Patent Description:
Optical isolators have important applications in telecommunications preventing reflected signals on fiber optic cables from producing unwanted signals. Isolators are also important when lasers are used, because reflected light can cause havoc with the operation of the laser itself. Most importantly, optical isolators and circulators are key building blocks for optical networks.

In the last years there has been considerable effort to introduce non-reciprocal components into integrated photonic circuits. Breaking the reciprocity is needed to implement any optical isolator, which allows light to go through in one direction but severely attenuates reflected light propagating in the opposite direction. Non-reciprocity has been introduced in integrated photonic systems via non-linear effects, via time dependent electro-optical modulation, and most commonly via magneto-optical effects. Although all attempts lead to non-reciprocal devices eventually, a practical implementation of an isolator or circulator is still lacking.

For the non-linear devices the generally low non-linear effect and the need for a high power pump source pose a major problem to on-chip integration. The isolators relying on electro-optical effects require a cumbersome space and time dependent modulation scheme and have so far only been shown to work with insufficient suppression ratios. Materials with strong magneto-optical activity prove to be very difficult to introduce into integrated photonic circuits. Either these materials show a very low effect or too high optical losses or they pose other problems like the crystal lattice mismatch between silicon and the magneto-optical material.

With regard to materials with magneto-optical activity, Michael Faraday observed in <NUM> that linearly polarized light propagating through matter parallel to a static magnetic field experiences rotation of the plane of polarization. This rotation of the plane of polarization is called Faraday rotation, and is most commonly achieved by introducing a magneto-optically active material into the system such as an iron garnet. It is however challenging to combine these materials e.g. with integrated silicon waveguides due to the large crystal lattice mismatch between silicon and garnet. Also the losses of these garnets are orders of magnitude higher compared to silicon.

The Faraday rotation of silicon is <NUM>°/cm/T at <NUM> which is roughly two orders of magnitude lower than what can be achieved with iron garnets. Assuming a biasing field of <NUM> T, a <NUM> long waveguide is needed to achieve the <NUM>° Faraday rotation necessary for an optical isolator.

It is an object of the present invention to provide an improved optical waveguide component for providing Faraday rotation to a plane of polarization of linearly polarized light propagating in a silicon waveguide component.

<NPL>), presents an optical waveguide component for providing Faraday rotation to a plane of polarization of linearly polarized light propagating in said waveguide component (TE mode isolator), wherein said waveguide component is made of silicon and having linear waveguide section and a magnetic field generator generating a magnetic field (<FIG>) parallel to the linear waveguide section.

<CIT> presents a device for rotating polarization plane of electromagnetic wave. <CIT> presents a magneto-optic rotator. <CIT> presents coupled waveguides for slow light sensor applications.

<NPL>, aims to show that the process of polarization conversion in the bend strictly depends on the characteristics of the straight waveguide. <NPL>, aims to demonstrate theoretically and experimentally how highly multimodal high index contrast waveguides with micron-scale cores can be bent, on an ultra-broad band of operation, with bending radii below <NUM> and losses for the fundamental mode below <NUM> dB/<NUM>°.

<CIT> presents an integrating elements for optical fiber communication systems based on photonic multi-bandgap quasi-crystals having optimized transfer functions. <CIT> presents a coupled-waveguide electro-optic switch based on polarisation conversion. <CIT> presents optical films and methods of making the same. <CIT> presents an on-chip optical polarization controller.

A polarization dependent isolator, or Faraday isolator <NUM> as shown in <FIG>, is made of three main parts, an input polarizer <NUM> (here polarized vertically), a Faraday rotator <NUM> and an output polarizer <NUM>, also called analyzer, here polarized at <NUM>°. Light traveling in the forward direction becomes polarized vertically by the input polarizer. The Faraday rotator will rotate the polarization by <NUM>°. The analyzer then enables the light to be transmitted through the isolator. Light traveling in the backward direction becomes polarized at <NUM>° by the analyzer. The Faraday rotator will again rotate the polarization by <NUM>°. This means the light is polarized horizontally (the rotation is sensitive to direction of propagation). Since the polarizer is vertically aligned, the light will be extinguished.

A magnetic field B applied to the Faraday rotator in <FIG> causes a rotation in the polarization of the light due to the Faraday effect. The angle of rotation is β and d is the length of the rotator. Specifically for an optical isolator, the values are chosen to give a rotation of <NUM>°.

