Since the early part of the 18th century, phase difference and path difference have been important optical concepts; in the case of coatings, however, they become crucial. First, we must emphasize that the concept of phase applies to harmonic waves. Except when the power is enormously high (outside the scope of this tutorial), the response of an optical coating to an electromagnetic stimulus is linear. This permits us to decompose any incident electromagnetic disturbance into a set of harmonic, or monochromatic, plane-wave components called the spectrum. It is easy to treat the spectral components separately and then assemble the individual responses into the resultant. Terms like spectral response, spectral sensitivity, and spectral variation are very familiar to us. Convention Contradictions
Our first major problem is connected with conventions. A convention is an agreed-upon rule for a parameter description, but given its nature, there is never a single correct one. All that is necessary is that a convention be followed once it is agreed upon. Unfortunately, conventions are rarely universally accepted, and in optics, with its many-threaded development, numerous conflicting conventions exist.
One example that particularly concerns us is how we interpret what happens in a reflection. When we look in a mirror, the image exhibits what is usually called a parity change. What is particularly important is that any right-handed set of reference axes becomes left-handed in reflection. The thin-film worker happily accepts this and adopts a convention that incorporates it, while the ellipsometrist prefers to retain the right-handed nature of the original reference. Thus, two important conventions coexist and, although it is easy to transfer results from one to the other, they nevertheless represent a considerable source of confusion. At normal incidence, for example, the thin-film worker sees no change in the reflection response to a linearly polarized harmonic wave as the direction of polarization rotates about the normal. The ellipsometrist, on the other hand, must create two orthogonal reference directions for the polarization and reverse one of them in reflection.
The thin-film convention is often known as the Abelès convention and the ellipsometric as the Mueller. Virtually everyone agrees that since it is the electric field of an optical wave that interacts with materials, concepts like amplitude reflection coefficient, phase shift, and polarization should apply to the electric, rather than the magnetic, field.
Let us further consider thin-film conventions. The concept of a single value of phase change is limited to light that does not change its polarization state. Fortunately, there are two linearly polarized modes, called the eigenmodes of polarization, that do not change their state in either reflection or transmission. Any arbitrarily polarized wave can be decomposed into a combination of these two modes, which can then be followed through the system separately. We express all of our results in terms of these modes.
For an oblique incidence, we can define the plane of incidence as that containing the direction of the incident wave and the surface normal. Fortunately, this plane also contains the directions of both the reflected and the transmitted, or refracted, waves. The two eigenmodes have electric-field direction parallel to the plane of incidence (p or TM polarization) or perpendicular to it (s or TE polarization) and retain their polarization in reflection and transmission.
Figure 1. When calculating phase differences, we use the reference system shown. The points O, the origin, and D, where the z-axis crosses into the substrate, are the reference points at which we calculate the phases of the various waves. We also need definitions of the reference directions for the electric fields of the polarization eigenmodes, s and p polarization, as indicated.
Before we can even begin to discuss any phase shifts, we must assign reference directions to the electric fields of both eigenmodes (see figure 1). Note that at normal incidence, the plane of incidence disappears and both s and p conventions collapse to an identical convention. Note also that the p direction, the s direction, and the direction of propagation form a right-handed set in incidence and transmission, but left-handed in reflection. The Mueller convention reverses the p direction in the reflected beam, so p and s directions must be chosen at normal incidence.
The concept of relative phase between two plane harmonic waves of identical frequency is still not entirely simple. If the waves are not propagating in exactly the same direction, then the relative phase depends on position. Any stated value must therefore be qualified by a further convention: a statement of where the phase is measured. figure 1 shows the usual reference axes for the thin-film community. We define phase change on reflection at point O on the front surface of the coating, and we measure phase change on transmission between the entry point O for the incident beam and the emergent point D for the transmitted beam. To these conventions, we add the fact that an increase in path length yields a lag in the phase of the wave. Reflections on Phase Change
For the remainder of this tutorial, let's concentrate on some aspects of phase change in reflection. It is usual to refer the ellipsometric parameters Ψ and Δ to the plane of incidence, so that their definition is
where ρp and ρs are the amplitude reflection coefficients for the p and s directions, respectively, and Ψ is taken in the first quadrant. If, however, we simply use the thin-film definitions ρp and ρs, we have a right-handed reference set of directions in incidence and a left-handed set in reflection. The ellipsometric convention is always right-handed, so we must add or subtract 180° from the definition of Δ in equation 1. This also applies to normal incidence. Note that we make no such correction in transmission.
