Abstract

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

1. Introduction

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 [15] 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 [1] 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 [2] and by Zakharyan [7] 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 [8].

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 [9] for waveguide gratings in 1996. In 1998 M.V. Berry [10] 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 [11] and waveguide Bragg gratings [12]. However, these asymmetrical Bragg or long period gratings in optical fibers or waveguides operate in the thick grating mode [13] 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 [14]. 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 (|K| = 2π/Λ) in respect to one another in either directions ( ± ):

n(x)=n0+n1cos(Kx);
α(x)=α0±α1sin(Kx)
where n0 is the average index in the grating area; n1 is the amplitude of the index modulation, α0 is the average absorption (α0 >0) or gain (α0 <0) constant and α1 is the amplitude of the gain/loss periodic distribution, x and z are the coordinates parallel and perpendicular to the grating slab respectively, as shown in 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:
2Ey(x,z)+k(x)2Ey(x,z)=0
where Ey(x,z) is the complex amplitude of the y-component of the electric field inside the grating region which is assumed to be invariant in the y direction and to oscillate with the angular frequency ω. The propagation constant k(x) is spatially modulated and related to the refractive index and absorption/amplification coefficient of the medium. According to [15] it is straightforward to derive the equations describing the amplitudes of the diffraction orders, Sm, as:
Ey(x,z)=m=+Sm(z)exp{j[(2πn0λsinθmK)x+2πn0λcosθz]}
For the typical case of scalar diffraction (where the wavelength of light is considered to be very small compared to the grating period (λΛ)) and for the case of a slow energy interchange between the modes, Sm, and assuming that the energy is absorbed and/or magnified slowly, we can neglect the second order derivatives, d2Sm (z)/dz2, and the multiwave coupled wave equations take on the following Raman-Nath form [15]:
cosθdSm(z)dz+j(πλmΛ2n0(μm)+α0)Sm+j2(κ+Sm+1(z)+κSm1(z))=0
where j = (−1)1/2, λ is the wavelength of the incident light, θ is the incident angle inside the grating medium,κ+and κare the coupling coefficients which in this type of gratings can be very different in magnitudes. They can be expressed as:
κ+=πn1/λ+α1/2;κ=πn1/λα1/2
The mismatch factor μ=2Λ(ε0)1/2sinθ/λ may take on any value. The Bragg condition is satisfied when μ takes on the value of an integer m, representing the m-th diffraction order. The Bragg condition is satisfied when the angle of incidence and reconstruction wavelength are such that μ becomes an integer. The boundary condition must satisfy Sm(0)=δ0mwhere δ0mrefers to the Kronecker symbol.

 figure: Fig. 1

Fig. 1 Geometry of asymmetrical diffraction on the combined gratings of refractive index and gain/loss modulations. When these two modulations are balanced,πn1/λ=α1/2, only diffraction into non-negative or non- positive orders occurs depending on the shift between the gratings.

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Equation (4) can thus be rewritten for the diffraction amplitudes Tm(z)=Smexp(α0z) in the dimensionless coordinatesξ=2πn1z/(λcos(θ)):

2dTm(ξ)dξ+j2(η+Tm+1+ηTm1)=jm(mμ)ρTm
whereη=1±α1λ/(2πn1)andρ=λ2/(Λ2n0n1) is the Nath factor.

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:πn1/λ=α1/2. 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:

