A theoretical analysis of asymmetrical diffraction in Raman-Nath, intermediate and Bragg diffraction regimes is presented. The asymmetry is achieved by combining matched periodic modulations of the phase and of the loss/gain of the material, which enables the breakdown of optical symmetry and redirects all resulting optical energy in only positive or only negative diffraction orders, depending on the quarter period shift directions between the phase and the loss/gain modulations. Analytic expressions for the amplitudes of the diffraction orders are derived based on rigorous multimode coupled mode equations in slowly varying amplitude approximation.
©2012 Optical Society of America
Optical gratings with periodic modulations of both the absorption coefficient and the refractive index have been studied in different diffraction regimes: in thick gratings (or Bragg) regime of diffraction [1–5] and thin gratings or Raman-Nath regime [6, 7]. The first theoretical description of light diffraction in an isotropic material with a periodic modulation of the refractive index and the absorption was given by Kogelnik  in Bragg regime. Considering periodic phase and amplitude modulations, the grating types are treated to be in phase. Later the diffraction problem was generalized by Guibelade for Bragg regime  and by Zakharyan  for Raman-Nath regime to describe out-of phase index and amplitude gratings. When considering absorption as an imaginary part of a refractive index, such periodic modulation can be considered as an imaginary grating. On the other side, optical amplification can be described by the absorption coefficient having the opposite sign. Amplification of light travelling through the grating can be produced by exciting (optically or electrically) an active medium. Such optical structures are thus referred to as active elements. Gain/loss reflective volume gratings have been studied mostly for applications in optical distributed feedback (DFB) lasers. In particular the lasing conditions of the periodical structures of index and gain have been investigated theoretically by H. Kogelnik and C.V Shank .
The concept of diffraction asymmetry (also referred to as non-reciprocity or unidirectionality) achieved through a phase shift between the index and the gain/loss gratings was first proposed by Paladian  for waveguide gratings in 1996. In 1998 M.V. Berry  studied a similar problem of an atom diffracted in free space by standing waves of light with the complex potential distribution having a purely absorptive imaginary part. The concept of non-reciprocity for reflective gratings has been extended to the case of grating assisted co-directional couplers  and waveguide Bragg gratings . However, these asymmetrical Bragg or long period gratings in optical fibers or waveguides operate in the thick grating mode  and are providing coupling between only two contra- or co-propagating modes. We are aware of one single attempt to experimentally investigate the active gratings with both refractive index and the gain/loss modulations in Raman-Nath diffraction regime by R. Baribassov et al . We give them a credit for very interesting experimental part, the theoretical portion remains to be significantly improved and clarified. The authors focused their attention solely on the asymmetry of the first ( ± 1) diffraction orders completely ignoring zero and higher orders that, unlike the Bragg regime, are typical for the Raman-Nath diffraction. To the best of our knowledge, there is no study on the diffraction phenomenon for the combined index and gain/loss modulation gratings from Raman-Nath regime through intermediate regime into Bragg diffraction regime.
2. Rigorous diffraction equation solution
In this paper, we analyze diffraction characteristics of active holographic gratings. The grating is assumed to be composed of the refractive index modulation and modulation of gain/loss with the same period but shifted by a quarter of period, Λ/4 (|| = 2π/Λ) in respect to one another in either directions ( ± ):Fig. 1 . For an incident plane wave with TE polarization (electric field perpendicular to the plane of incidence) the scalar wave equation can be presented in the following form:15] it is straightforward to derive the equations describing the amplitudes of the diffraction orders, Sm, as:15]:
Equation (4) can thus be rewritten for the diffraction amplitudes in the dimensionless coordinates:
We will analyze the particular grating configuration for which its behavior riches its maximum asymmetry when the contributions from index and gain-loss gratings are equal:. We will refer to such grating as the perfectly asymmetrical grating. In this perfectly asymmetrical case the diffraction equations can be presented in a compact, dimensionless form:
3. Raman-Nath diffraction regime
The obtained equations can be analyzed within the Raman-Nath limits. There are multiple criteria of the occurrence of Raman-Nath regime of diffraction. Each of these criteria are evaluated in  showing that each of them are met within 1% diffraction efficiency when the condition is satisfied, where is the quality factor of the grating, defined as. It can also be expressed through the Nath factor , it means that . It is well known that for an pure index modulation grating, the amplitudes of the diffraction orders are described by Bessel function of the correspondent order,(Fig. 2(a) ) or in the case of pure imaginary gratings with absorption or gain modulation, by the modified Bessel functions(Fig. 2(b)) for zero average loss/gain factor (α0 = 0). Similarly to the gain/loss grating, the perfectly asymmetrical grating provides amplified diffraction into the various orders, however, the main difference consists of in the fact that energy is diffracted in only positive (or only negative) orders. For the Nath-factor of zero value the expressions in Eq. (10) are reduced toFig. 2(c). As we can see, the solutions are very different from the ones for traditional index modulation gratings. In addition to full diffraction asymmetry, the incident wave seems to be unaffected as nondiffracted light in the zero order without any amplitude or phase variation passing through the grating (red, solid horizontal line in Fig. 2(c)). The diffraction efficiency in the first order dominates for small values of the grating strength (), then the light gets diffracted gradually into the higher orders.
