Block Defocused Spherical Fabry-Perot Interferometer

Anton Persin and DaliborVukicevic

A solid Fabry-Perot interferometer with a quasi-confocal resonator has been constructed. The essentialcharacteristics of this interferometer are discussed in relation to its advantages for spectral analysis.These advantages include high resolving power (>107) with satisfactory instrumental finesse (>30) instatic mode. Experimental results show that this instrument can be used for analysis of spatial coher-ence along the cross section of the incoming light beam.

Introduction

A Fabry-Perot interferometer with spherical mir-rors has been widely used for about 15 years.' Incomparison with a plane-parallel Fabry-Perot inter-ferometer, the spherical Fabry-Perot interferometer(hereafter designated SFP) has the advantage ofbeing more suitable for very high resolution spectros-copy. Whereas for a plane-parallel interferometerthe relationship between light power and resolutionhas the form of a constant product, for a sphericalFabry-Perot interferometer the relationship betweenthese two quantities is expressed in the form of aconstant quotient.2 ' 3 This specific characteristic ofthe spherical Fabry-Perot interferometer makes pos-sible the use of high resolving power at low levels ofincident light. Bradley 4 was first to show that Con-nes's confocal interferometerl, 2 can be modified byslight defocusing into an instrument with almost lin-ear dispersion over a few interference orders. Thedispersion of a plane-parallel Fabry-Perot interfer-ometer decreases with the second power, while in thecase of a confocal SFP interferometer it decreaseswith the fourth power of the radius of the inter-ference rings. The defocused SFP interferometer(DSFP) has been successfully used as an instrumentfor high resolution spectroscopy of pulsed lasersources.3-i Due to its versatility it has been widelyapplied in various experimental arrangements. 7 -9

It is well known that a plane-parallel Fabry-Perotinterferometer requires accurate adjustment and per-manent control of mirrors to attain very high paral-lelism. This, however, imposes certain restrictions

to its application. To overcome this difficulty, Pe-terson and Yariv10 have developed a plane-parallelFabry-Perot interferometer shaped as a glass blockwith mirrors evaporated on its plane-parallel surfac-es. Auth" has shown that such an instrument canbe used not only in static mode but also in scanningmode, provided the glass block is being heated orcooled, respectively.

In the present paper we intended to develop andtest a block defocused SFP interferometer whichwould have characteristics of a spherical interfer-ometer and which would be reliable and simple touse.

Review of Theoretical Considerations of a DefocusedSFP Interferometer

High-resolution DSFP interferometers have beenextensively studied by several authors.1 -6 "12 -2' Herewe give a brief review of the most important resultsobtained.

Interference rings of a DSFP interferometer are lo-calized in the vicinity of the plane of symmetry A-Abetween two mirrors (Fig. 1). The rings are causedby third-order aberrations intensified by multiple re-flections.'

In a paraxial approximation, a directly transmit-ted ray and a four times reflected ray are spatiallyidentical in the meridional plane. Taking into ac-count the geometrical factors, the following expres-sion is obtained for the difference in optical paths ofthese rays:

W = 4(R + p) - 4pr2 /R2- / R3.

Both authors were with the Institute Ruder Boskovic, 41000 Za-greb, Yugoslavia, when this work was done; D. Vukicevic is now atthe Institute of Physics of the University, 41000 Zagreb, Yugosla-via.

Received 28 December 1971.

(1)

Here, R is the radius of curvature of the two mirrors,p is the deviation from confocality, and r is the zonalradius. By adjusting the mirrors so as to attain alow negative value of p, the dispersion dr/dX will bequasi-linear in the vicinity of the zonal radius r de-termined from Eq. (1) by

February 1973 / Vol. 12, No. 2 / APPLIED OPTICS 275

AM2

Fig. 1. Geometry of a DSFP resonator.

r,2

+ (2/3)pR = 0. (2)

By analogy to the plane-parallel Fabry-Perotinterferometer, the instrumental characteristics ofthe DSFP interferometer are determined by the Airyfunction.3 ' 20 However, if the wavefronts are calcu-lated by repetitive numerical tracing of the wavenor-mals, an instrumental function is obtained thattakes into account the wedge etalon effect. In fact,a spherical Fabry-Perot interferometer can be pic-tured as a rotationally symmetrical wedge whoseangle increases from zero on the axis of symmetry upto a maximum value for the maximum zonal radius.This leads to the appearance of spurious interferencefringes.19 -2 ' Bradley and Mitchell19 have shownthat secondary fringes in a DSFP interferometerwith mirrors of 10-cm radii of curvature can be ob-served only outside the region of linear dispersion.In the case of smaller radii of curvature secondaryrings also occur inside this region.