2dTm(ξ)dξ+jTm1=jm(mμ)ρTm
It is obvious from this equation and the boundary conditions Tm(0)=δ0m that all negative diffraction orders are null: Tm = 0 for n<0, and that the zero order efficiency has a constant value, i.e.T0(ξ)=1. The solution can be derived by defining:
Tm(ξ)=Rm(ξ)exp(+jξm(mμ)ρ/2)
where Rm(ξ) has the following form:
Rm(ξ)=j20ξRm1(ξ)exp(jξ(2mμ1)ρ/2)
with relatively simple solutions for at least the first few diffraction orders:
R0(ξ)=1;R1(ξ)=2jsin((μ1)ξρ/4)ρ(μ1)exp(j(μ1)ξρ/4);R2(ξ)=j2ρ2(sin((μ2)ξρ/2)2(μ1)(μ2)exp(j(μ2)ξρ/2)sin((μ3)ξρ/4)(μ1)(μ3)exp(j(μ3)ξρ/4));R3(ξ)=jρ3(sin(3(μ3)ξρ/4)3(μ1)(μ2)(μ3)exp(j3(μ3)ξρ/4)sin((μ4)ξρ/2)(μ1)(μ3)(μ4)exp(j(μ4)ξρ/2)+sin((μ5)ξρ/4)(μ2)(μ3)(μ5)exp(j(μ5)ξρ/4))
The higher order amplitudes can also be derived in a similar way, but their expressions are quite cumbersome and will not be presented in this paper.

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 [13] showing that each of them are met within 1% diffraction efficiency when the condition Qξ1 is satisfied, whereQ is the quality factor of the grating, defined asQ=2πλz0/n0Λ2cos(θ). It can also be expressed through the Nath factor ρ=2Q/ξ, it means that ξ(2/ρ)1/2. 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,Jm(ξ)(Fig. 2(a) ) or in the case of pure imaginary gratings with absorption or gain modulation, by the modified Bessel functionsIm(ξ)(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 to

Tm(ξ)=Rm(ξ)=(ξ)m2mm!;form=0,+1,...+N
which are very good approximations for non-zero values of ρwhen ξ(2/ρ)1/2. The diffraction efficiencies, |Sm(ξ)|2=|Rm(ξ)|2, are presented in Fig. 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 (ξ<4), then the light gets diffracted gradually into the higher orders.

 figure: Fig. 2

Fig. 2 Diffraction efficiencies in zero and higher diffraction orders by the index grating (a), the gain/loss grating with zero average gain/loss value (b), and the perfectly asymmetrical grating (c) also with zero average gain/loss value.

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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 [15], 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.

R0(ξ)=1;R1(ξ)=jξ2;atμ=1R2(ξ)=2jρ(ξ4sin(ξρ/4)ρexp(jξρ/4))atμ=2R3(ξ)=jρ2[ξ8+(ξ8cos(ρξ/2)12ρsin(ρξ/2))exp(jξρ/2)]atμ=3
As can be seen from these equations, only the first order diffraction amplitude is independent of the Nath-factor. The evolution of the second (m = 2) and third (m = 3) order amplitudes at Bragg angles (μ=m) as they propagate through the grating along the dimensionless coordinateξ are shown in Fig. 3 for different values of the Nath factorρ. 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 ρ=3,6 and 12 respectively. Diffraction for ρ10occurs in the Bragg regime [16], and our calculation shows that diffraction in higher (than the first) orders are becoming insignificant when ρ 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.

 figure: Fig. 3

Fig. 3 Diffraction efficiencies in the second (a) and third (b) diffraction orders for different values of the Nath factor (ρ) at Bragg angles. The diffraction efficiency for the third order is magnified 10x, 100x and 1000x for ρ=3,ρ=6,andρ=12respectively.

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Such diffraction distributions for first (red curves), second (blue curves) and third (green curves) orders as functions of the mismatch factor μ (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 μand the dimensionless grating thicknessξ. 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 lowρvalue 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 atμ=1. With the light propagating through the grating withρ=1.5 (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.

 figure: Fig. 4

Fig. 4 Diffraction efficiencies of the first (red), second (blue) and third (green) diffraction orders as a function of the mismatch factor μ for different values of the Nath factor (ρ) and the dimensionless grating thicknessξ. The proportional contribution of the second and the third diffraction orders at μ=1is shown in percentage in each plot.