In <FIG> is shown the evolution of the polarization for a folded Faraday rotator. In part A of <FIG> is shown a standard Faraday rotator as used for an optical isolator. The left part <NUM> depicts a wave travelling in forward direction; the right part <NUM> shows a wave traveling in backward direction. When a wave k is launched into the waveguide <NUM> with a polarization angle of <NUM>°, the polarization will be rotated clockwise to <NUM>° in this example. For the reverse direction in waveguide <NUM> the wave vector k and bias field are opposite. Thus the initial polarization of the wave vector in guide <NUM> is <NUM>°, and the polarization rotation with respect to the wave vector k is counter-clockwise. Thus the polarization is rotated further to an angle of <NUM>°. This is a typical situation for the Faraday isolator, where light reflected in the output port is rotated by <NUM>° in respect to input polarization and thus can be blocked by a polarizer.

In part B, two Faraday rotators <NUM>, <NUM> are connected by a bend <NUM> that preserves the polarization state. The net Faraday rotation is in this case zero. In part C, a similar Faraday rotator assembly is shown as in B, but now the rotators <NUM>, <NUM> are connected with a birefringent bend <NUM>, that accumulates a <NUM>° phase-shift between the vertical and horizontal polarization. The polarization rotation in the two parts adds up to <NUM>° or <NUM>°, depending on the direction of the <NUM>° phase- shift.

Birefringence is the property of optically non-isotropic transparent materials that the refractive index depends on the polarization direction, i.e. the direction of the electric field. For example, it is observed for crystalline quartz, calcite, sapphire and ruby and in nonlinear crystal materials like LiNbO<NUM>, LBO and KTP. Often birefringence results from non-cubic crystal structures. In other cases, originally isotropic materials (e.g. crystals with cubic structure and glasses) can become anisotropic due to the application of mechanical stress, or sometimes by application of a strong electric field; both can break their original symmetry. In optical fibers, birefringence may result from an elliptical shape of the fiber core, from asymmetries of the fiber design or from mechanical stress, e.g. caused by bending. In the specific case of integrated optical waveguides, form-birefringence can be easily achieved by designing suitable waveguide shapes (e.g. a rectangular strip waveguide), so to ensure different effective index for the two orthogonally polarized (quasi TE and quasi TM) eigenmodes. Also in the case of integrated optics, strain and bends can play an important role in mode birefringence, but a key advantage of integrated waveguides is that, once all parameters are known, changing the waveguide shape enable a total control of the birefringence, also in presence of strain and bends.

<FIG> shows for example what happen to Faraday rotation when the waveguide is designed to have zero birefringence both in the straight and bent sections. The waveguides before and after the bend behave analogous to the straight waveguide with the left arm in <FIG> behaving like the forward travelling case in Fig. 1b and the right arm like the backward travelling case. The one difference is that the bend preserves the polarization state in respect to wave vector and does not mirror it. From this follows that the right arm will reverse the Faraday rotation of the left arm such that the net rotation is zero. To have a nonzero Faraday rotation in a bent waveguide, the bend has to mirror the polarization. This can be done by introducing birefringence into the bend as shown in <FIG>. Here, the vertical and horizontal polarization are phase shifted with respect to each other by <NUM>°, which makes the case of <FIG> analogous to <FIG>.

Typical magneto-optically active materials include CeYIG, γ-Fe<NUM>O<NUM>, orthoferrites and CoFe<NUM>O<NUM> nanoparticles, which have Faraday rotation values (°/cm) of -<NUM>, <NUM>, <NUM> and <NUM>, respectively, with a magnetic field of <NUM> Tesla and a wavelength of λ = <NUM>,<NUM>. In contrast, silicon shows a Faraday rotation value of merely <NUM> °/cm, but has very low optical losses and a figure of merit (FoM) which is <NUM> orders of magnitude larger than the best MO materials. FoM is defined as the ratio of Faraday rotation to loss factor.

Despite this much better ratio silicon is usually not considered as a candidate for a magneto-optical material in optical isolator components for optical networks, because in order to achieve a reasonable Faraday rotation, a propagation length of several centimeters is required.

