficient *α*. Perfect coupling at *x* = 0 corresponds to *α* = 1 and perfect over-coupling at *x* = 0 corresponds to *α* = 2. We observe from Eq. (12) that the the acoustooptic coupling bandwidth ∆*x* is approximately given by

for *α* = 1 and *α* = 2.

The group delay of mode 1 due to the acoustooptic interaction region is given by

where τ_{d} > 0 corresponds to a delay and τ_{d} < 0 corresponds to an advance, compared to propagation of mode 1 in a section of length *L* in an unperturbed fiber. Thus the sign of the delay can be changed by changing the sign of ∆_{tg}.

## 3. Simulation of a single acoustooptic interaction region

Having reviewed the transmission coefficient for a uniform grating, we will as a first example consider the propagation of a Gaussian pulse through such an acoustooptic interaction region. Light contained in mode 2 is here assumed to be removed after the acoustooptic interaction region using a mode stripper [20].

#### 3.1. Almost full coupling

The case with full coupling, *α* = 1, is not so interesting, since then the transmission is 0 for *x* = 0. Instead, we consider the case with somewhat less than full coupling, *α* = 0.8. Figure 2 shows |*T*|^{2}, *ϕ*, and τ_{d} for this case. It is clear from the figure that for positive ∆*t _{g}*, the situation is similar to propagation close to a resonance in a passive two-level medium [24, 25], except that here the advance is due to coupling of energy between two modes [26,27].

We then simulate transmission of a Gaussian pulse through the acoustooptic interaction. The input pulse is assumed to have a duration τ_{p}, and a carrier frequency *ω*
_{0}. The output after propagation through the acoustooptic interaction is calculated from the transmission coefficient according to Eq. (30) in the Appendix. Figure 3 shows the result. The output pulse is damped, as expected from Fig. 2(a), and it is delayed or advanced compared to the input, depending on the sign of ∆*t _{g}*: The pulse is advanced if ∆

*t*is positive, and vice versa. This is in contrast to a pulse centred at a Lorentzian absorption feature in a two-level medium, which is always advanced. We also note that the pulse is somewhat distorted due to spectral reshaping and dispersion. Pulse distortion due to acoustooptic coupling is however not a fundamental limitation, as this effect may be eliminated e.g. by the approach in Sec. 4.

_{g}#### 3.2. Full over-coupling

Figure 4 shows |*T*|^{2}, *ϕ*, and τ_{d} for perfect over-coupling (*α* = 2). We have also simulated propagation of a Gaussian pulse in this case. The results are presented in Fig. 5. We observe that the damping is small, as expected from Fig. 4(a). We also note that the pulse is either delayed or advanced, depending on the sign of ∆*t _{g}*: A positive ∆

*t*leads here to a delay, and vice versa. Physically, a positive ∆

_{g}*t*means that the group velocity of mode 2 is lower than the group velocity of mode 1. Over-coupling means that the pulse spends some time in the slow mode 2, leading to a delay compared to mode 1. The same reasoning applies for a delay when ∆

_{g}*t*is negative.

_{g}#### 3.3. Fundamental limitations on maximum delay/advance

There are several physical constraints for slow and fast light devices [2,3,28,29]. One important fundamental constraint for media where the group velocity is controlled by changing the linear refractive index, is the Kramers-Kronig relations [30–33]. For example, one can show using these relations that for passive media with negligible absorption in a given frequency range of interest, the group velocity must be less than the vacuum velocity of light [30]. However, not all slow light devices satisfy the Kramers-Kronig relations. An example is non-minimum phase filters, such as optical all-pass filters, where there is no unique relationship between the transmission phase and magnitude [1]. This allows for more freedom in design, compared to devices that are restricted by the Kramers-Kronig relations.