Spurious rings can be observed even if the focusingsystem or the recording camera are not exactly fo-cused onto the surface on which the observed ringhas been formed. Because of this, the identificationof rings presents considerable difficulties.

Whereas a plane-parallel Fabry-Perot interferome-ter is very sensitive to tilting of the mirrors, the mir-rors of a DSFP interferometer should be located atan appropriate separation if the mode matchingtechnique is to be avoided.

Various transversal modes excited by an incidentelectromagnetic wave in the spherical resonator of aDSFP interferometer give rise to the appearance ofspurious peaks. These can be avoided in two ways.By the mode matching technique the geometricalproperties of the Gaussian beam of incident light arematched with the geometrical properties of the basicmode TEMooq of the interferometer. An alternativeway of avoiding spurious peaks is to locate the mir-rors of the DSFP interferometer in a nearly confocalposition, so that spurious peaks are overlapped bypartial degeneration of transversal modes. Underthe assumption that partial degeneration shouldcause overlap among M transversal modes and thatthe interferometer should have finesse N, the param-

eter of deviation from confocality would satisfy thecondition

1E1 < 1/N(M + 1), (3)

wherem + n < M

and

e = (d - R)/R.

Here, m and n are integral indices of transversalmodes, and d is the separation between the sphericalmirrors.

Unlike a plane-parallel Fabry-Perot interferom-eter, a SFP interferometer is not translationally invar-iant. As illustrated in Fig. 1, a ray of light I from arotationally symmetric input beam travels along thepath of a ray I* entering at a diametrically oppositepart. Ray I*, after two successive reflections insidethe resonator, becomes spatially identical with ray I,and interference results. The retardation of the di-ametrically conjugated ray following the incomingray would be decreased by a factor of 2, comparedwith the retardation of the ray that has suffered fourreflections. Accordingly, the spatial coherence alongthe cross section of the input beam will give rise toalteration in intensity of the interference maxima.The intensity of the interference maxima will in-crease for an even integer, while it will decrease foran odd integer.

When the SFP interferometer is used for analysisof spatial coherent light whose degree of spatial co-herence is i, its free spectral region and instrumentalfinesse are increased by a factor of 2.

Design of a Block DSFP Interferometer

The block etalon was made of optical glass with anindex of refraction n = 1.5163. The plates of theetalon had the shape of spherical surfaces. The ra-dius of curvature of both surfaces was 5.0434 cm.The surfaces were polished and then coated withmultilayer dielectric coating with reflectance in therange up to 0.99 at a wavelength of 6328 A. Thecoefficient of reflection for a multilayer dielectricmirror2 3 evaporated on an etalon of index of refrac-tion ne in a medium of index of refraction n1 is in-variant to the substitution of ne by n1 and ni by nedue to the symmetry of the relation for this coeffi-cient.

The separation between the vertices of the spheri-cal surfaces was 5.0325 cm, or the departure fromconfocality was -0.0109 cm. The solid state DSFPinterferometer was mounted on a flat plate held inposition by three micrometric screws (Fig. 2), whichbear on the optical axis of the interferometer into theaxis of the input light beam.

Testing and Calibrating of the Interferometer

The advantage of the D$FP interferometer is thatits mirrors are permanently in adjustment; in fact,

276 APPLIED OPTICS / Vol. 12, No. 2 / February 1973

surface perpendicular to the direction of the inputbeam. Figures 4(a) and 4(b) were taken in the sameway but by focusing onto the central rings. A densi-togram of the interferogram 4(a) is shown in Fig. 5.Figure 6 shows the dependence of the frequency ofinterference rings on their radius. This dependenceis linear over at least two orders of interference.

The interferometer described in this paper has afree spectral range of 985 MHz. In the region of thezonal radius re = 2.28 mm (region of quasi-lineardispersion) the inversed dispersion was dX/dr = 1.24x 10-20 A/mm and the resolving power R = 2 x107, with finesse N = 35.

The absorption of the given glass material per cen-timeter of path at a wavelength of 6328 A was 0.012cm-'. It appears that the absorption affects the in-strumental function due to the reduced effectivenumber of interference beams. A modification ofMitchel's computer program24 is in progress in thislaboratory. It is hoped that this might contribute tothe theoretical improvements of the instrumental fi-nesse of the DSFP interferometer.