The magneto-optical activity of silicon can be exploited through either of two effects: the magneto-optical phase shift or Faraday rotation. Upon deciding which effect of the two is the more appropriate, it is important to know that silicon is usually the material with the highest refractive index used in integrated photonic systems and therefore serves as waveguide core material. The high index contrast between silicon in the core and the cladding materials such as silica ensures that the optical wave is to a large part concentrated in the silicon. This is advantageous as it ensures compact dimension and high optical field strengths. However, a magneto-optical phase shift needs asymmetry in the electro-magnetic field distribution. Because the field is concentrated in silicon, the field distribution is symmetric and the magneto-optical phase shift is very small.

To the contrary, conventional Faraday rotation does not need such asymmetry and, for this reason, the present invention studies the potential of silicon waveguides as Faraday rotators.

Referring now to <FIG>, showing two possible ways to wrap up a photonic waveguide: (A) in meander form, and (B) in spiral form. To use a silicon waveguide for an optical isolator one needs <NUM>° Faraday rotation. If one assumes for simplicity that field is completely confined in the silicon, it would mean that the waveguide has to be <NUM> long if a <NUM> T magnetic biasing field is applied. Assuming a more realistic <NUM> T our waveguide would be <NUM> long. These and much longer lengths are feasible with silicon waveguides, but the form factor of such long straight waveguide are not be desirable, as they spoil the very concept behind integrated optical circuits. Shortening of the length or footprint of the waveguide, for example by meandering (<FIG>) or spiraling (Fig. 3Bb) is thus necessary.

In <FIG>, the waveguide <NUM> is being wound in a meandering form e.g. on a plane of a component, adjacent linear sections 31a, 31b that are parallel to each other, but constitute counter-propagating waveguide sections, are combined with bent sections <NUM> providing a phase-shift between the parallel sections 31a, 31b.

In <FIG>, the waveguide <NUM> is wound as a double spiral having counter-propagating counter clock-wise and clock-wise branches <NUM> and <NUM> respectively. Each opposite linear section 36a, 36b of the same turn is combined with bent sections 37a, 37b, each providing a phase-shift that compensates for the cancellation of Faraday rotation between counter-propagating waves in adjacent parallel linear sections of the waveguide, in the same manner as in <FIG>.

In <FIG> is shown in more detail a spiral for Faraday rotation, and <FIG> shows an experimental setup for a spiral with birefringent bends. A study on an integrated silicon Faraday rotator operating at <NUM> wavelength is now discussed, with reference to <FIG> and <FIG>.

The Faraday rotation can be understood as a coupling between the Transverse electric TE mode and the Transverse magnetic TM mode of a waveguide. In a TE mode there is no electric field in the direction of propagation, and in TM modes there is no magnetic field in the direction of propagation. To achieve a rotation of <NUM>°, <NUM>% of the power in one mode needs to be coupled over into the other mode. For a birefringent waveguide the maximum power fraction that can be coupled from one state to the other is given by: <MAT> where Δβ is the mismatch of propagation constants between the TE- and TM-mode and κ is the coupling constant between the two modes, which is for strong mode overlap equal to the Faraday rotation given in rad/m. A standard silicon-on-insulator single mode waveguide of <NUM> height and around <NUM> width is for this reason not suitable as a Faraday rotator. It features a relatively strong form birefringence leading to a Δβ of <NUM>•<NUM><NUM> <NUM>/cm. For a biasing field of <NUM> T this results in a coupling efficiency of only η=<NUM>•<NUM>-<NUM>.

Instead, a silicon waveguide with a square <NUM>×<NUM>µm2 cross section may be used. The large area ensures a low birefringence even if cladding and substrate have differing refractive indices, thanks to high confinement of the mode in the silicon core. Further, the fundamental TE- and TM-mode have a large field overlap for this geometry, yielding a maximum possible coupling constant κ. Care has to be taken that for such a multimode system only the two fundamental modes are excited and the coupling to higher order and thereby unwanted modes is suppressed. This was previously achieved by carefully designing bends and coupling section as well as ensuring a low sidewall roughness, also resulting in low propagation losses in the magnitude order of <NUM> dB/cm. In several centimeters long devices, such loss levels are mandatory to achieve.

