Multiple-beam cyclic interferometer

Miguel Cervantes M.

A multiple-beam cyclic interferometer with unidirectional trajectories is presented and discussed conceptual-ly. The main feature of the device is the utilization of controllable losses in the setup to modify convenientlythe form of the fringes. Sufficient reduction of the losses results in finer and more contrasting fringes thanthose of a Fabry-Perot with comparable surface reflectance. The design is derived from a two-beaminterferometer, and the procedure canamplitude.

The longitudinal-mode selector demonstrated bySmith1 is a multiple-beam reflection interferometer,which has been called the Fox-Smith 2A4 interferometer(FSI). This device and its variants 3 -5 are useful forsingle-frequency operation of gas lasers. Basically, itsconfiguration is like that of a Michelson interferome-ter except that the beam splitter is oriented at 900 fromthe normal position, as shown in Fig. 1(a). In princi-ple, the device is a variant of the Fabry-Perot (F.P.)interferometer in which light is introduced into thecavity by a beam splitter. 6

In this paper we consider passive devices that aresimilar to the FSI, the main difference being that herepower losses in the setup are deliberately introduced ina controlled manner to modify the instrumental func-tion of the device. Figure 1 illustrates (a) the FSI, and(b) multiple-beam versions of Mach-Zehnder, and (c)Sagnac interferometers. In Figs. 1(a) and (b) twocomplementary fringe patterns are produced when thelosses are negligible. In Figs. 1(c) and 2 a single pat-tern is produced whose parameters are a function ofthe losses of the setup.

We discuss theoretically the properties of the devicedepicted in Fig. 2, which serves to convey the concept.

Generally, in every interferometric device, energylosses that do not contribute to the formation of fringesare undesirable because they reduce the modulationand strength of the useful signal. The mechanisms

The author is with Universidad de Sonora, Centro de Investiga-cion en Fisica, A.P. A-88, Hermosillo, Son., 83190, Mexico.

Received 16 April 1987.0003-6935/88/101952-04$02.00/0.© 1988 Optical Society of America.

be extended and used with other two-beam devices with division of

that cause them include reflection losses, scattering,and absorption by the material and coatings employed.We assume that, in addition to the above mentionedloss sources inherent in every real-world setup, lossesare intentionally introduced by the operator in a con-trolled manner with the purpose of modifying the formof the fringes.

In Fig. 2(a) multiple beams of interference are pro-duced when the source F generates monochromaticplane waves, in the S or P state of polarization, that areincident on the beam splitter BS. Here L representsan optical element that reduces the amplitude of thetraveling waves traversing it. This is discussed later.

We define the dimensionless parameter B = EE,where E is the scalar amplitude of the electric field ofthe incident wave on L and E' that of the wave aftertraversing L. A fraction (1 - B2) of the energy isremoved by L in each pass. The term loss is used todenote the amount of energy lost in each round tripand does not contribute to formation of the fringepattern. We assume for simplicity that the plane mir-rors A, B, and C are perfect reflectors and that thelosses of the entire setup are caused solely by L.

The phase difference between any two consecutivebeams depends on the optical path length of the closedtrajectory OABC and the phase changes that occur onreflection on the mirrors and the beam splitter em-ployed.

To calculate the resulting amplitude at 0', we con-sider for simplicity a beam splitter consisting of an air-glass boundary such as the hypotenuse of the right-angle prism shown in Fig. 2(b). The surface of a glassin air would give a beam splitter that, although sensi-tive to polarization, has properties that change slowlywith wavelength and have no phase variations. Its lowefficiency can be compensated by using a material ofhigh index of refraction.

The reflection losses at the adjacent faces are alsoput into the coefficient B.

1952 APPLIED OPTICS / Vol. 27, No. 10 / 15 May 1988

(a)

C)

()

Fig. 1. Schematics of the (a) Fox-Smith interferometer; (b) multi-ple-beam Mach-Zehnder; and (c) three-mirror Sagnac.

