## Abstract

A highly dispersive waveguide structure is proposed to efficiently compress and expand ultra short pulses in a package forming a sufficiently small footprint. A sub-wavelength grating is fashioned into a ridge waveguide to take advantage of multiple dispersive effects and spread the mode over a significantly larger area than a standard single-mode waveguide. The structure is designed to take advantage of the amplified dispersion near cutoff. Modal analysis is performed on two variations of the structure using a finite element solver package. The predicted dispersion is sufficient to double the width of a 1 ps pulse within the width of a standard 5 inch (127 mm) wafer. A theoretical analysis of the grating component composing the structure confirms that the dispersion values are fully reasonable.

©2004 Optical Society of America

## 1. Introduction

Telecommunications, imaging systems, military uses for high-power lasers, and numerous other optical applications continue to grow the need for methods of controlling ultra-short pulses. In particular, the ability to use dispersive media and structures to broaden the pulse allows one great control of individual frequency components and can provide a means to amplify specific frequency components individually. With the continuing development of high-energy broad-spectrum lasers, such fine control over frequency components, coupled with the ability to expand and compress pulses significantly aid in the ability to shape pulses for myriad applications.

The difficulty with this process comes in the method used to broaden or compress the pulse. Conventional techniques make use of dispersive gratings or prisms. Frequency components are spread apart as the pulse winds through a complex optical system consisting of multiple components that separate and recombine the pulse and allow differing frequencies to travel slightly different paths. While these methods are relatively simple, the gratings must be of an exceptional quality, and the space required to achieve sufficient broadening or compression is impractical for most applications.

There are also a number of fiber and waveguide-based methods. One example makes use of chirped Bragg gratings in optical fibers [1]. At the Bragg frequency, the grating has very high reflectance. Chirped gratings cause different frequency components of the pulse to be reflected from different points in the fiber, thus spreading the pulse. Other methods incorporate Bragg gratings used in transmission [2–3] or self phase modulation in a QPM structure [4,5].

Although numerous methods exist, most have such small dispersion as to require a significant length of fiber or space in a laboratory. Further, the loss for some of these structures is significant enough to raise fears regarding damage due to the dissipation of the lost energy into the compressing structure, particularly for the high intensities present in typical short pulses. Some of the less lossy methods incorporate difficult and complex alignments that would ideally be eliminated. A highly dispersive waveguide-based device with few components is more stable and better suited to fiber-based and communication-oriented applications among others.

## 2. Dispersion and pulse broadening

The principles governing dispersion and pulse broadening are covered in great detail in most basic optics textbooks, however, it is illustrative to highlight a few basic details. The propagation constant of light is conventionally written as:

where n_{eff} is either the refractive index of the medium or the effective index of the guiding structure. If the pulse of light is assumed to have a Gaussian shape of waist, τ_{0}, after propagating a finite distance, z, the new beam waist has a dependence on the second derivative of the propagation constant with respect to frequency. This second derivative is a function of the material or structure itself, and not of the beam of light. Hence, it is defined as the dispersion of the medium:

The broadened beam waist is defined by:

where:

Note that there is also a frequency chirp caused by the dispersive effects, which is expressed by:

If the pulse incident on the dispersive structure is already chirped with a sign opposite of the frequency derivative of the propagation constant for the structure the pulse will experience compression instead of broadening.

To increase the rate at which a pulse is spread or compressed, the dispersion must be increased. For example, consider a 1 ps Gaussian pulse at 1.55 microns. Expanding the pulse to 2 ps within the confines of a 4-inch wafer requires a structure with dispersion on the order of 6800 ps/nm×km. A typical optical fiber has a dispersion of around 10–20 ps/nm×km near this wavelength.

In bulk media, dispersion is caused by the dependence of material index on wavelength. In waveguide structures dispersion is also caused by the two additional effects: The difference in propagation constants for different modes, and the dependence of a single mode’s propagation constant on frequency. We will explore dispersion in single-moded structures, which will allow us to ignore modal dispersion. Material dispersion is generally quite small in comparison with the dispersive values we are hoping for. Also, it is only dependent on the material used, so this type of dispersion cannot be adjusted through modifications to the waveguide shape. Hence we will restrict our considerations to waveguide dispersion and explore the ways different frequencies interact with the waveguide shape.