However, a <NUM> Faraday rotator might be too long to fit on to a chip. A much more compact footprint of the rotator can be achieved by wrapping the waveguide into a spiral <NUM> as shown in <FIG>. As explained above, such an arrangement poses the problem that the Faraday rotation in the sections where the wave propagation direction and magnetic bias are parallel is compensated by the rotation in the sections where the two are counter-propagating, i.e. antiparallel with regard to the direction of propagation of light. This can be compensated by introducing a phase shift of Δϕ=<NUM>° between TE- and TM-mode after each turn. This novel design technique ensures that the Faraday rotation further adds up when the propagation direction with respect to the external B-field is reversed.

In <FIG>, the generally linear sections 41a and 41b (purple) of the spiral <NUM> feature no or low birefringence, while the bends <NUM> (green) feature a birefringence which does lead to a phase shift of Δϕ=<NUM>° when passing each bend. One U-turn then adds up to Δϕ=<NUM>°. This total phase shift per U-turn ensures that the Faraday rotation further adds up when the propagation direction with respect to the external magnetic field, indicated with an arrow Bbias, is reversed.

The birefringence of the bent sections may in some embodiments be based on total internal reflection mirrors, in order to achieve minimum footprint.

To prove the validity of then inventive concept, the inventors designed a spiral with birefringent bends and <NUM>×<NUM>µm2 linear or straight sections. The total length of the linear sections amounts to <NUM> and the footprint of the spiral is <NUM>×<NUM>µm2. An isolator was built by placing one polarizer oriented <NUM>° at one end facet of the spiral and one oriented <NUM>° at the other end facet (<FIG>). A biasing field of <NUM> Tesla was applied. The isolation ratio is the forward transmission divided by the backward transmission. To ensure a defined input state for both directions, single mode fibers leading from the laser source to the device were used. To check the measurement the forward and backward transmission was measured for both orientations of the magnetic biasing field, which should invert the isolation ratio.

The resulting spectra are shown in <FIG>, showing the isolation spectrum for both orientations of the magnetic biasing field. As expected, the isolation ratio is inversed for a reversed magnetic bias. The wavelength dependence of the spectra is due to the dispersion of the birefringence in the bends. Thus, the condition of Δϕ=<NUM>° is only met for a limited wavelength range. Unfortunately, the maximum isolation ratio is only <NUM> dB. The reason for this is the residual birefringence in the linear sections, caused by either strain or slight deviations from the square cross-section. This poses two problems: Firstly, it limits the maximum power fraction that can be coupled from one mode to the other (Eq. <NUM>) and, secondly, the linear sections connecting the <NUM>° bends add an additional phase difference on top of the intended <NUM>° for each bend. The first problem can be mitigated by increasing the magnetic biasing field and thereby increasing κ. Alternatively, both problems can be resolved by suppressing the unwanted birefringence. If the birefringence is due to shape inaccuracies the aspect ratio of the square cross-section can be altered with a thin conformal coating done with atomic layer deposition. The strain birefringence can thus be reduced by suppressing the birefringence in the linear section of the spiral, e.g. by thermally tuning the device.

In the following, a practical design using the above mentioned concept will be elaborated upon. An important choice is the waveguide profile. Typically, a silicon waveguide layer is <NUM> high, and the width of the waveguide is adjusted to allow for a single mode for each polarization. However, such waveguides are not well suited for Faraday rotation. In <FIG> the quasi-TE and the quasi-TM mode of such a waveguide is depicted. The field distribution of both modes is very different which leads to strongly different propagation constants or angular wavenumber β, given by the equation: <MAT>.

But an efficient Faraday rotation is only possible if both modes have roughly the same propagation constant. Another disadvantage of a single mode waveguide is that a large part of the field is outside the silicon core where it cannot contribute to the polarization rotation as the material is inactive. This calls for a large and symmetric cross-section, such as depicted in <FIG>. In a <NUM> × <NUM> waveguide, the field is confined in the silicon, and the field distributions of the two modes look identical leading to identical propagation constants β.

However, a large cross-section causes another problem, as the waveguide allows for higher order modes beside the two fundamental modes mentioned above. The number of modes in a waveguide distinguishes multi-mode waveguides from single-mode waveguides.

Higher order modes will spoil the functionality of the Faraday rotator if they are excited. Two sections of the total system are critical in terms of avoiding higher order mode coupling:.

however a more elegant and compact way is to gradually increase the curvature of the bend, for example to have the shape of the first section of an Euler spiral, referred to as an Euler bend. See co-pending <CIT> for reference. This gradual change in the curvature significantly reduces coupling to higher order modes. It has been shown that for a waveguide cross-section of <NUM>×<NUM><NUM> the effective bending radius can be as small as <NUM> and the coupling to higher order modes may still be below <NUM> dB per bend.

