Internal mechanism of perfect-reflector-backed dielectric gratings to achieve high diffraction efficiency

. The work started 20 years ago [Appl. Opt. 42 , 6255 (2003)] investigating the physical mechanism of multilayer dielectric reflection gratings to achieve 100% diffraction efficiency is extended to offer much deeper insight than before. How different diffraction amplitudes of the top surface corrugation contribute to the −1st-order efficiency of such a grating is shown analytically using a minimum set of real parameters . The two diffraction amplitudes transmitted through the corrugation play a dominant role in enabling DE = 100%. The necessary and sufficient condition for 100% efficiency is derived, and a very simple sufficient condition is also given. Moreover, the role of the reflection phase of the perfect-reflector, including the contribution due to optical path between the corrugation and the reflector, is emphasized.

Since the initial proposal of Svakhin et al [1], highefficiency, reflection gratings made on top of Multi-Layer Dielectric substrates (MLD grating) have found important applications in laser pulse compression and beam combining systems.The modelling of these gratings relies on numerical tools based on electromagnetic theory of gratings, and for design researchers resort to numerical optimization that conceals the physics of the problem.
The underlying physics of an MLD grating (MLDG) is cooperation of diffraction by the top surface-relief (or volume) grating and the optical interference taking place in the connection layer (see Fig. 1).However, few papers have been devoted to study the detailed physical mechanism of MLDGs.Shore et al [2] first raised the issue.As to the work of Tishchenko and Sychugov [3], although its interpretation of complete destructive interference between the direct 0 th -order and the round-trip 0 thorder above the MLDG is unimpeachable, its attribution of achieving a high diffraction efficiency (DE) to leaky wave excitation and energy accumulation inside the MLDG is only a speculation.The leaky-wave-excitation theory is later inherited in [4] and [5].Hu et al [6] made a simplified modal analysis of MLDGs, which is based on previous results of high DE transmission gratings; however, because of the employed crude approximations their work does not reveal physical mechanism of MLDGs well.In 2003, using symmetry considerations Wei and Li [7] laid down a good framework and obtained some useful insights, but with hindsight I think the work stopped prematurely.In this paper, I extend the work of [7] much further and present new results under the following assumptions in [7].
As in [7], an MLDG is divided into three parts (Fig. 1): the Top Grating (TG), the Connection Layer (CL), and the Perfect Reflector (PR).The physical model of [7] rests on three assumptions: 1) the entire MLDG is lossless and the reflector is perfect (reflectance = 1); 2) the wavelength-to-grating-period ratio permits only two reflected propagating orders in cover (air) and two transmitted propagating orders in CL; 3) the evanescent waves in CL can be neglected.Among these assumptions, the first two can be reasonably well secured.The third is fundamental, and its validity can be justified on physical ground and verified by numerical tests.In addition, two more modelsimplifying assumptions are made in [7]: 4) symmetric grating profile and 5) Littrow incidence.On basis of assumptions 1) -3), for TE or TM incidence from the air side, the 2×2 normalized S matrix of the MLDG is given by Eq. ( 5) of [7]: where R = diag[exp(iφ−1), exp(iφ0)] with φi, i = −1, 0, being the sum of the reflection phase at the CL-PR interface and the phase delay due to round-trip optical path of the ith diffraction order in CL, rud, rdu, tuu, and tdd are the 2×2 submatrices of the normalized 4×4 S matrix of TG, S (TG) , which can be calculated by using a rigorous numerical method.The approximate model represented by Eq. ( 1) is highly accurate for most practical cases, but it is still too complicated to reveal the physical mechanism of an MLDG to achieve high DE.

    
(5) Equation ( 5) is one of the key results of [7].Its physical interpretation is given by Fig. 3 of [7].However, rud = rdu = 0 is a poor approximation for a high-index grating and the equalities are never true in practice.
In the present work, under assumptions 1) -5), without the above approximation, the DE of an MLDG is written as , Note the following two features of ζ.First, Δθ depends only on the absolute values of τ and τ' and their phase difference.Second, φ appears only in Δχ.It can be shown that, as a periodic continuous function of φ, min ( ) 0 max ( );       (12) therefore, there is always a φ that makes Δχ = 0 and 2 2 min ( ) is the necessary and sufficient condition for existence of a φ that makes DE = 100%.The two φ values in (−π,π] that satisfy (13) can be calculated in closed form.
The above new results have many merits.The DE of an MLDG is related directly to the minimum number of real parameters of the S matrix elements of the TG and the phase φ.It allows us to study contributions of different diffraction amplitudes of the TG to achieving high DE.For example, it follows from Eqs. ( 8) and (12), if |τ| = |τ'| and arg(τ/τ') ≠ mπ for an integer m, then DE = 100% is possible by adjusting the thickness of the CL.This result completely gets rid of the need for the strong approximation rud = rdu = 0 imposed in [7].A corollary of this theorem is if |τ| 2 = |τ'| 2 > 1/4, then DE = 100% is possible.My numerical tests have shown that for gratings of a commonly seen groove profile and a refractive index 1 < n < 2.5, with air incidence, as the groove depth increases, sooner or later the condition |τ| 2 = |τ'| 2 > ¼ is reached.
The explicit functional form of χ(φ) allows us to see how DE depends on the reflection phase of the PR and the thickness tCL of the CL.In design of a single-polarization grating, when (13) is satisfied, tCL can be freely adjusted to reach the condition ζ = ± π.For a polarizationindependent grating, the φ required by ζ = ± π for TE polarization is in general different from that for TM polarization; therefore, tCL alone cannot simultaneously provide 100% DE for both polarizations.This partially explains why design tolerance of a polarizationindependent MLDG is much smaller than that of a singlepolarization grating.It also suggests a way to increase the design tolerance of the former by adjusting the phase difference of the PR between the two polarizations [8].
In summary, the work started in [7] has been extended to gain much deeper insight into the physical mechanism of an MLDG to achieve 100% DE.How different diffraction amplitudes of the TG contribute to the −1 storder DE of an MLDG is shown.The two transmitted diffraction orders of the TG play a dominant role in enabling DE = 100%, although the transmitted and reflected orders are interrelated by the unitarity of S (TG) .The necessary and sufficient condition for 100% DE is derived, and a very simple sufficient condition is also given.The role of the compound reflection phase φ in making DE = 100% for a single-polarization grating is revealed, and the importance of the phase difference between the TE and TM polarizations in designing a polarization-independent MLDG is emphasized.

Fig. 1 .
Fig. 1.Division of a multilayer dielectric grating into three parts: the top surface-relief grating (TG) of arbitrary symmetric profile, the connection layer (CL) of thickness tCL, and the perfect reflector (PR) of 100% reflectance.
arg(a+/a−); therefore, DE = 100% if and only if ζ = ± π. Taking all symmetry properties of the grating problem into account, I have succeeded in deriving this expression for