## 3. Dispersive waveguides

We turn now to the realm of waveguide structures. Ridge waveguides operating near cutoff tend to have a rather significant amount of dispersion. Figure 1 shows the dispersion for a 400 nm×200 nm silicon ridge waveguide on a silicon dioxide substrate as determined by the effective index method. The structure must be this small in order to maintain a single-mode profile. The dispersive magnitude is relatively high, and it can be increased by a factor of 5 or more by shrinking the ridge down to 320 nm×160 nm since it pushes the operation very close to modal cutoff. However, such a structure has some definite drawbacks. With such a minute waveguide size, coupling light into the structure becomes significantly more problematic. Although this can be done, there is an additional issue. The high index contrast results in a highly confined mode in a very small area. Since the purpose of the dispersive structures is to allow us to expand and compress ultra-short pulses, we can expect to interact with pulses of a modestly high intensity. Confining the pulse to such a tiny area virtually guarantees that nonlinear effects will play a significant role.

While the coupling and nonlinear issues can be accounted for, it would be advantageous to have a much more reasonably sized waveguide. SiON has a significantly lower index than Si and could provide one alternative. However, we present another option. Instead of restricting ourselves to simple ridge structures, we consider making use of a grating-based structure in the shape of a ridge. The high dispersion of a ridge-like waveguide can be taken advantage of, but the grating nature lowers the effective index of the ridge and forces the mode to spread over a much larger area. If the period is too large, the structure will act as a diffraction grating and spread the light in multiple orders. However, for an appropriate choice of structure size and period, only the fundamental diffraction order will propagate and the structure forms a single-mode waveguide. Figure 2 demonstrates a structure with a 275 nm period, 150 nm fins, and 1.25 micron depth. 6 periods are used in this structure.

Comsol’s FEMLab software implements a finite element solver which is used to evaluate the waveguide propagation constants as a function of wavelength for the two different types of modes, TE and TM. The TE-like modes are an expression of the wave equation in terms of H_{z} and are based on the assumption that there is no component of the E-field along the z-axis. The TM-like modes are solutions of the formulation of the wave equation in terms of E_{z} with the assumption that the H-field is entirely transverse to the z-axis. The formulation for the TE-like case is:

and for the TM-like case is:

Note that this definition for the TE and TM modes is not specifically identical to the more traditional definition in the waveguide community where the TE mode is defined in terms of E-fields parallel to the grating vector (in the horizontal direction in all the waveguide profile diagrams) and the TM mode is based on H-fields parallel to the grating vector (with the E-fields primarily lying in the vertical direction in the waveguide profile diagrams). However, for modes relatively far from cutoff the two definitions are nearly equivalent. For the remainder of this paper, “TE”, “TM”, “TE-like”, and “TM-like” will refer to the FEMLab definition of the modes, while “true TE” and “true TM” will refer to the traditional definitions used in the waveguide community as defined above.

The mode equations are expressed as eigensystems and are solved for the eigenmode propagation constant, β. The results were additionally verified using an independent solver based on the finite difference method of [6]. The finite difference solver was based on a fixed rectangular grid and thus had a significantly lower accuracy than the FEMLab results, but the results agreed to within a few percent, which gives us confidence in the FEMLab simulations. Additionally, the simulations were tested for convergence based on grid sampling to further ensure the accuracy of the simulations. Dispersion is calculated using the difference approximation to the second derivative of the propagation constants. Although this assumes negligible material dispersion, we know that material dispersion provides nowhere near the dispersive magnitude we require, and hence we seek a structure with very large waveguide dispersion.