<FIG> depicts an exemplary implementation of the inventive design. A silicon Faraday rotator <NUM> is shown in <FIG> with non-birefringent quadratic linear waveguide sections <NUM> and <NUM> depicted (green). The bent birefringent parts <NUM> are also depicted (red). In <FIG> is shown the layout of an exemplary birefringent section <NUM>. The width of the waveguide is narrowed down followed by an Euler bend, and again widened.

The Faraday rotation is generated in those waveguide sections <NUM> of <FIG> that are parallel to the biasing field Bbias. The needed <NUM>° birefringence of each turn is achieved by two sections <NUM> each adding <NUM>° phase shift between the TE- and TM-mode. These two sections are connected with a non-birefringent waveguide <NUM>. This waveguide <NUM> will not contribute to the Faraday rotation as the direction of wave propagation is orthogonal to the biasing field. The birefringent section <NUM> consists of three parts: a taper 73a from the quadratic waveguide cross-section to one with a large aspect ratio, an Euler bend 73b and a taper 73c back to the quadratic cross-section. The total birefringence Δϕ<NUM> + Δϕ<NUM> + Δϕ<NUM> of the three parts 73a - 73c in <FIG> gives a total polarization rotation of <NUM> °.

An Euler bend 73b with large aspect ratio is for two reasons beneficial. Firstly, it creates the needed birefringence. Secondly, it allows for a smaller bending radius. The reason for this are that a more narrow waveguide will have less undesired modes that can be coupled and that the narrowing of the waveguide will increase the propagation mismatch between the modes and thus suppress coupling. <FIG> shows a possible implementation of an integrated silicon Faraday rotator. With a <NUM> T biasing field Bbias, the active waveguide needs to be <NUM> long to achieve the required <NUM>° polarization rotation. The rotating sections have a <NUM> × <NUM> cross-section. The bending radius of the Euler bend was chosen to be <NUM> with a <NUM> × <NUM> waveguide cross-section. The tapers are <NUM> long. The total device has a footprint of <NUM> × <NUM><NUM>, which is comparable to devices relying on iron garnets which use ring resonators (typically <NUM> × <NUM><NUM>) and Mach-Zhender interferometers (typically <NUM> × <NUM><NUM>, <NUM> × <NUM><NUM>).

Thus, despite the more than two orders of magnitude smaller of Faraday rotation that silicon provides in comparison to iron garnets, the exemplary silicon based device is only slightly larger. The reason for this is twofold:.

To ensure the functionality of the device two questions are of importance: Firstly, how much birefringence is allowed in the polarization rotating section and, secondly, how accurately has the birefringence in the bend to be adjusted. The former constraint originates from the fact that a mismatch in propagation constant limits the maximum power that can be coupled from one mode to the other. The maximum efficiency can be calculated by
<MAT>
where Δβ is the mismatch of propagation constants between the TE- and TM-mode and κ is the coupling constant between the two modes, which is for strong mode overlap equal to the Faraday rotation given in rad/m. For an isolator the polarization needs to be rotated by <NUM>°. This means that <NUM>% of the power needs to be coupled from the TE- to the TM-mode or vice versa. Accordingly, η needs to be larger than <NUM>. This constraint leads to an upper bound for the allowable birefringence of <MAT>.

For a silicon waveguide and a biasing field between <NUM> - <NUM> Tesla, the coupling constant κ lies between <NUM> rad/m and <NUM> rad/m, which is also the upper bound for Δβ. Although the birefringence in the waveguide in <FIG> is ideally zero, it might still be birefringent due to strain occurring in the manufacturing process or the substrate. If this is the case, the birefringence has to be counter-acted by an aspect ratio of waveguide section that slightly deviates from a square, or by post-fabrication tuning described in the following section.

