In a previous Letter1 we described how to demonstrate the phase behavior of Wood anomalies using a Babinet compensator. In Fig. 1 we see the fringes obtained, along the spectrum of a quartz iodine lamp, with the Babinet between crossed polarizers when the analyzer is on the spectrum. The spectrum is also shown for the light polarized normal to the grooves. The absorption band is a Wood anomaly of the grating.
Fig. 1. Wood anomaly top, the light distribution. Bottom, the equal chromatic order fringes if the polarizer, the Babinet, and the analyzer are on the entrance slit. Middle, the fringes when the phase delay of the grating is added to those of the Babinet (analyzer
over the spectrum, Babinet, and polarizer at the entrance slit).
Fig. 2. Fringes of equal chromatic order with polarizer and Babinet at the entrance slit and the analyzer on the spectrum for varying incidence angle. Top, the fringes are bent downward in the anomaly. Middle, the fringe system is broken. Bottom, each fringe is bent
upward connecting with the next one.
Such strong anomalies appear only for certain groove profiles, wavelengths, and incidence angles. So if, for example, the incidence angle is changed, the resonance becomes weaker but the phase has a very interesting behavior. If the grating rotates about an axis parallel to the grooves, just as for spectral scanning, the fringe system (Fig. 2) changes its form and the fringes which are broken in the middle system are connected downward in the upper system and upward in the lower system.
If a bright interference fringe corresponds to a phase difference of 2ΗΠ and this fringe in the left part of the spectrum connects with one of the same or next order in the right part, something strange must occur when the system is broken.
We shall give an explanation by means of a precarious analogy with a theory2 developed for constant λ and take the incidence angle to be the variable, contrary to the case of a
Fig. 3. Different positions of the pole and the zero of the diffracted field in the complex λ plane. Physically only the real axis is significant. The phase of the diffracted field is given approximately by the angle between the vectors λ0 and λp. (a), (b), and (c) correspond to
top, center, and bottom of Fig. 2, respectively.
photograph of the spectrum. We leave the challenge to justify this analogy or to find a better explanation for the theoreticians. We will also suppose that the component polarized parallel to the grooves has a completely regular phase behavior so that it is a suitable reference beam.
Neviere2 finds that the diffracted light may be described by
where z = sinθ with θ as the incidence angle, Z0 and zp are complex values for which there is a zero and a pole of the diffracted field E. Physically only real values of z are of interest.
We suppose that it is also possible to obtain a similar equation as a function of the wavelength (this is our tentative supposition) and write
where B' is also a slowly varying function of λ; λ0 and λp are complex. These values are also functions of the incidence angle and hence of z. In Fig. 3 we represent three possible positions of the pole and the zero which correspond to the fringes of Fig. 2.
The phase variation is sufficiently described by (λ – λ0)/(λ – λp) and it is given by the angle between (λ – λ0) and (λ – λp).
Thus, moving along the Re(λ) axes from small to large λ in Fig. 3(a) we first have increasing values of the phase, later near Re(,λu) ≅ Re(λp) it decreases rapidly to negative values,
and finally for large λ the phase tends to zero. For the case of Fig. 3 (c) there is a steady increase of the phase toward 2π. In the case of Fig. 3(b) the connectin of the phase before and after the resonance wavelength λ = λ0 is not defined, corresponding to the broken system. So when the grating is rotated λo changes its position and the strongest resonance is observed where λ0 crosses the real axis.
The break in the fringes and their connection with the neighboring ones are due to the fact that there the normal component of the diffracted field becomes zero.
This work was supported by CONICET and SUBSIDIO UBA.
References 1. J. M. Simon and M. C. Simon, "Diffraction Gratings: a Demon
stration of Phase Behavior in Wood Anomalies," Appl. Opt. 23, 970 (1984).
2. M. Neviere, "The Homogeneous Problem," R. Petit, Ed. (Springer-Verlag, Heidelberg, 1980), p. 123.