Figure 2. We can best calculate the effect of a lack of uniformity on the wavefront by adding imaginary incident material (gray region) to the front surface of the coating, so that the reference surface for phase calculations is completely free from uniformity errors.
Uniformity of an optical coating is an important attribute in a well-corrected system. To calculate the effect of a lack of uniformity on a wavefront, we must both consider the movement of the reflecting surface from its ideal position and note any change in its phase shift on reflection. We can derive these parameters readily. The easiest way to deal with them without running into sign problems is to introduce a variable thickness of incident medium material at the front of the coating, so that this new front-reference surface is everywhere of the correct figure (see figure 2). In such a case, any phase shift calculations will include the extra, doubly traversed path and can be immediately translated into a change in the figure of the wavefront. We can handle transmittance errors similarly.
An opaque metal layer will introduce no change with thickness in the phase shift at the surface; therefore, we can express the effect on the reflectance figure as a wavefront error that is exactly twice the uniformity error. For a dielectric coating, the effect becomes more complicated. The most common all-dielectric reflector is the quarter-wave stack, which consists of alternate high- and low-index quarter-wave layers. If the outermost layer is of high index, then the phase change on reflection is 180°. If the thickness of the coating is locally reduced as a result of nonuniformity, then the reflecting surface drops away from its ideal position, inducing a corresponding phase lag. As a result of thickness reduction, the phase change on reflection at the coating surface moves into the third quadrant and partially compensates for the path error. For example, a 1% change in the thickness of a 21-layer silicon dioxide/tantalite quarter-wave stack at a wavelength of 1000 nm yields a wavefront error of 0.053λ instead of 0.06λ. The rate of change of phase with thickness does not vary greatly over the useful high-reflectance range of the quarter-wave stack.
Unfortunately, the process that reduces the effect of a uniformity error in the quarter-wave stack can greatly increase the effective figure error in other types of reflecting coatings. The wavefront error is not just a function of the physical position of the reflecting surface but also of the phase change on reflection at the surface. In certain types of coatings, the phase effect can be dominant.
Figure 3. Because of the phase effects, the calculated wavefront error for a 1% uniformity error (thickness reduction) in a 27-layer broadband reflector for the visible region varies significantly from what would be expected from simple uniformity considerations.
The quarter-wave stack has a limited spectral width. Broadband reflectors, therefore, essentially consist of two or more quarter-wave stacks deposited in series, although the layer thicknesses may also be tapered gradually through the structure. Over part of the reflecting range, the light must penetrate deep into the interior before it is reflected, adding considerable path difference and consequent change in phase. In such a region, rapid changes in phase with thickness can occur, introducing large differences in the actual wavefront error (see figure 3).
The coating shown in the figure consists of a 27-layer broadband reflector for the visible region, with a uniformity error of -1%. If no phase effects existed in the coating, the error would translate into a slowly varying negative value close to -0.1λ. This is shown as the lower curve. If we take phase into account, however, we obtain the upper curve.
Note that not only does error change rapidly with wavelength, but considerable magnification occurs along with several sign changes. Ramsay and Ciddor illustrated that this is not an academic matter. Almost 40 years ago, they reported a component that was at the same time concave, flat, and convex, depending on the wavelength. The implications are clear.1
There is much more to phase than can be included here.2, 3 None of these effects is mysterious. They are all unambiguously calculable, but accurate results require knowledge and consistent application of the conventions. oeReferences
1. J. Ramsay and P. Ciddor, Applied Optics 6, p. 2003 (1967).
2. P. Baumeister, Optical Coating Technology, SPIE Press, Bellingham, WA (2004).
3. H. Macleod, Thin-Film Optical Filters, Third ed., Institute of Physics Publishing, Bristol and Philadelphia (2001).
Angus Macleod is president of Thin Film Center Inc., Tucson, AZ.