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When the Nath factor is getting larger, ρ=3, 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,ρ=6, 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 whenρ>10 [16]. 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+θ andθ, 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 (μ=+1;μ=+2;) in the negative angle directions, as in the negative Bragg anglesμ=1; μ=2;etc., however, as the grating strength and theρfactor are large enough, there is practically no diffracted light in these negative directions, as shown schematically in Fig. 5(e) and 5(f).

 figure: Fig. 5

Fig. 5 Comparative diffraction on the index grating (a, b, c, d) and the perfectly asymmetrical grating (e, f, g, h) for different incidence directions and different incidence angles.

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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 incidenceθ=0that can be analyzed whenμ=0, the equations for the diffraction order amplitudes are reduced to the following forms:

R0(ξ)=1;R1(ξ)=2jsin(ξρ/4)ρexp(jξρ/4);R2(ξ)=2jρ2(sin(ξρ)4exp(jξρ)sin(3ξρ/4)3exp(j3ξρ/4));R3(ξ)=jρ3(sin(9ξρ/4)9exp(j9ξρ/4)sin(2ξρ)6exp(j2ξρ)+sin(5ξρ/4)15exp(j5ξρ/4))
In contrast to the more general case of non-normal light incidence, as we can see from these equations, the amplitudes are periodic functions of the grating strengthξ, or dimensionless grating thickness. The period isΔξ=4π/ρ, and all diffraction orders going through constructive or destructive interference reach their maxima and minima for the same values ξ(m)max=4πm/ρand ξ(m)min=2π(2m1)/ρ, where m – integer, respectively. Unlike for the case of non-normal incidence, where there is no limitation on the amplitude growth (unless we account for the gain medium saturation), the diffraction order amplitudes in the case of normal incidence achieve their maxima at specific values of the grating strength (grating thickness) as seen in 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 [10]. 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 ρvalues within the Raman-Nath diffraction regime as in Fig. 6(a) for ρ=0.25where 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 ρ=1.5(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 forρ=2, 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].

 figure: Fig. 6

Fig. 6 Diffraction efficiencies of the first (red), second (blue) and third (green) orders for normal light incidence (θ=0) as a function of the grating strength ξ for different values of the Nath factor (ρ).

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5. Conclusion

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 strengthξ0.15πhave 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 [17].

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.

Acknowledgment

We are very grateful to Prof. Joseph Goodman, Prof. Michael Berry and Dr. Radan Slavik for fruitful discussions.

References and links

1. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

2. E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16(2), 173–178 (1984). [CrossRef]  

3. C. Neipp, I. Pascual, and A. Belendez, “Experimental evidence of mixed gratings with a phase difference between the phase and amplitude grating in volume holograms,” Opt. Express 10(23), 1374–1383 (2002). [CrossRef]   [PubMed]  

4. M. Fally, M. Ellabban, and I. Drevensek-Olenik, “Out-of-phase holographic gratings: a quantitative analysis,” Opt. Express 16(9), 6528–6536 (2008). [CrossRef]   [PubMed]  

5. M. V. Vasnetsov, “Oscillations conditions in a gain grating in the Bragg diffraction regime,” Opt. Commun. 282(10), 2028–2031 (2009). [CrossRef]  

6. M. Chang and N. George, “Holographic dielectric Grating: theory and practice,” Appl. Opt. 9(3), 713–719 (1970). [CrossRef]   [PubMed]  

7. G. G. Zakharyan and A. V. Galstyan, “Mixed phase and absorption this gratings diffraction,” Opto-Electron. Rev. 15(1), 20–26 (2007). [CrossRef]  

8. H. Kogelnik and C. V. Shank, “Coupled‐wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972). [CrossRef]  

9. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996). [CrossRef]   [PubMed]  

10. M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998). [CrossRef]  

11. M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. 29(5), 451–453 (2004). [CrossRef]   [PubMed]  

12. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005). [CrossRef]   [PubMed]  

13. T. K. Gaylord and M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20(19), 3271–3273 (1981). [CrossRef]   [PubMed]  