Let 6 be the phase difference between two consecu-tive beams given by

27rLA 0'

where L is the optical length OABC, 0 is the total phasechange produced by the mirrors, and X is the vacuumwavelength. The phase change produced by the beamsplitter is implicit in the Fresnel coefficients r,r',t,t'associated with the refractions at the interface. Ac-cording to this, the resultant electric field for an inci-dent wave E0 exp(iwt) is given by

E = E exp(iwt) [ Btt' exp(ib) _1I - Br exp(ib)-rwhere we assume that the number of interfering beamsis infinite.

Since the interface is nonabsorbing this produces anormalized flux density given by

I [(r - B)2 + 4Br sin2 (6/2)][(1 - Br)2 + 4Br sin2 (5/2)]

(1)

(b) /1 9'a 0~~~~

Fig. 2. Schematics of (a) ring laser analog of the FSI, (b) using anair-glass boundary as a beam splitter. Angles are exaggerated.

t.0*

2 8

(a) (b)

This function is described in Figs. 3(a), (b), and (c).Particularly, when B approaches the value of r, Eq.

(1) takes the form of the complementary Airy function,which is essentially the instrumental function of theFSI7 and also of the setup shown in Fig. 1(b). When Btends to unity, i.e., losses vanish, I tends to 1, and thefringes disappear.

The fringe parameters are calculated immediatelyfrom Eq. (1). Thus

Imax = [Br +J2 for a = (2m +1)r,

min [B-r 2 for = 2m7r.

The contrast factor defined as the ratio of maxima tominima of intensity is

C= [(Br - 1) - (B + r)12(Br+ 1)(B-r)

Clearly, the minimum intensity at the center of a darkfringe tends to zero as B nears the value of r. Conse-quently, the contrast factor tends to infinity as shownin Fig. 4.

The sharpness of the fringes is conveniently mea-sured by their average width defined by the widthbetween the points on either side of the minimum

1.0

0.5

0.08 0(c)

Fig. 3. Fringe profiles as a function of the field reflectance: (a) for B = 0.9, (b) B = 0.7, and (c) B = 0.5.

15 May 1988 / Vol. 27, No. 10 / APPLIED OPTICS 1953

Zir

B=0.5

C.25 .36 .49 .64 RI R

30

B =.7 5 .7 .9 .9

15

1.0 r.4 .5 .6 .7 .8 .9 1.0

Fig. 4. Contrast vs field reflection coefficient for B = 0.5, 0.7, and0.9.

where the irradiance takes the average value of themaximum and minimum 8 as shown in Fig. 5.

According to this, the fringe finesse is obtained:

F= r 1 mw9 71/2 * (2)

2 i 2(B'r 2 + 1)J

This function is shown in Fig. 6 with the ideal finessecurve of a F.P. etalon. Note that for a given value of Bthe corresponding curve increases monotonically withr, lying above the F.P. curve for values of r smaller thanB. The curves intersect, and then the F.P. curve in-creases more rapidly. The exact value of r at which theintersection occurs can be calculated from Eq. (2) andthe ideal F.P. finesse curve given by 7rr/(1 - r2).

In Fig. 6 the abscissa of the intersection points is inthe vicinity of the B value, because the criteria used todefine the fringe width are different. If we had cho-sen, the half-power point to define the finesse, as in thecase of the F.P. curve, they would have intersected at r= B, because in the interferometer B plays the role ofthe amplitude reflection coefficient of the second sur-face of the etalon, while r is that of the first one. Theimportant difference is that B comprises both the inev-itable losses of every real setup and those that can becontrolled to modify the form of the fringes. Fromthis viewpoint, a device made up of relatively lossycomponents can yield fringes characterized by minimaof zero and very high visibility by adjusting the valuesof r and B, so that their magnitudes are equal or at leastvery close to each other. The finesse is affected moreseverely by the increase of the losses, and it is propor-tional to the magnitude of the product Br as seen in Eq.(2).

Various methods can be implemented to introducelosses in a controlled fashion. Optical elements can beinterposed in the path of the interfering beams thattake out a certain amount of energy each time lighttraverses them. Generally, these elements increasethe optical path of the closed trajectory ABCO and,therefore, can be used simultaneously to tune the out-put of the interferometer. This may be consideredadvantageous or undesirable depending on the use inquestion. Tuning the output of the interferometer

mo x. a.,,,, >

(Et - I .a f1 (2m+1br

Fig. 5. Average width of a fringe is defined as the width at which theirradiance takes the average value between the maximum and mini-

mum I'.