It is interesting to investigate whether Kramers–Kronig type relations apply for the acous-tooptic transmission coefficient *T*, that is, whether ln |*T*| and *ϕ* form a Hilbert-transform pair. To establish such relations between the amplitude and phase response, *T* must be zero-free in one half-plane, either Im *ω* > 0 or Im *ω* < 0. Systems with such responses are usually called minimum or maximum phase [34]. For sufficiently weak coupling, it can be shown that the transmission is a minimum or maximum phase response. However, for strong coupling, it is easy to verify that there will be zeros in both half-planes. In the present case, for *α* = 0.8, *T* is a minimum or maximum phase response, depending on the sign of ∆*t _{g}* (Fig. 2), while for

*α*= 2 it is not (Fig. 4).

## 4. Simulation of two acoustooptic interaction regions

We here show how to obtain large delays or advancements using two acoustooptic interaction regions, separated by a length ∆*z*
_{2} of unperturbed fiber. The idea is to couple light from mode 1 to mode 2 in the first acoustooptic interaction region. The unperturbed fiber section is used to obtain a large delay/advancement caused by the difference in group velocity for mode 2 compared to mode 1. The light is then coupled back to mode 1 in the second acoustooptic interaction region. By turning the acoustic waves on/off, one can switch on/off the delay/advancement. One obtains a delay/advancement if the group velocity of mode 2 is lower/higher than the group velocity of mode 1. Note that causality is not violated, even though one may obtain an advancement of several pulse lengths. The advancement is simply due to the higher group velocity of mode 2, compared to mode 1.

A related concept is the conversion-dispersion delay scheme [35], where the centre wavelength of the input pulses is shifted and the pulses are subsequently injected into a medium with large dispersion, resulting in a delay depending on the wavelength shift. A second wavelength converter then returns the centre wavelength to its initial value.

It can also be noted that a similar setup as the one presented in this section, i.e. two long-period gratings separated by a section of unperturbed fiber, has previously been used as a dispersion-compensating module [36].

To analyze the setup mathematically, we use the transfer-matrix approach [16], where the fiber is divided into three sections, and the field after the last section is determined from the field before the first section. Let ∆*β*
_{n}, κ_{n}, and ∆*z _{n}* =

*z*

_{n+1}−

*z*be the phase-mismatch, coupling coefficient, and length, respectively, of section

_{n}*n*. The transfer matrix for section

*n*is then given by

where

Here

where

In the present case, we have

where the normalized coupling coefficient in each section is given by *α*
_{1} = *α*
_{3} = 1, and *α*
_{2} = 0, giving **T**
_{2} = **I**. The normalized phase-mismatch is ∆*β*
_{1}∆*z*
_{1} = ∆*β*
_{3}∆*z*
_{3} = −π*x* and ∆*β*
_{2}∆*z*
_{2} = −*πx∆z*
_{2}/∆*z*
_{1}. Performing the matrix multiplication we find that the transmission coefficient for the three sections is given by

$$-\genfrac{}{}{0.1ex}{}{1}{1+{x}^{2}}{\mathrm{sin}}^{2}(\genfrac{}{}{0.1ex}{}{\pi}{2}\sqrt{1+{x}^{2}})\phantom{\rule[-0ex]{.2em}{0ex}}\mathrm{exp}\left[\mathrm{i\pi x}\left(1+\Delta {z}_{2}/\Delta {z}_{1}\right)\right].$$

In the limit when ∆*z*
_{2} ≫ ∆*z*
_{1} and *x* ≪ 1, we observe from Eq. (23) that the transmission coefficient approaches

The corresponding group delay can be found by inserting Eq. (24) into Eq. (14), giving

The physical interpretation of Eq. (25) is that when ∆*z*
_{2} ≫ ∆*z*
_{1} and *x* ≪ 1, the delay/advance is due to the difference in group velocity between mode 1 and mode 2 in the unperturbed fiber section. It is clear from Eq. (25) that the delay/advance can be made arbitrarily large by choosing a correspondingly large value of ∆*z*
_{2}/∆*z*
_{1}. However, there are practical limitations on the maximum length ∆*z*
_{2} due to dispersion and loss in the unperturbed fiber section, as discussed in Sec. 4.1.