Fig. 2. A block DSFP interferometer in a PVC shield. The op-tical axis is adjusted to the axis of the input light beam by means

of micrometric screws.

these are fixed to the required position during theshaping of the spherical surfaces and remain perma-nently adjusted.

It has been verified experimentally that this ad-justment is satisfactory. Similarly to any otherSFP, a block DSFP interferometer can be used forthe determination of the degree of spatial coherenceof the incident light. This can be attained by re-cording interference rings with the full input aper-ture [Fig. 3(a)] and with one half of it blocked off[Fig. 3(b)]. The degree of spatial coherence was cal-culated from the intensity of rings of the same orderof interference relative to each other. Figure 3(a)was taken with the full input aperture; it shows adecrease in intensity of rings of every second order,in contrast to the rings shown in Fig. 3(b), whichwere taken with the semicircular input aperture.The modal structure of light of a laser having twolongitudinal modes separated by 468 MHz was ana-lyzed. Figures 3(a) and (b) show the appearance ofsecondary maxima. The intensity of secondarymaxima is appreciably high in the central region,since in both cases the camera was focused on thesharp peripheral rings. The interference image doesnot occur in one plane but is localized on a convex

(b)Fig. 3. System of interference rings of a quasi-confocal DSFPinterferometer. The modal structure was analyzed with two lon-gitudinal modes (a) with the full aperture of the beam; (b) with

one-half of it blocked off.

February 1973 / Vol. 12, No. 2 / APPLIED OPTICS 277

1 2 3 4r(mm)

(b)Fig. 4. In the central part the dispersion is almost linear. Twolongitudinal modes were taken over two orders of interference:(a) the free spectral range is increased by a factor of 2 whenusing a beam of coherent light and the full aperture; (b) same as

(a) but with the semicircular aperture.

It

0 1 2 3 4r(mm)

Fig. 5. Densitogram of interferogram 4(a).

Fig. 6. Dispersion of light is constant over at least two orders ofinterference.

Conclusion

The characteristics of a DSFP block interferome-ter show that this instrument can be successfully ap-plied for the investigation of the modal structure andspatial coherence of laser beams. It can be used as areference instrument suitable for any experimentalarrangement. The high resolving power and satis-factory instrumental finesse provide, through the op-tical system for microphotography, direct insightinto the instantaneous modal structure and spatialcoherence of laser light. Because of these advan-tages it can be employed for diagnosis of gas andpulse lasers of high radiation power.

References

1. P. Connes, Rev. Opt. 35, 37 (1956).2. P. Connes, J. Phys. Rad. 19, 262 (1958).3. M. Hercher, Appl. Opt. 7, 951 (1968).4. D. J. Bradley, Nature 215, 499 (1967).5. D. J. Bradley, Opt. Acta 15, 431 (1968).6. D. J. Bradley and C. J. Mitchel, Appl. Opt. 8, 710 (1969).7. S. F. Jacobs, J. N. Bradford, and J. W. Berthold, Appl. Opt.

9, 2477 (1970).8. D. M. Paul and D. A. Jackson, J. Phys. E. 4, 170 (1971).9. D. A. Jackson and D. M. Paul, J. Phys. E. 4, 173 (1971).

10. D. G. Peterson and A. Yariv, Appl. Opt. 5, 985 (1966).11. D. C. Auth, Appl. Opt. 8, 1125 (1969).12. D. A. Jackson, Proc. Roy. Soc. (London) 263A, 289 (1961).13. D. R. Herriott, Appl. Opt. 2, 865 (1963).14. D. Herriott, H. Kogelnik, and R. Kompfner, Appl. Opt. 3, 523

(1964).15. R. L. Fork, D. R. Herriott, and H. Kogelnik, Appl. Opt. 3,

1471 (1964).16. P. Toschek and B. Wolf, Z. Angew. Phys. 22, 185 (1967).17. J. R. Johnson, Appl. Opt. 6, 1930 (1967).18. J. R. Johnson, Appl. Opt. 7, 1061 (1968).19. D. J. Bradley and C. J. Mitchel, Philos. Trans. Roy. Soc.

London 263, 209 (1968).20. D. J. Bradley and C. J. Mitchel, Appl. Opt. 8, 707 (1969).21. M. Hercher, Appl. Opt. 8, 709 (1969).22. K. Kinosita, J. Phys. Soc. Japan 8, 219 (1953).23. M. Born and E. Wolf, Principles of Optics (Pergamon, Lon-

don, 1965).24. C. J. Mitchel, Private communication.

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