Advanced FECO Topics

A. The Typical Experiment

The following animation shows what basic information is available from Fringes of Equal Chromatic Order (FECO) during a typical SFA experiment.

Figure 1: Animation of FECO's during a realistic loading-unloading cycle.
Figure 1: Animation of FECO's during a realistic loading-unloading cycle.

As one can see in Figure 1, the surfaces are initially separated, i.e. not in contact. The cylindrical shape of the surfaces (Ag mirrors) is expressed as curved primary fringes. As the surfaces are moved closer (i.e. the surface separation is decreased) the primary fringes shift in wavelength towards smaller wavelengths (left).

At some point, the surfaces jump into adhesive contact - that is when the attractive force gradient exceeds the spring constant of the force measuring spring. The surfaces spontaneously flatten due to surface finite energy. Even at zero external load there is a finite contact area, as theoretically predicted by the well known JKR theory of contact mechanics. The flattening of the surfaces is readily visible from the primary FECO's.

Further increasing of the external load results in a greater contact area, which can be directly seen in the FECO pattern.
Upon unloading, the contact area is decreased until a certain minimal area is reached (at negative external load) where the surfaces suddenly separate.

Upon close inspection of the animation, one can distinguish a faint background modulation which is due to 'secondary' and 'tertiary' fringe patterns.

B. The Electric Field inside a Three-layer Interferometer

(Note: Due to high data quantity, download times for this section may be longer than usual.)
The analytical solution for primary FECO's in a three-layer interferometer are readily obtained using the assumption of high reflectance mirrors. In the limiting case of maximum reflectance (i.e. R->1), the electric field inside the interferometer is represented as a standing wave. Typically, the real reflectance is smaller than one (R~95%), which is enough to produce a traveling wave which transports energy through the interferometer (transmission). I have used the computer to obtain an animated representation of the electric field inside the three-layer interferometer. Such an animation is unique for a given geometry, wavelength and position.

The following figure displays again FECO's of symmetrical surfaces in flattened contact. For each point in the following figure one can animate the electric field. I have made a numerated selection of interesting points for which the electric field is animated below.

Enlarged view: Figure 2: symmetrical mica-mica in flat contact in air.
Figure 2: symmetrical mica-mica in flat contact in air.

The here following animations represent the electric field inside the three-layer interferometer corresponding to the specific positions indicated in Figure 2. The electric field inside a three-layer interferometer is reality the sum over all excited wavelength. To simplify the problem, I have animated the electric field at only one specific wavelength at a time.

In the animations, white ligth is shined from the left. The highly reflecting mirrors are located at the borders (left and right) and are assumed to exhibit no phase change upon reflection. The mirror reflection coeficients are set to be 90%.

Plain sinusoidal lines represent the amplitude components, white for the positive direction and black for the backwards reflected component. Note that the white component is always slightly bigger than the black component, since one has non vanishing transmission, i.e. transport of net energy. The sum of both components, which is represented as bright gray area, coresponds hence not exactly to a standing wave. An important consequence is that nodes as well as antinodes are travelling from the left to the right, i.e. in the direction of the energy flow. This motion of the nodes and antinodes is slow in intervals where the amplitude is big and can be very fast when the amplitude gets small.

The background gray-level is chosen to represent the refractive index of the respective medium.

The first two animations correspond to two different cases of resonance; i.e. they are on the exact fringe positions of FECO with chromatic order N=7 and N=8 respectively.

Reminder: The chromatic order is a measure (in units of 2 pi) for how much the complex phase of a plane wave is advanced through a full cycle in the interferometer, that is the sum of twice the optical distance from mirror to mirror plus the phase change on reflection on both mirrors and, where applicable, phase change on transmission on inner interfaces.

(1) N=7, x=0µm
(1) N=7, x=0µm

This is the electric field at the exact wavelength of the FECO with chromatic order N=7 inside the contact zone at x=0µm.

(2) N=8, x=0µm
(2) N=8, x=0µm

This is the electric field at the exact wavelength of the FECO with chromatic order N=8 inside the contact zone at x=0µm.

The following two animations demonstrate how the electric field is altered, if one sits on the resonance frequency outside the contact zone, that is when an air gap is introduced and one has a real three-layer interferometer instead of only one layer as above. As one can see, the single components of the electric field may well be discontinous at the newly created inner interfaces as long as the sum of them meets the continuity condition.

(3) N=7, x=50µm
(3) N=7, x=50µm

This is the electric field at the exact wavelength of the FECO with chromatic order N=7 slightly outside the contact zone at x=50µm.

(4) N=8, x=50µm
(4) N=8, x=50µm

This is the electric field at the exact wavelength of the FECO with chromatic order N=7 slightly outside the contact zone at x=50µm.