14. R. Birabassov, A. Yesayan, and T. V. Galstyan, “Nonreciprocal diffraction by spatial modulation of absorption and refraction,” Opt. Lett. 24(23), 1669 (1999). [CrossRef]  

15. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985). [CrossRef]  

16. M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980). [CrossRef]  

17. L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008). [CrossRef]  

References

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  1. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  2. E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16(2), 173–178 (1984).
    [Crossref]
  3. C. Neipp, I. Pascual, and A. Belendez, “Experimental evidence of mixed gratings with a phase difference between the phase and amplitude grating in volume holograms,” Opt. Express 10(23), 1374–1383 (2002).
    [Crossref] [PubMed]
  4. M. Fally, M. Ellabban, and I. Drevensek-Olenik, “Out-of-phase holographic gratings: a quantitative analysis,” Opt. Express 16(9), 6528–6536 (2008).
    [Crossref] [PubMed]
  5. M. V. Vasnetsov, “Oscillations conditions in a gain grating in the Bragg diffraction regime,” Opt. Commun. 282(10), 2028–2031 (2009).
    [Crossref]
  6. M. Chang and N. George, “Holographic dielectric Grating: theory and practice,” Appl. Opt. 9(3), 713–719 (1970).
    [Crossref] [PubMed]
  7. G. G. Zakharyan and A. V. Galstyan, “Mixed phase and absorption this gratings diffraction,” Opto-Electron. Rev. 15(1), 20–26 (2007).
    [Crossref]
  8. H. Kogelnik and C. V. Shank, “Coupled‐wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972).
    [Crossref]
  9. L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
    [Crossref] [PubMed]
  10. M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998).
    [Crossref]
  11. M. Greenberg and M. Orenstein, “Irreversible coupling by use of dissipative optics,” Opt. Lett. 29(5), 451–453 (2004).
    [Crossref] [PubMed]
  12. M. Kulishov, J. M. Laniel, N. Bélanger, J. Azaña, and D. V. Plant, “Nonreciprocal waveguide Bragg gratings,” Opt. Express 13(8), 3068–3078 (2005).
    [Crossref] [PubMed]
  13. T. K. Gaylord and M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20(19), 3271–3273 (1981).
    [Crossref] [PubMed]
  14. R. Birabassov, A. Yesayan, and T. V. Galstyan, “Nonreciprocal diffraction by spatial modulation of absorption and refraction,” Opt. Lett. 24(23), 1669 (1999).
    [Crossref]
  15. T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
    [Crossref]
  16. M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
    [Crossref]
  17. L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
    [Crossref]

2009 (1)

M. V. Vasnetsov, “Oscillations conditions in a gain grating in the Bragg diffraction regime,” Opt. Commun. 282(10), 2028–2031 (2009).
[Crossref]

2008 (2)

M. Fally, M. Ellabban, and I. Drevensek-Olenik, “Out-of-phase holographic gratings: a quantitative analysis,” Opt. Express 16(9), 6528–6536 (2008).
[Crossref] [PubMed]

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

2007 (1)

G. G. Zakharyan and A. V. Galstyan, “Mixed phase and absorption this gratings diffraction,” Opto-Electron. Rev. 15(1), 20–26 (2007).
[Crossref]

2005 (1)

2004 (1)

2002 (1)

1999 (1)

1998 (1)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998).
[Crossref]

1996 (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

1985 (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

1984 (1)

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16(2), 173–178 (1984).
[Crossref]

1981 (1)

1980 (1)

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[Crossref]

1972 (1)

H. Kogelnik and C. V. Shank, “Coupled‐wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972).
[Crossref]

1970 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Azana,

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Azaña, J.

Bélanger, N.

Belendez, A.

Berry, M. V.

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998).
[Crossref]

Birabassov, R.

Chang, M.

Clausen, A. T.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Drevensek-Olenik, I.

Ellabban, M.

Fally, M.

Galili, M.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Galstyan, A. V.

G. G. Zakharyan and A. V. Galstyan, “Mixed phase and absorption this gratings diffraction,” Opto-Electron. Rev. 15(1), 20–26 (2007).
[Crossref]

Galstyan, T. V.