Figures 6–8 show simulated output for a Gaussian pulse propagating through the three sections. The simulations are performed for different values of *τ _{p}*/∆

*t*and ∆

_{g}*z*

_{2}/∆

*z*

_{1}. The light contained in mode 2 is assumed to be removed after the second acoustooptic interaction region.

We make several observations from Fig. 6. Firstly, we observe that the main part of the output pulse is delayed/advanced approximately 6∆*t _{g}*. This is somewhat more than predicted from Eq. (25), and is due to the additional delay/advance in the two acoustooptic coupling sections, as compared to the contribution to the delay/advance from the unperturbed fiber section alone. Secondly, we observe a small satellite pulse experiencing approximately the same delay as in the absence of acoustooptic coupling. The explanation for this is that a part of the input pulse is not coupled in the first and third fiber section, and propagates with the group velocity of mode 1 through the unperturbed fiber section. Finally, we observe that the energy of the output is less than the energy of the input. This is due to the fact that |

*T*|

^{2}is not unity over the entire input pulse spectrum, meaning that some of the light is contained in mode 2 after the three fiber sections.

It is clear from Fig. 7 that pulse distortion due to spectral reshaping can be avoided by choosing a pulse duration *τ _{p}* ≫ |∆

*t*|, such that the pulse spectrum fits into the the central spectral region where the transmission spectrum is close to unity. Furthermore, by choosing ∆

_{g}*z*

_{2}≫ ∆

*z*

_{1}, we can obtain a large delay/advancement, as shown in Fig. 8.

#### 4.1. Numerical example with relevant experimental parameters

We here estimate the performance of an optical buffer based on acoustooptic coupling between two co-directional modes. We use the physical parameters from Lee et al. [22], who demonstrated a tunable polarization filter using a torsional acoustic wave. The torsional wave coupled light between the two polarizations of the LP_{01} mode of a birefringent fiber. They obtained a FWHM bandwidth of 4.8 nm at 1.55 μm for a 60 cm long interaction region.

Inserting the numerical values above, we find that ∆*t _{g}* = 1.3 ps, where we have used that the FWHM bandwidth corresponds to ∆

*x*= 1.60. This gives a pulse duration of 13 ps, assuming

*τ*= 10 ∆

_{p}*t*.

_{g}Loss and dispersion limit the maximum length ∆*z*
_{2} of the unperturbed fiber section. Both loss and dispersion may vary significantly with wavelength, fiber parameters, and for different modes. The loss of the fundamental mode can be as low as 0.2 dB/km at 1.55 *μ*m in a single mode fiber, while the loss of higher order modes is typically higher. As an estimate, we assume that the loss is 0.5 dB/km for both polarizations of the LP_{01} mode in this example. By demanding that the total loss in the unperturbed fiber section must be less than 3 dB (corresponding to 50% loss), the maximum length ∆*z*
_{2} is 6 km. Dispersion becomes important for fiber lengths longer than the dispersion length *L _{D}*. The dispersion coefficient is typically |

*β*

_{2}| ≈ 23 ps

^{2}/km for the fundamental mode at 1.55

*μ*m in a standard step-index fiber. We here assume that the dispersion is |

*β*

_{2}| = 100 ps

^{2}/km for both modes. The resulting dispersion length is then

*L*=

_{D}*τ*

_{p}^{2}/|

*β*

_{2}| = 1 .7 km. Thus, we see that dispersion limits the maximum length ∆

*z*

_{2}for the parameters above.

Taking the length of the unperturbed fiber section to be ∆*z*
_{2} = 1.7 km, we obtain a delay of *τ _{d}* = 3.7 ns, corresponding to a relative delay of 285 pulse lengths for the 13 ps pulse. It can be noted that the sign of the delay can be changed by switching the input polarization in this case.