Also the deviation of polarization rotation in bends from <NUM>° can lead to efficiency decrease. The accuracy with that the birefringence in the bend needs to be adjusted can be calculated in a similar fashion. One can define an effective birefringence of one Faraday section and a <NUM>° bend
<MAT>
where ΔβEB and LEB are the birefringence and length of the Euler bend. Further, ΔβT(l) is the length dependent birefringence of the taper integrated over its length and LFR is the length of the Faraday active section. The reasoning behind this effective birefringence is that as long as the Faraday rotation of each section is small, the deviance from <NUM>° birefringence in the turn can be interpreted as a mode mismatch between the TE- and TM-mode in the polarization rotating section. Equation <NUM> and <NUM> are also valid for Δβeff, yielding the same upper bounds. The fact that the length of the bending section occurs in the numerator and the length of the polarization rotating section occurs in the denominator means that the smaller the ratio of the former to the latter length is the more relaxed the constraints for ΔβEB and ΔβT(l) will be. Nonetheless, if one estimates the allowable error in birefringence by assuming that ΔβT(l)= ΔβEB, for the design in <FIG> this means that ΔβEB must be adjusted to an accuracy of <NUM> rad/m. For a <NUM> × <NUM> waveguide profile, where the change of birefringence with width is <MAT> at <NUM> wavelength, this corresponds to an accuracy in width of <NUM>. This may not be feasible with current fabrication techniques, and may call for a post-fabrication trimming.

Post-fabrication trimming may be done by tuning by temperature and trimming by oxidization, for example. The former method utilizes the fact that the refractive index of the silicon waveguide core and the silica cladding slightly change with temperature. For silicon the thermo-optic coefficient dη/dT = <NUM> · <NUM>-<NUM> <NUM>/ K, for silica it is <NUM> · <NUM>-<NUM> <NUM>/K. The altered refractive index causes a change in birefringence that can be calculated with
<MAT>.

For the <NUM> × <NUM> waveguide profile <MAT> is -<NUM>(m · K)-<NUM>. For an adjustable temperature range of ΔT = <NUM>K the birefringence has a tuning range of: <MAT>. This means one can compensate for a change in width of: <MAT>. Larger errors in width can be compensated by oxidizing the outer part of the waveguide. For this approach, one may omit the silica top cladding in the bend and oxidize the surface of the waveguide, see <FIG>. If the edges of a waveguide's cross section are oxidized, the aspect ratio will change slightly and cause a change in its birefringence. Only the open faces are oxidized and thus the aspect ratio and with that the birefringence changes. For the waveguide in <FIG> this yields an oxide thickness dependent change in birefringence of:
<MAT>
This means that by oxidizing <NUM> of the outer silicon, one can compensate for an error in width of <NUM>, which is well within the limits of current manufacturing processes.

In conclusion, the birefringence of an inventive optical waveguide section may be controlled by at least one of the following measures, selected from the bulleted list:.

The examples presented above assumed a uniform magnetic field in plane with the waveguides and aligned with the sections where Faraday rotation occurs. This can be achieved in practice by using suitably designed permanent magnets.

In <FIG>,a spiral (not shown) like the one in <FIG> is placed on a component chip <NUM> which is provided with a conductor <NUM> leading current through the center of the spiral. A magnetic field B is thus generated in a closed loop around the center of the spiral, turning it to a Faraday rotator component.

<FIG> shows an alternative approach to avoid cancellation of the Faraday rotation, based on making the magnetic field B to bend or follow the waveguide spiral shape on a chip <NUM>. In principle it could be realized with a simple electric wire <NUM>, but as strong magnetic fields in the order of <NUM> Tesla are needed for this practical Faraday rotation, it would be difficult to achieve with a single wire. A more promising approach is shown in <FIG>, where a toroidal wire <NUM> and an iron core <NUM> extending to the outside of the toroid, to come as close as possible to the waveguides. The spiral waveguide could also be sandwiched between two such electromagnets.

Claim 1:
An optical waveguide component (<NUM>, <NUM>) for providing Faraday rotation to a plane of polarization of linearly polarized light propagating in said waveguide component (<NUM>, <NUM>), wherein said waveguide component (<NUM>, <NUM>) is made of silicon and said waveguide component (<NUM>, <NUM>) has folded or wound sections (<NUM>) that are parallel to an externally applied magnetic field, wherein said waveguide component (<NUM>, <NUM>) comprises birefringent bent sections (<NUM>, 37a, 37b), characterised in that said birefringent bent sections comprise bends based on total internal reflection mirrors, in order to achieve minimum footprint.