The diffraction efficiency expressions described by Eq. (11) are essential for understanding Raman-Nath diffraction regime. Furthermore, the general solution of the rigorous diffraction Eq. (4) where only the slowly varying second order derivatives are neglected , provides a more general understanding of the multi-order diffraction from Raman-Nath through intermediate and into Bragg diffraction regime.
4. Intermediate and Bragg diffraction regimes
4.1 Arbitrary incidence
The expressions for the diffraction amplitudes are simplified when the Bragg condition is satisfied for each diffraction order individually.Fig. 3 for different values of the Nath factor. The amplitudes strongly depend on the Nath factor. The higher diffraction order, the stronger this dependence. In Fig. 3(b) the plots are magnified 10x, 100x and 1000x in order to make them visible for 6 and 12 respectively. Diffraction for occurs in the Bragg regime , and our calculation shows that diffraction in higher (than the first) orders are becoming insignificant when increases. However, as the Bragg condition cannot be satisfied for all orders at the same time. Therefore, it is interesting to plot out the angular bandwidth of the diffraction efficiency.
Such diffraction distributions for first (red curves), second (blue curves) and third (green curves) orders as functions of the mismatch factor (which is proportional to the sinus of the angle of incidence and the light wavelength) are shown in Fig. 4 for different values of the Nath factor and the dimensionless grating thickness. In fact, these different distributions reflect different regimes of diffractions starting from Raman-Nath regime with its broad angular and wavelength selectivity as it is shown in Fig. 4(a) for lowvalue and shallow gratings. The second and third order proportional contributions in respect to the first diffraction order (in %) are shown on each plot. There is about 15% contribution from the second order and 0.23% from the third order into the first order diffraction at. With the light propagating through the grating with (Fig. 4 from (a) to (c)) the spectral bandwidth is narrowing, however the contribution from the second order still remains above 10% due to emerging spectral side-lobes having their respective peaks over the angular Bragg conditions.
When the Nath factor is getting larger, , the diffraction regime is moving into the intermediate mode with increasing angular selectivity and significantly lower contribution from the higher diffraction orders down to about 3% for the second and 0.01% from the third orders respectively (Fig. 4(d-f)). Finally, for even larger Nath factors,, most of the light is concentrated in the first diffraction order (Fig. 4(g-i)) with only 0.76% contribution from the second order with very high angular and wavelength selectivity. Formally this is still in intermediate diffraction regime, the Bragg regime beginning only when . Therefore our results show that the perfectly asymmetrical grating can provide strong diffraction predominately into the first diffraction order while preserving the zero order amplitude absolutely nondepleted.
In traditional index gratings or even in imaginary gratings of absorption or gain-loss modulation light diffraction is always symmetric in respect to the normal to the grating slab interface. The diffraction is fully symmetric between the incident angles and, as shown in Fig. 5(a) and 5(b). In the case of the perfectly asymmetrical grating, seen in Fig. 4(a), there is some residual light from the positive first and the second orders () in the negative angle directions, as in the negative Bragg angles etc., however, as the grating strength and thefactor are large enough, there is practically no diffracted light in these negative directions, as shown schematically in Fig. 5(e) and 5(f).
The diffraction asymmetry on the perfectly asymmetrical grating is also effective on diffraction from the opposite side of the grating (g, h), as shown in Fig. 5 in comparison with the index grating (c, d).
4.2 Normal incidence
For normal incidencethat can be analyzed when, the equations for the diffraction order amplitudes are reduced to the following forms:Fig. 5, and then all amplitudes simultaneously decrease to zero. This constitutes a remarkable mode interaction that is worthwhile a further study. Interestingly, zero order diffraction remains constant. The process is very similar to diffraction of an atom beam by standing waves of light . As it already has been said, the incident wave are passing through the grating without any changes and at the same time producing strong diffraction only in positive or only negative diffraction orders.