N .3650 -

40

30

20

10

.6

.49 .64 .81

.7 .8 .9

.98 R50

40

30

)0 20

_ 10

.99 r

Fig. 6. Ideal finesse curves for various values of B, from 0.75 to 0.99,and F.P. (dashed line). The intersection points with the F.P. curve

have abscissas at -r = B, respectively.

without altering the value of B can be done simply byenlarging the geometric path ABCO.

A plane parallel plate produces reflection losses thatcan be varied by changing the angle of incidence oflight on it. If the lateral displacement suffered by thelight is not desired, another similar plate can be usedwith opposite tilt to cancel said displacement. Eachplate contributes an increment of the optical path bythe amount t(gi cos' - cosI), where t is the thickness ofthe plate, ,u is its refractive index, and 1,1' are theexternal and internal angles of incidence, respectively.The incidence of p-type polarized light at the polariz-ing angle ensures zero losses to a first approximation,and from that level they can be gradually increased.Two methods we can think of that give rise to absorp-tion losses are now described.

Two thin wedges can be arranged to form a planeparallel plate, much like a Babinet compensator,whose thickness may be changed by displacing themlongitudinally. The wedges can be made from absorp-tion filters with suitable density and the appropriateangles to conform requirements.

A variable absorption filter can also be obtainedfrom a liquid solution contained in a cell formed by

1954 APPLIED OPTICS / Vol. 27, No. 10 / 15 May 1988

plate parallel walls. Thus the absorbed energy de-pends on the concentration of the solution. Presum-ably this method enables us to produce primarily ab-sorption changes with relatively small changes of thepath length, since the refractive index suffers littlechange.

These schemes are suggested as a means to intro-duce losses in a continuous and easily controllable way.However, it is expected that many others can be suitedto the specific situations of the user.

Summarizing: We have analyzed the response of aring laser analog of a passive Fox-Smith interferome-ter to monochromatic plane wave illumination. Thedevice relies on the use of controlled flux losses tomodify the form of the fringes in a single output in-stead of the normal two-pattern device.

The fringe parameters are a function of the reflec-tance of the beam splitter used in the setup and theround-trip energy loss. Fringe finesse is comparablewith that of F.P. interferometers, but an enhancementcan be attained by sufficient reduction of the loss level.Extremely high-contrast values require only appropri-ate matching of the above-mentioned coefficients.

In principle, the device works as a flexible complexreflector by which one can selectively discriminatespectral lines, for example, according to the spectralabsorptance of the element L. Conversely, an analysisof the output power can be used to obtain informationabout the spectral absorptance of element L.

This work was carried out while the author was onsabbatical leave at Centro de Investigaciones en Op-tica, A.C., Mexico.

References

1. P. W. Smith, "Stabilized, Single Frequency Output from a LongLaser Cavity," IEEE J. Quantum Electron. QE-1, 343 (1965).

2. A. G. Fox, U.S. patent 3,504,299.3. P. W. Smith, "Single Frequency Lasers," in Lasers, Vol. 4, A. K.

Levine and A. J. De Maria, Eds. (Marcel Dekker, New York,1976), Chap. 2.

4. P. W. Smith, "Mode Selection in Lasers," Proc. IEEE 60, 422(1972).

5. D. C. Sinclair, "A Confocal Longitudinal Mode Selector for SingleFrequency Operation of Gas Lasers," Appl. Phys. Lett. 13, 98(1968).