Finally, we note that the delay/advancement can be switched on/off by turning on/off the acoustic waves in the two interaction regions. The group velocity of torsional acoustic waves in optical fibers is 3764 m/s [21], so for a 60 cm long interaction region the switching time is approximately 159 *μ*s.

#### 4.2. Practical limitations

As noted in Sec. 4.1, both loss and dispersion limit the length of the unperturbed fiber section. It is desirable to use a fiber where the higher order mode is far from cutoff, to obtain a tightly confined higher order mode with low loss. This could make it difficult to remove light contained in the higher order mode after the second acoustooptic interaction region using a mode stripper based on macrobending loss. However, as shown in Fig. 8, the power transmission can be made to approach unity across the pulse spectrum. One could therefore drop the mode stripper after the second acoustooptic interaction region, at the expense of obtaining a residual amount of light in mode 2. Residual light in mode 2 is undesirable as it may introduce crosstalk. Such crosstalk may also be caused by non-perfect coupling in each acoustooptic coupling section. This effect accumulates when cascading a large number of delay stages.

In Ref. [22], approximately 98% of the light was coupled at resonance in a single acous-tooptic interaction region. Assuming 98% coupling efficiency in each acoustooptic interaction region, the effective loss due to non-perfect coupling is about 4% for one delay stage. This limits the maximum number of delay stages to 15–20, neglecting loss and dispersion in the unperturbed fiber section.

## 5. Discussion

It has been shown in Sec. 4 how to obtain a delay/advance of several pulse lengths. An advantage with this approach compared to an optical buffer based on coupling into a fiber loop is that here the coupling happens between two modes in a *single* fiber.

A further advantage is that it could find applications within wavelength-division multiplexing systems, by choosing the channel spacing larger than the acoustooptic coupling bandwidth. Several channels can then be addressed independently by superposing acoustic waves of different frequency.

A limitation of the present approach is that the difference in group index between the two coupled modes is typically very small (~ 10^{-3}), implying that long fiber sections are needed. This is in contrast to slow light systems based on e.g. EIT, where a group index of ~ 10^{7} can be obtained.

Although we consider a long-period grating due to an acoustic wave in this paper, the results apply to long-period gratings in general. However, it is crucial for the long-period grating to be dynamically tunable to switch on/off the delay. This is readily achieved using acoustooptic coupling.

## 6. Conclusion

An optical buffer based on acoustooptic coupling between two co-propagating modes is analyzed numerically. We find that both pulse delays and advances of several pulse lengths can be obtained using two acoustooptic coupling regions separated by a section of unperturbed fiber.

By considering realistic experimental parameters, we find that the proposed setup should be best suited for pulses in the picosecond range, where dispersion and loss in the unperturbed fiber section places a limit on the maximum achievable delay/advance.

## A. Expression for the transmitted field

The amplitude of an arbitrary optical pulse at position *z* = 0 can be written as

where

For a Gaussian pulse envelope modulated by a carrier wave with frequency *ω*
_{0}, we have

where *τ _{p}* is the duration of the pulse.

From Eq. (28) we find that

The transmitted field, relative to propagation in an unperturbed fiber, can then be written as

$$=\genfrac{}{}{0.1ex}{}{\sqrt{\pi}}{2}\genfrac{}{}{0.1ex}{}{{\tau}_{p}}{\Delta {t}_{g}}\mathrm{exp}\left(-i{\omega}_{0}\Delta {t}_{g}\tau \right){\int}_{-\infty}^{\infty}T\left(x\right)\mathrm{exp}\phantom{\rule[-0ex]{.2em}{0ex}}\left[-{\left(\genfrac{}{}{0.1ex}{}{\pi}{2}\genfrac{}{}{0.1ex}{}{{\tau}_{p}}{\Delta {t}_{g}}x\right)}^{2}\right]\mathrm{exp}\left(-\mathrm{i\pi x\tau}\right)dx,$$

where we have introduced the normalized time

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