Similarly to the case of oblique incidence, the incoming wave is diffracted into multiple diffraction orders for small values within the Raman-Nath diffraction regime as in Fig. 6(a) for where diffraction into the second and third order higher than into the first one. As the Nath factor is getting larger, the first order diffraction is becoming predominant (Fig. 6(b)) and finally for (Fig. 6(c)), less than 5 percent diffracts into the second order, and less than 0.2% into the third order. This ratio is reduced to 2.8% and 0.05% respectively for, i.e. practically all light is redirected into zero and the first diffraction orders. We are theoretically not in the Bragg regime at normal incidence, but practically all light is distributed only in the zero and the fundamental order, as in the case of thick gratings in a waveguide [11, 12].
Before concluding, we emphasize the fact that the gain micro-structures presented in this paper need to be excited (for example by a pump laser) so that the wave to be diffracted can also be amplified by the medium, in a similar way to EDFA (Erbium Doped Fiber Amplifiers) pumped a 980nm laser light coupled in the cladding to amplify the C band guided in the core of the fiber around 1.5 micron wavelength. In our case the gain media is also periodically patterned via a lithographic process similar to standard IC fabrication (optical lithography and etching), as well as the underlying periodic phase micro-structures.
The results presented in this paper were obtained for zero average loss/gain factor (α0 = 0). However, this factor can be controlled independently and can be used for designing particular diffraction pattern of the perfectly asymmetric gratings. This amplifying ability provides us with a very convenient tuning mechanism. Indeed, doping materials, whether these are rare-earth ions for the near infrared, or incorporated dye molecules for the visible spectrum or heterostructures in semiconductors, absorb light when they are not pumped by an external source. Therefore, by controlling the level of the pumped power, the diffraction efficiency can be tuned practically from zero (no pumping) up to very strong asymmetrical diffraction mostly into the single first diffraction order.
There are many different dynamic diffractive or holographic elements presented in literature, which work either as switchable (switching from one state of the others), efficiency tunable (tuning efficiency from a minimum to a maximum), functional tuning (tuning a specific functionality such as a focal length or a grating period), and the reconfigurable elements (where the optical function and/or the efficiency can be completely reconfigured). Our grating (or hologram) structure is thus comparable to tunable efficiency gratings (or holograms), such as H-PDLC (Holographic-Polymer Dispersed Liquid Crystal) holograms, for which the diffraction efficiency in the fundamental order can be tuned from nearly zero to a maximum value close to 90%, for a single polarization. However, our solution does not require any liquid crystals layers (and the various environmental constraints associated to this technology), and can be fabricated by standard lithography at wafer scale. Moreover, our solution yield the asymmetry discussed in this paper which cannot be produced by standard efficiency tunable holograms such as H-PDLCs.
Our technological solution enables us to design cascaded arrays of such gratings since the nondepleting light in the zero order can be used as an incident wave onto a next grating, and thus produce a very desirable constant efficiency cascaded gratings system. Such cascaded grating architectures are very desirable in numerous applications today, such as optical telecom (OADM), high uniformity laser or LED edge lit display backlights, fiber or free space laser sensor arrays, parallel read-out of optical storage, etc....
Finally, the complex amplitude of the first diffractive order that represents the grating response in the spectral domain is in the form of sinc-function. Such function has the rectangular Fourier impulse response (rect(x) = 1 for |x| ≤ 0.5 and 0 otherwise). In the temporal domain an input Gaussian pulse with the appropriately short duration diffracted into the first order will be transformed into flat top waveform with the duration proportional to the grating thickness. This approach has already been employed using very weak index gratings. Such gratings with the strengthhave spectral characteristics very close to sinc-shape. The proposed gratings permanently preserve their sinc-shape in the first diffraction order for any grating strength values and through the all diffraction regime (from Raman-Nath to Bragg). Flat-topped pulses are expected to play a major role in future high-speed all-optical signal processing systems due to their tolerance of timing jitter. The generation of such pulses from those with a Gaussian profile currently represents a significant research effort .
We have described in this paper the diffraction characteristics of a hybrid index and gain/loss modulation grating (or hologram) with a quarter period shift between the index and the gain/loss modulations with the same period. We have analyzed the resulting diffraction efficiencies from Raman-Nath through intermediate and up to Bragg diffraction regimes. We have also derived a set of equations predicting diffraction efficiencies for arbitrary incidence angle. The diffraction model is based on multimode coupled wave theory in the slowly varying amplitude approximation.
A unique characteristic of such gratings is that they can diffract energy in only either positive or negative orders, while yielding a perfectly non depleted zero order. Such novel grating structures can be fabricated via traditional lithography, and is under experimental study.
We are very grateful to Prof. Joseph Goodman, Prof. Michael Berry and Dr. Radan Slavik for fruitful discussions.
References and links
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