Ring interferometers with unit transmittance

Herman van de Stadt Astronomical Observatory, Zonnenburg 2, 3512 NL Utrecht, The Netherlands. Received 19 March 1985. 0003-6935/85/152290-03$02.00/0. © 1985 Optical Society of America.

In most interferometers the energy of a single input beam is divided between two output beams; for example, the Mi-chelson interferometer (Fig. 1) directs energy not only toward output A but also toward B from both interferometer arms. But there is a certain class of ring interferometer with only a single output beam. It will not produce an interference effect and will have unity transmission. Although an interferometer without interference may seem to be a contradiction in terms, we will show that similar situations occur in electrical systems, which are called all-pass networks, loss-free systems, or equalizers (Ref. 1, pp. 239 and 249).

In Fig. 2 we present a ring interferometer with a single

Fig. 1. Michelson interferometer with two output beams A and B.

Fig. 2. Ring interferometer with a single output beam A. The am­plitudes at positions 1, 2, 3, and 4 are E+

1,, E-1, t1E+ + r1E- exp(iφ),

and E- exp(iφ).

2290 APPLIED OPTICS / Vol. 24, No. 15 / 1 August 1985

output beam. Since all input energy leaves the interferometer through a single output beam, energy conservation does not permit destructive interference; thus there is no interference. We will derive this characteristic of unit transmittance by following the amplitudes of the interferometer beams in two different ways. First, we follow the amplitudes of individual beams leading to a geometric series for the total amplitude transmittance (Ref. 2, p. 324). Second, we define the effective amplitudes including all multiple-beam contributions. This constitutes an elegant way of describing more complicated interferometers similar to the description of multilayer thin films (Ref. 2, p. 59).

The amplitudes of individual beams depend on the reflec­tance r and transmittance t of the beam splitter in Fig. 2. The phase difference between multiple beams depends on the optical path length of one round trip in the ring interferom­eter, defined by a phase φ. There are opposite signs of r for beams incident from the left and reflected upward and beams incident from below and reflected to the right (Ref. 3, p. 92). The total amplitude transmittance is

where we again used a minus sign for r1 (Ref. 3, p. 92). Eliminating E- from Eqs. (3) and (4) and using r2

1 + t21 = 1

yield Eq. (1). The remarkable property of this interferometer is its 100% intensity transmittance irrespective of beam splitter properties and phase difference. In contrast with other interferometers, there will be no interference effect noticeable in the output beam of this interferometer.

We next describe a similar but more complicated ring in­terferometer with two beam splitters shown in Fig. 3. We define the shortest phase length between the two beam splitters as φ1, the phase length of the upper part of the in­terferometer as φ2 and the lower part as φ3. Effective total amplitudes are marked in Fig. 3 at five places: E+

1 and E-1 at

positions marked 1 and 2; E+2 and E-

2 at positions 3 and 4; E3 at position 5. The amplitude of the beam incident from below on the left-hand beam splitter is E-

2 exp(iφ3), and the ampli­tude of the beam incident from above on the right-hand beam splitter is E-

1 exp(iφ). The following relations between E+1

E-1 E+

2 E-2 and E3 hold:

If we assume lossless optical components (no absorption, no scattering), we have r2

1 +t21 = 1, and we find for this converging

series

Since the numerator and denominator in Eq. (1) are complex conjugates the intensity transmission is

Eliminating the various amplitudes using r21 + t2

1=1 and r22

+ t22 = 1 yields the total amplitude transmission

An alternative way of deriving Eqs. (1) and (2) is the use of effective amplitudes. In Fig. 2 the total incident and upward reflected amplitudes on the beam splitter are marked 1 and 2 and are defined as E+ and E-. Here E- includes not only the reflected part of the incident amplitude E+ but also the transmitted fraction of all round trip beams. The effective amplitude of the beam incident on the beam splitter from below is marked 4 and is E~ exp(iφ) while the effective am­plitude of the beam at the interferometer exit A is marked 3 and is t1E+ + r1E- exp(iφ). The total amplitude transmit­tance is

On the other hand the relation between E- and E+ is

The nominator and denominator in Eq. (6) are again complex conjugates. Thus the intensity transmittance is unity, as in Eq. (2) and as required for energy conservation. In the lim­iting case r2 = 0, t2 = 1, φ1 = 0, and φ2 + φ3 = φ, Eq. (6) reduces to Eq. (1).

We have demonstrated the property of unit transmittance by building the ring interferometer of Fig. 2 using a He-Ne laser as a light source. When aligning the different compo­nents, one easily achieves the situation where the directly transmitted beam combines with the beam that made one round trip through the interferometer. The two beams pro-

Fig. 3. Unit transmittance interferometer with two beam splitters and phase lengths φ1, φ2, and φ3 between them. The amplitudes at

positions 1, 2, 3, 4, and 5 are E+1, E-

1, E+2, E-

2, and E3. Fig. 4. Ring interferometer with unit reflectance.

1 August 1985 / Vol. 24, No. 15 / APPLIED OPTICS 2291

duce an interference pattern of high contrast, which can be projected on a screen with a magnifying lens. Next the fringe density of the interference pattern is reduced by improving the alignment of the optical components, until finally a single uniformly illuminated spot is obtained for perfect superpo­sition of all beams with multiple round trips. Note that the intensity is not only uniform but also remains constant even when the round trip path length changes many wavelengths, e.g., due to temperature variations. This is in contrast to the Michelson interferometer of Fig. 1, where the intensity of beam A varies between zero and maximum whenever the path length difference changes one-half wavelength.

The ring interferometer of Fig. 4 demonstrates the principle of unit reflectance interferometers. The same formalism as used above can be used to derive the following expression for the amplitude reflection coefficient:

Equations (1), (6), and (7) are equivalent to the equations for the transfer impedance of all-pass networks as given in Ref. 1. On p. 239 such an all-phase network is defined as having the property that zeros and poles of the transfer function are negatives of one another. This condition is fulfilled in our equations where numerator and denominator are complex conjugates.

The property of unit transmittance or unit reflectance of ring interferometers according to Figs. 2, 3, and 4 refers to continuous electromagnetic waves. The behavior is quite different for a short light pulse entering the ring interferom­eter. A single light pulse is split in a series of pulses with de­creasing intensity. The delay is determined by the round trip time. This property may have useful applications in mode locking devices or in optical delay devices. However, in this Letter we mainly wanted to explain the phenomenon of unit transmittance or unit reflectance of ring interferometers for continuous electromagnetic waves.

In conclusion, we have shown that the phenomenon of lossless ring interferometers can be explained with a formalism in which the numerator and denominator are complex con­jugates. This is the optical analog of loss-free systems in electrical network theory.

References 1. H. W. Bode, Network Analysis and Feedback Amplifer Design

(Van Nostrand, Reinhold, New York, 1959). 2. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,

1970). 3. E. Hecht and A. Zajac, Optics (Addison-Wesley, Reading, Mass.,

1977).

2292 APPLIED OPTICS / Vol. 24, No. 15 / 1 August 1985