The electric field distributions for the nano dispersion amplified waveguide (nano-DAWG) operating at 1.55 microns are shown in Fig. 3. It is interesting to note that the majority of the energy for the TE case is actually in the substrate instead of the fins, while the converse is true of the TM mode. In some ways this seems counterintuitive, but in reality it is consistent with what appears to be occurring. In the TE case the short period of the fins results in an effective medium above the substrate layer that has too low of an effective refractive index to support a mode by itself, but it is sufficient to allow a mode to propagate just beneath it in a similar manner to modes formed in a silicon layer just beneath a small silicon ridge. In the TE case, the energy is obviously much more tightly confined, but it is still spread over a significantly larger area than the single moded ridge waveguide mentioned earlier. Additionally, one should note that shorter wavelengths have propagation constants further from cutoff, and thus they become more centralized in the fins instead of the substrate in similar manner to the energy distribution for the presented TM case. This means that the structure could potentially be used in an amplifier type configuration where the pump beam is used to excite a signal in a neighboring medium.

After solving for the propagation constants, and hence the effective refractive indices, at a range of incident wavelengths, the software evaluates dispersion through a difference approximation to the second derivative. The TE mode has remarkably high calculated dispersion, while the TM case, although having a relatively high dispersion, is not single moded. The dispersion for the principle modes in both cases is shown in Fig. 4.

Of additional interest is the extreme birefringence this structure exhibits. The refractive index for the principle TE mode at a wavelength of 1.55 microns is 1.4663, while the index for the TM mode at that wavelength is 2.7078! The equation for the birefringent beat length,

gives a beat length magnitude of 1.2485 microns. Typical polarization-maintaining fibers (PMF) have beat lengths on the order of 2–3 mm.

The silicon-air interface introduces an index contrast that is quite large, and hence the mode is highly confined. Using SiON (index of 2.0) for the fins instead of Silicon (index of 3.47) allows the energy to spread out across the interfaces to a much greater degree and increases the dispersion further. Further, the lower guiding index means that the fins must be significantly larger to maintain a single-moded guiding structure, which makes fabrication a degree easier. The TE case is single-moded for the SiON nano-DAWG with a period of 825 nm. The duty cycle is maintained between the two cases (which gives 450 nm width fins), as is the grating height (1.25 microns). The electric field distributions are displayed in Fig. 5.

Once again, dispersion for the TE and TM modes is calculated from the propagation constants. Figure 6 shows a plot of dispersion for each case. Interestingly, the dispersion is double that of the first nano-DAWG design. Also, the calculated beat length for this design at a wavelength of 1.55 microns is 6.556 microns.

As can be seen, the nano-DAWG dispersion is nearly a full order of magnitude greater than that of the ridge waveguide structures, and 3 orders of magnitude over optical fibers! We now consider numerically the effect of the structure. Assume that a nano-DAWG of the presented specifications is fabricated along the length of a standard 4-inch (101.6 mm) wafer. Incident light is in the form of a 1 ps Gaussian pulse at 1.55 microns central wavelength. From Eqs. (3) and (4) we can predict that the pulse will be broadened to 1.71 ps upon leaving the structure. If the nano-DAWG is fabricated along the length of a full 5-inch (127 mm) wafer, the 1 ps pulse will be broadened to a full 2 ps.

## 4. Theoretical analysis

There are several interesting characteristics in these structures. First, the grating provides an additional dispersion effect. The effective index for the true TE mode in a grating structure includes a higher order dependency on the ratio of wavelength to grating period [7]:

where

$${\epsilon}_{P0}={\left(\frac{f}{{n}_{g}^{2}}+\frac{\left(1-f\right)}{{n}_{c}^{2}}\right)}^{-1}$$

f is the grating fill factor, Λ is the grating period, n_{g} is the refractive index of the grating, and nc is the refractive index of the surrounding region. While this expression indicates the presence of a higher order grating dispersion, it is not entirely accurate in this case due to the high contrast materials and the extremely small size of the grating period. As such, to fully analyze these structures, many more terms from the permittivity expansion need to be incorporated. We turned to a rigorous coupled wave (RCW) algorithm [8] to evaluate an effective index for the nano-DAWG structures. The effective index and corresponding dispersion for an infinite grating using the design parameters for the first structure is shown in Fig. 7.