6. G. Hernandez, Fabry-Perot Interferometers (Cambridge U. P.,London, 1986).

7. Reference 3, p. 102.8. This criterion is considered more appropriate in this case due to

the form of the fringes. For the F.P. etalon conveniently charac-terized by high surface reflectance the loss level is comparativelylow. The intensity maximum is close to 1, and the minimum isclose to 0 in the transmission patterns. (In practice these limitsare never reached, since there is always a certain amount ofenergy absorbed.) In this situation, choosing the half-powerpoints to determine the fringe width is fair and justifiable. In ourcase, the maxima and minima of intensity can be relatively farfrom 1 and 0, and yet the fringe width can be comparatively small,as illustrated in Figs. 2(a), (b), and (c). Thus the half-powercriterion is unfair at best and sometimes not applicable becausethe maximum may be less than one-half or the minimum greaterthan one-half. Furthermore, the latter situations can be compat-ible with those involving high finesse and a contrast factor, as canbe seen in Eqs. (4) and (6). The chosen criterion coincides withthe conventional one when the maxima and minima approach 1and 0, respectively, for then the average value tends to one-half.

Books continued from page 1942

Pattern Recognition in Practice, II. Edited by EDZARD S.GELSEMA and LAVEEN N. KANAL. North-Holland, ElsevierScience Publishers, Amsterdam, 1986. 572 pp. $79.25.

This volume contains a collection of forty-five papers presented atthe Pattern Recognition in Practice II conference held in Amster-dam in June 1985. The first conference in this series was also held inAmsterdam in 1980, and the next is due for May 1988.

As the title suggests, most of the papers focus on practical uses ofpattern recognition. This field is very broad, and so the book isdivided into the following topical sections: Image processing tech-niques; Knowledge based and model driven systems; 3-D reconstruc-tion; Applications; Feature extraction, classification and mapping;Population classification; and Pattern recognition and artifical in-telligence. North-Holland has done a fine job of printing the papersdirectly from the original manuscripts, and a useful subject index isincluded. Not surprisingly, most of the papers come from Europe,especially The Netherlands and West Germany. Indeed, some U.S.readers may be unaware of the extensive European contributions toimage analysis and pattern recognition. This text ought to serve as auseful introduction to those readers. Quite a large proportion of thepapers deal with medical imaging applications.

Before looking at some of the papers individually, it is worthpointing out that many of the papers are sure to be heavy going formost of us. Even someone with a background in image processing isunlikely to understand many of the papers without doing extensivereading of the references. The papers are quite specialized, and soreaders will probably only appreciate those papers in their field ofexpertise. In addition, it is a little disappointing that the collectioncontains no review or tutorial papers.

An example of the problem discussed above is the paper by J. J.Gerbrands, E. Backer, and W. A. G. van der Hoeven on edgedetection by dynamic programming. This paper compares the per-formance of the dynamic programming approach to several textbookedge operators using the well-known Abdou and Pratt criteria foredge detector quality. Despite the impressive results for edges innoise, the reader who is new to this technique would almost certainlybe forced to view this paper as a stepping stone to the earlier worksreferenced.

The latter half of the volume contains several fascinating papersdealing with pattern recognition applications. T. Kaminuma, R.Minamikuwa, and I. Suzuki describe a hardware and softwaresystem for reconstructing the development of cell nuclei from aseries of images which spatially and temporally resolve developingembryos. Two-dimensional image slices through the eggs are re-corded by varying the focal point and depth of focus of the micro-scope system. A sequence of these slices recorded at 80 frames/minis then processed for extracting such information as nuclei type andcell lineage. Classification of biological macromolecules recorded inextremely noisy electron micrographs is considered by M. van Heel.The goal here is to take a large set of micrographs and average themfor increasing the SNR. Each micrograph is aligned with respect tothe others by linear and rotational cross-correlation. Data compres-sion and noise smoothing are achieved by computing the principalcomponents of the image set. These are then used to classify eachimage into one of several classes according to a hierarchical ascen-dancy scheme. B. Dubuisson, P. Malvache, and D. Grenier usestatistical pattern classification in monitoring nuclear reactors.The major problem here is the lack of real data with which toexperiment, since (thankfully!) catastrophic failures are extremelyrare. Nevertheless, the authors have implemented a pattern recog-nition system on-line in a nuclear reactor and report good results.Signals from the reactor are analyzed by three different classifica-tion schemes working in parallel, and the results are fed to an expert

continued on page 1966

15 May 1988 / Vol. 27, No. 10 / APPLIED OPTICS 1955