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

T. K. Gaylord and M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20(19), 3271–3273 (1981).
[Crossref] [PubMed]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[Crossref]

George, N.

Greenberg, M.

Guibelalde, E.

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16(2), 173–178 (1984).
[Crossref]

Jeppesen, P.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled‐wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972).
[Crossref]

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kulishov, M.

Laniel, J. M.

Magnusson, R.

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[Crossref]

Moharam, M. G.

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

T. K. Gaylord and M. G. Moharam, “Thin and thick gratings: terminology clarification,” Appl. Opt. 20(19), 3271–3273 (1981).
[Crossref] [PubMed]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[Crossref]

Mulvad, H. C. H.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Neipp, C.

Orenstein, M.

Oxenlowe, L. K.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Pascual, I.

Plant, D. V.

Poladian, L.

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled‐wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972).
[Crossref]

Slavik, R.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Vasnetsov, M. V.

M. V. Vasnetsov, “Oscillations conditions in a gain grating in the Bragg diffraction regime,” Opt. Commun. 282(10), 2028–2031 (2009).
[Crossref]

Yesayan, A.

Yongwoo Park, J.

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

Zakharyan, G. G.

G. G. Zakharyan and A. V. Galstyan, “Mixed phase and absorption this gratings diffraction,” Opto-Electron. Rev. 15(1), 20–26 (2007).
[Crossref]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

IEEE J. Sel. Top. Quantum Electron. (1)

L. K. Oxenlowe, R. Slavik, M. Galili, H. C. H. Mulvad, A. T. Clausen, J. Yongwoo Park, Azana, and P. Jeppesen, “640 Gb/s timing jitter-tolerant data processing using a long-period fiber-grating-based flat-top pulse shaper,” IEEE J. Sel. Top. Quantum Electron. 14(3), 566–572 (2008).
[Crossref]

J. Appl. Phys. (1)

H. Kogelnik and C. V. Shank, “Coupled‐wave theory of distributed feedback lasers,” J. Appl. Phys. 43(5), 2327–2335 (1972).
[Crossref]

J. Phys. Math. Gen. (1)

M. V. Berry, “Lop-sided diffraction by absorbing crystals,” J. Phys. Math. Gen. 31(15), 3493–3502 (1998).
[Crossref]

Opt. Commun. (2)

M. V. Vasnetsov, “Oscillations conditions in a gain grating in the Bragg diffraction regime,” Opt. Commun. 282(10), 2028–2031 (2009).
[Crossref]

M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Criteria for Raman-Nath regime diffraction by phase gratings,” Opt. Commun. 32(1), 19–23 (1980).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Opt. Quantum Electron. (1)

E. Guibelalde, “Coupled wave analysis for out-of-phase mixed thick hologram gratings,” Opt. Quantum Electron. 16(2), 173–178 (1984).
[Crossref]

Opto-Electron. Rev. (1)

G. G. Zakharyan and A. V. Galstyan, “Mixed phase and absorption this gratings diffraction,” Opto-Electron. Rev. 15(1), 20–26 (2007).
[Crossref]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

L. Poladian, “Resonance mode expansions and exact solutions for nonuniform gratings,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54(3), 2963–2975 (1996).
[Crossref] [PubMed]

Proc. IEEE (1)

