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J. Eur. Opt. Society-Rapid Publ. 22, 1( 2026)
transfer between the 0th and �1st orders when scanning the incidence angle. The structure, as it is well-known in resonant waveguide grating( RWG) will then support guided modes and will exploit waveguide resonance in high refractive index dielectric [ 10 ] with the aim to use this device as sensor. The main difference with other works [ 11 – 13 ] is that the incident light, studied in TE polarization, is coming from the backside of the grating( from the substrate) in order to avoid disturbance effects from the external medium to be probed( Fig. 1). In this case, resonant reflected orders will be also retrieved from the same side( substrate) than the incident beam one.
As mentioned in [ 14 ] in a theoretical point of view, the waveguide and the grating must be in a high refractive index material for the energy transfer to occur between the two reflected �1st and 0th orders only, when scanning the incident angle in the vicinity of the Littrow angle( h L, angle for which the reflected �1st order is superposed to the incident light). This condition is essential to avoid any transmitted order from having a propagating character in the external medium and to couple the evanescent diffracted orders to the guided modes. Several other parameters are conditioning the optical energy transfer existence, in particular the grating geometry. The following subsections aim to describe how the guided modes could propagate in the resonant structure and how they can be excited by the grating to obtain optical energy transfer between the two reflected orders.
2.1 Mode’ s propagation in the waveguide
The resonant grating is partially etched in a high refractive index medium layer n wg deposited on the substrate of refractive index n s, as represented in Figure 2. The grating shows here a square profile with a depth d, aperiodK and a duty cycle( 1�f). The underneath layer( buffer layer), as part of the waveguide, has a thickness W g. The superstrate refractive index is n super( Fig. 2a) and corresponds to the refractive index of the medium to be probed( liquid or air).
To understand how the resonant structure behaves, an equivalent structure is considered in Figure 2b. Itiscom- posed of only one high refractive index plane layer, sandwiched between the substrate and superstrate. Its total thickness is noted w eqTOTAL = W g + d eq where d eq is the equivalent thickness corresponding to the top grating and which can be written as:
d eq n wg ¼ d n eq; ð1Þ
where n eq is the equivalent refractive index of the plane layer that is substituted for the grating. n eq is dependent of the incident wave polarization( TE or TM) according to Rytov’ s formulas [ 15 ]. In the case of air as superstrate( n super = n air = 1) and with f = 0.5, it follows for TE polarization: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u t n 2 wg þ 1 w eqTOTAL; TE ¼ w g þ d eq; TE ¼ w g þ d = n wg: ð2Þ
2
To create a guided mode in the waveguide equivalent structure of refractive index n wg, theplanewavepropagatingin
Figure 1. Excitation of the guided mode by the dielectric diffraction grating.
the direction of the wave vector k ~ wg( jj k ~ wg jj ¼ 2p n k wg) under a angle should interfere constructively with itself( Fig. 2c). This condition leads to the dispersion equation of the lth guided TE mode when introducing the mode effective refractive index n e ¼ n wg cosa:
lp ¼ 1 2 uwg þ u wg þ 2p s super k w pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
eqTOTAL n wg2 � n 2 e; ð3Þ
where uwg and u wg are the phases due to the total reflection s super
at the substrate and superstrate boundaries, expressed for TE polarization by:
8 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! n 2 e � n2 s
>< >: uwg s
¼�2atan |
n 2 wg � n2 e |
; |
|
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! |
|
|
n 2 e � n2 super n 2 wg � n2 e |
: |
u wg super ¼�2atan
The number of modes supported by the equivalent structure can be determined versus waveguide thickness w eqTOTAL and wavelength k from the dispersion equation( 3). For TE polarization, the l-mode cut-off happens when n e = n s( n s > n super) below which value the mode could not propagate anymore:
rffiffiffiffiffiffiffiffiffiffiffiffiffi
w
eqTOTAL
¼ k TE cut�off n 2 s �n 2 super n 2 wg�n 2 s
lp þ atan |
|
|
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi |
: |
ð4Þ |
2p |
n 2 wg � n2 s |
Under an appropriate incident angle, the TE 0 mode could be excited in the resonant structure for l = 0 if the waveguide equivalent thickness is above its corresponding cut-off. For different incident angles, other modes( l > 0) can be guided if increasing the waveguide equivalent width. It can lead to optical energy transfer between the 0th and �1st orders if the grating geometry is adapted to excite the mode. The grating coupling to the guided modes is described in the following paragraph.
2.2 Guided modes excitation by the grating diffraction orders
The TE modes will propagate in the waveguide at a speed associated to their effective indices n e = n wg cosa and will be