Note: It 'looks like' odd order fringes have a node in the center and even order fringes have an antinode in the center. As we already know this is not true since we have a travelling wave. Sometimes, however, the different sensitivity of odd and even fringes to refractive index in the gap is associated with the fact that even order fringes exhibit a higher electric field in the gap and are hence more sensitive to refractive index. Unless the medium exhibits non-linear response this is of course complete nonsense! In contrary, it seems that the components of the electric field of an odd fringe are more affected by the refractive index of the gap.

To make the picture complete, the following animation shows how the electric field looks like when one is not in resonance, i.e. not on a FECO.

(5) N=7.5, x=0µm
(5) N=7.5, x=0µm

This is the electric field between the resonances of chromatic order N=7 and N=8, but inside the contact zone at x=0µm.

C. Phasechange on Reflection and the Mysterious Anharmonic FECO of Order N=0

A plane wave, when reflected on an interface, experiences a phase change. The phase change on the confining (outer) mirrors of our interferometer is of particular interest because it has an important effect on the calculation of surface separations in the SFA (i.e. for the transformation of fringe positions into real-space separation). In complex representation, the phase change is nothing else but the argument of the (complex) reflection coefficient.

The refractive index of silver has a finite imaginary part, hence a silver mirror exhibits a phase change depending of wavelength and mirror thickness. The phase change of a silver mirror lies between 0° for a very thin layer and asymptotically reaches -134° for a thick layer.

The following animation demonstrates what happens to the FECO's when the phase change is varied from 0° to -180° in ten steps of -18° each. The absolute value of the reflection coefficient is, however, kept constant |r|=80%.

Figure 3: Appearance of the anharmonic N=0 fringe as the phase change on reflection on the mirrors gets negative.
Figure 3: Appearance of the anharmonic N=0 fringe as the phase change on reflection on the mirrors gets negative.

At first, one can nicely see the basic resonance of the interferometer, that is the fringe with chromatic order N=1. It's wavelength coincides exactly with twice the optical distance inside the interferometer; i.e. n:=n1=n2=1.5 and T:=T1=T2=1µm and inside the flattened contact area one has T3=0 so that the optical distance is 2*T*n=3µm=30'000Å, hence the fringe of chromatic order N=1 at the wavelength 60'000Å, which is in the far infrared.

The fringes that are left (shorter wavelength) of the base resonance represent the harmonics (N>1) of the optical resonator.

The introduction of a negative phase change on the mirrors creates the remarkeable possibility of an anharmonic resonance with chromatic order N=0. The existence of such a resonance, which, at first may seem unphysical, is relatively easy to explain. The condition that the total phase must be a multiple of 2 Pi to cunstructively interfere with itself is actually met for a plane wave, if, for travelling from one mirror to another the total advance of phase compensates exactly for the amount of the negative phase change on reflection, so that the total phase after two reflections is 0°. For this to work out, the sum of the phase changes of the two mirrors must be negative. For a phase change of exactly -180°, the mysterious FECO (N=0) becomes a sub-harmnonic resonance which has NOT the same electric field as the base resonance (N=1) in the case of 0° phase change. This is because the sub-harmonic FECO has the properties of an even fringe, whereas the base resonance (N=1) has an odd chromatic order. Therefore, their sensitivity towards the refractive index of the gap medium is different.

A phase change of exactly -180° occurs for example at the interface of two non-absorbing media, if traveling into the medium of higher refractive index.

The following animation illustrates the electric field of such a N=0 fringe with a tiny air gap and for a mirror phasechange of -2.34 rad which is similar to the phasechange observed on silver/mica mirrors in the visible.

Figure 4: Animated transvese electrical field of the sub-harmonic N=0 fringe for a wavelength-independent phase change of -2.34 rad at the mirrors (left and right extremes).
Figure 4: Animated transvese electrical field of the sub-harmonic N=0 fringe for a wavelength-independent phase change of -2.34 rad at the mirrors (left and right extremes).

From this animation, it can be readily seen that, due to energy transport, the electric field is not a 'standing wave' and all attempts to explain different sensitivities to gap refractive index in terms of 'nodes' or 'antinodes' is in fact absurd. This can also be observed in the above shown animations of N=7 and N=8 harmonic fringes, but is less obvious due to image magnification and variable travel velocities of nodes and antinodes throughout one periode.

Note: Provided with highly reflecting mirrors exhibiting a negative phase chenge close to -180°, it is theoretically possible to accurately measure a say 5nm gap between the mirrors using visible light of 100x bigger wavelength! This is using the N=0 subharmonic fringe of chromatic order N=0, which clearly demonstrates the power of interferometry at its extreme.

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