T. K. Gaylord and M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73(5), 894–937 (1985).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 Geometry of asymmetrical diffraction on the combined gratings of refractive index and gain/loss modulations. When these two modulations are balanced, π n 1 /λ= α 1 /2 , only diffraction into non-negative or non- positive orders occurs depending on the shift between the gratings.
Fig. 2
Fig. 2 Diffraction efficiencies in zero and higher diffraction orders by the index grating (a), the gain/loss grating with zero average gain/loss value (b), and the perfectly asymmetrical grating (c) also with zero average gain/loss value.
Fig. 3
Fig. 3 Diffraction efficiencies in the second (a) and third (b) diffraction orders for different values of the Nath factor (ρ) at Bragg angles. The diffraction efficiency for the third order is magnified 10x, 100x and 1000x for ρ=3 , ρ=6 ,and ρ=12 respectively.
Fig. 4
Fig. 4 Diffraction efficiencies of the first (red), second (blue) and third (green) diffraction orders as a function of the mismatch factor μ for different values of the Nath factor (ρ) and the dimensionless grating thickness ξ . The proportional contribution of the second and the third diffraction orders at μ=1 is shown in percentage in each plot.
Fig. 5
Fig. 5 Comparative diffraction on the index grating (a, b, c, d) and the perfectly asymmetrical grating (e, f, g, h) for different incidence directions and different incidence angles.
Fig. 6
Fig. 6 Diffraction efficiencies of the first (red), second (blue) and third (green) orders for normal light incidence ( θ=0 ) as a function of the grating strength ξ for different values of the Nath factor (ρ).

Equations (14)

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n(x)= n 0 + n 1 cos( Kx );
α(x)= α 0 ± α 1 sin( Kx )
2 E y (x,z)+k (x) 2 E y (x,z)=0
E y (x,z)= m= + S m (z)exp{ j[ ( 2π n 0 λ sinθmK )x+ 2π n 0 λ cosθz ] }
cosθ d S m (z) dz +j( πλm Λ 2 n 0 (μm)+ α 0 ) S m + j 2 ( κ + S m+1 (z)+ κ S m1 (z) )=0
κ + =π n 1 /λ+ α 1 /2; κ =π n 1 /λ α 1 /2
2 d T m (ξ) dξ + j 2 ( η + T m+1 + η T m1 )=jm(mμ)ρ T m
2 d T m (ξ) dξ +j T m1 =jm( mμ )ρ T m
T m (ξ)= R m (ξ)exp(+jξm(mμ)ρ/2)
R m (ξ)= j 2 0 ξ R m1 ( ξ ) exp(j ξ (2mμ1)ρ/2)
R 0 (ξ)=1; R 1 (ξ)=2j sin((μ1)ξρ/4) ρ(μ1) exp(j(μ1)ξρ/4); R 2 (ξ)= j2 ρ 2 ( sin((μ2)ξρ/2) 2(μ1)(μ2) exp(j(μ2)ξρ/2) sin((μ3)ξρ/4) (μ1)(μ3) exp(j(μ3)ξρ/4) ); R 3 (ξ)= j ρ 3 ( sin(3(μ3)ξρ/4) 3(μ1)(μ2)(μ3) exp(j3(μ3)ξρ/4) sin((μ4)ξρ/2) (μ1)(μ3)(μ4) exp(j(μ4)ξρ/2)+ sin((μ5)ξρ/4) (μ2)(μ3)(μ5) exp(j(μ5)ξρ/4) )
T m (ξ)= R m (ξ)= (ξ) m 2 m m! ; for m=0,+1,...+N
R 0 (ξ)=1; R 1 (ξ)=j ξ 2 ; at μ=1 R 2 (ξ)= 2j ρ ( ξ 4 sin(ξρ/4) ρ exp(jξρ/4 ) ) at μ=2 R 3 (ξ)= j ρ 2 [ ξ 8 +( ξ 8 cos(ρξ/2) 1 2ρ sin(ρξ/2) )exp(jξρ/2) ] at μ=3
R 0 (ξ)=1; R 1 (ξ)=2j sin(ξρ/4) ρ exp(jξρ/4); R 2 (ξ)= 2j ρ 2 ( sin(ξρ) 4 exp(jξρ) sin(3ξρ/4) 3 exp(j3ξρ/4) ); R 3 (ξ)= j ρ 3 ( sin(9ξρ/4) 9 exp(j9ξρ/4) sin(2ξρ) 6 exp(j2ξρ)+ sin(5ξρ/4) 15 exp(j5ξρ/4) )

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