4.1

Modal behavior of a waveguide

The modal behavior of a dielectric step-index waveguide is fully determined by its couple Δn – thickness. The index difference between the film and the substrate will determine the guiding conditions. Low index difference will induce so-called “weak guiding conditions”, which means an important part of the propagating field exist in the substrate and can be affected by its absorption along the waveguide length. A thicker film must then be deposited to allow modes propagation. High index difference will lead to a strong confinement of the electromagnetic film into the guide. This induces a smaller thickness of the film but increases the coupling issues.

In addition to that, we show that the number of propagating modes in a waveguide with given Δn and W decreases when the wavelength increases. For instance, a 10-modes waveguide at λ=0.632 μm will progressively loose its higher modes one by one with increasing wavelength. Above a certain value (called the cut-off wavelength), only the fundamental mode can propagate: we have a single-mode waveguide. Considering those notions, we understand that we can return the problem and determine the index difference Δn (by choosing the appropriate materials) and the thickness W of the deposited film in order to obtain a single-mode waveguide at a specific wavelength (or range).

4.2

M-lines method for samples characterization

This method concerns only the dielectric waveguides. A short theoretical reminder is made before presenting this method. The propagation of light into a dielectric waveguide is characterized by a set of effective indexes. A given mode propagates into a gradient index waveguide with the wave vector given in Eq. 1.

The quantities k_x(x) and k_z are the transverse and longitudinal component, the vacuum permeability, ε(x) the material permittivity distribution in the x direction. The refractive index can be written as in Eq.2

and Eq. 1 can be rewritten

The quantity k²_x/k²₀ is defined as the effective index n_eff of the mode, which can be interpreted as the refractive index “seen” by a plane wave corresponding to a specific mode and propagating along the z axis. In the geometrical approach of electromagnetic waves propagation, the angle of the propagating ray θ(x) measured with respect to the z axis is obtained from Eq. 4

and thus we have

The propagation constant is given in Eq. 6.

Solving the Maxwell’s equation in the waveguide succeeds to discrete solutions for the propagation constant, results in discrete solutions for the effective index as well.

The principle of the m-lines method is to excite with a monochromatic radiation the different modes of a slab waveguide sample using a prism coupling technique. Fig. 3 shows the principle of the experiment.

Fig. 3.

Principle of the m-lines method

When a prism is placed on the surface of a slab waveguide and we have an incident beam in total reflection conditions at the base of the prism, theory shows that light can be coupled through the evanescent field existing in the air gap between the layer and the prism base if the phase matching conditions of Eq.7 [6] are fulfilled.

The quantities n_p and n_film are the refractive index of the coupling prism and the thin layer. θ_p and θ_film are the beam angular direction. In those conditions, rays of the focalized input beam filling the phase matching conditions will be coupled into the waveguide and “black lines” will be detected at discrete angular positions α_i. From the measurement of α_i, we retrieve the corresponding mode effective index through Eq.8.

A is the angular aperture of the prism, np its refractive index. Note that in order to excite successive modes from the fundamental one, it is necessary to use a coupler with higher index with respect to the film to be characterized. This can be checked out knowing the refractive index of film bulks since the index of the thin film will be lower than the bulk material. The method gives access to the set of effective indexes of the waveguide (called also mode indexes). Processing those data allows retrieving the index profile and the thickness of the waveguide.

For gradient index stuctures, the index profile can be retrieve using the original procedure proposed by White [8] and based on the WKB method. Index profile n(x) for TE modes is computed solving Eq.9.

Quantity x(m) is the turning point in a gradient index waveguide, m is the mode number, n_eff(m) is the effective index of mode m and k the modulus of the wave vector. ϕ₀ and ϕ₁ are the phase changes at the turning points x(m) and -x(m) which are taken equal to π/4 in [8]. In our study, we have an a priori knowledge that the index profile is a step. The method to retrieve the film index is then slightly changed. Phase changes computed at waveguide-air and waveguide-substrate interfaces are different due to sharp change in index. In [9], Tien et al. solved the equation

where W is the film thickness. ϕ₀ and ϕ₁ are the phase shifts respectively at the air-film and film-substrate interface. Knowing two effective indexes, we compute the index n_film and then the thickness W.

4.3

Characterization in the visible and NIR

Considering what previously said, we start to characterize samples in the visible and near infrared (NIR) where a multimode behavior is expected. The advantage of this procedure is to extract the thickness of the thin layer which is constant with respect to the wavelength. We characterize two type of samples which are AMTIR-1 (Chalcogenide glass) and Zinc Selenide (ZnSe) films. Fig.1 and Fig.2 show they are suitable for measurement from visible to mid-infrared range. ZnSe is deposited on Zinc Sulfide (ZnS) substrate while AMTIR-1 is deposited on a silica substrate. We proceed to measurements at λ=0.632μm and λ=1.323μm. As a coupler we use a TiO₂ prism with indexes n=2.864 at 0.632μm and n=2.715 at 1.323μm with TE polarization. The prism aperture is A=45.1458°. In Tab.2. are given the bulk indexes of the substrate used for computation.

Tab. 2:

index of the substrates

The measurements are carried out with a broadband source placed behind a monochromator to select the desired wavelength. We report in Tab.3 and Tab.4 the experimental values of effective indexes for the near infrared for AMTIR-1 and ZnSe films.

Tab. 3:

Angular position of TE guided modes for AMTIR-1 sample on silica substrate at λ=1.323μm

Tab. 4:

Angular position of TE guided modes for ZnSe sample on ZnS substrate at λ=1.323μm

Based on Eq.12, we compute the refractive index and thickness of the films. For the chalcogenide sample we obtain n_TE =2.301 ± 0.0011 and e=2.98μm ± 0.014. For the ZnSe sample we obtain n_TE =2.441 ± 0.0012 and e=3.83μm ± 0.033.

In Fig.5, we show an example of m-lines outputs obtained during the measurements

Fig. 5.

M-lines outputs for a chalcogenide slab waveguide at 13 μm (left) and 0.632μm (right)

4.4

Characterization in the thermal infrared

Previous measurements on ZnSe sample have shown that this structure, initially designed for the midinfrared range, is suitable for guidance and that there is no important drawbacks on the technology. It has also permitted to measure the film thickness, which will be used for later measurement in the thermal infrared.

Indeed, since we expect to be single-mode at 10.6μm, we obtain a single line: therefore the knowledge on the film thickness is mandatory to compute the refractive index according to Eq.12. The setup we implement in the frame of our project uses the 10.6 μm line of a CO2 laser. We use a Germanium prism with n=4.0 and A=45°. The layout of the setup is shown in Fig.6.

Fig. 6.

The m-lines setup for mid-infrared

This setup has been developed at LAOG and is to date under validation. Next step is to complete the phase with a measurement at 10.6μm and eventually extend it to other pertinent wavelength of the Darwin band.