Although the dispersion is of a similar magnitude to the nano-DAWG dispersion, its slope is in the opposite direction. However, the transverse electric field vectors in Figs. 3(a) and 5(a) suggest that much of the field for the TE modes is not actually contained in the ridge. Further, the slope of the dispersion in Fig. 1 is of the same sign as what we see from the model of the nano-DAWGs. This suggests that the structures are actually acting much more like ridge waveguides very close to cutoff, which we would expect to have high dispersion. For comparison, a model was made of a ridge waveguide of the same dimensions as the first nano-DAWG. The cladding and substrate region were kept the same as in the fin structure, but the grating region was replaced by a solid rectangular guiding region with an index of 1.567, the effective index of the grating at a wavelength of 1.535 as predicted by the RCW.

The dispersion of this structure gives the limit for the nano-DAWG structure where the grating effect goes to zero. The dispersion is the dotted line in Fig. 8.

The shape of the curve is very similar to the curves for the nano-DAWGs, though the magnitude is somewhat smaller. To analyze this further, the constant refractive index of the ridge structure was replaced by the index values predicted from the RCW analysis at each wavelength analyzed. The propagation constants were then found and analyzed at each wavelength to determine dispersion. The modal analysis predicts that there are no propagating modes for wavelengths above 1.55 µm. Below that cutoff point, this representation provides an upper limit for the nano-DAWG dispersion wherein the grating effect reaches the highest magnitude. Dispersion for the propagating mode is plotted as the dashed line in Fig. 8.

The dispersion for the infinite grating limit is much higher than what was predicted for the nano-DAWGs. However, the fact that the structure is non-guiding where the nano-DAWG actually supports a mode, suggests that the RCW analysis is not an entirely accurate indicator of the effective index in the guiding structure, which one might expect since it contains only six grating periods. But the infinite grating limit and constant index limit give an upper and lower bound on the expected dispersion for the actual structure. As the figure demonstrates, the modeled nano-DAWG dispersion falls comfortably between the two limits as we would expect.

Essentially then, there are two effects present in the nano-DAWG structures. The dominant one is the waveguide effect for a ridge-like waveguiding structure with an extremely low index contrast. The grating nature allows us to fabricate a ridge with a substantially lower index contrast than could be obtained with standard materials. The second effect is the grating dispersion. This induces a decrease in the effective index as a function of wavelength, which amplifies the tendency of the waveguide to approach cutoff at higher wavelengths and causes the slope and magnitude of the dispersion to be increased even further.

These combined effects give further insight into the nature of the dispersion plots in Figs. 3(a) and 5(a). The first nano-DAWG design actually has an extrema in the dispersion curve. It seems that below 1.54 µm the structure is far enough from cutoff that the waveguide effect is no longer dominant and the grating effect plays a larger role, giving the curve a negative slope. Also, the second design actually has a significantly larger dispersion despite lower index materials and a significantly larger structure. However, if the configuration approximates a ridge waveguide near cutoff, we might expect this since the energy is spread over a much larger area than in the first design. Hence, in light of a low index contrast ridge-like waveguide approximation, the dispersion curves do make a great deal of sense.

## 5. Conclusion

Nano dispersion amplified waveguides appear to provide remarkably high dispersion. With the appropriate mode incident on the structure, a substantial degree of compression can take place within an area the size of standard 4-inch wafer. The nano-DAWGs are suitably small to be integrated into a reasonably sized device and appear to provide an excellent solution for many pulse shaping applications. Additionally, the structures offer a huge birefringence giving a beat length nearly three orders of magnitude smaller than that of typical PMFs. This opens the door to myriad further applications relating to separation of the different polarization states. It is also worth noting that although the mode is spread over a fairly broad area and that the majority of it is actually present in the substrate for the TE mode. As such, there is a distinct possibility for these devices to provide applications relating to local-field effects and local-field enhancement as well as optical amplifiers that incorporate both the dispersive effects and the unique mode profile.

The modal analysis performed using a finite element solver appears to match the trends that a theoretical analysis would predict. The dispersion values were also checked using two different solver methods to ensure accuracy and convergence. Although the dispersive properties of the structures are remarkably high, they are not significantly higher than a simple ridge structure with an ultra-low index contrast. The grating nature of the guide allows one to fashion ridge-like waveguides operating very close to cutoff, which enhances the dispersive nature of the structure. As such, nano-DAWGs appear to offer an innovative and attractive approach for future pulse compression, expansion, and shaping devices.

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