Interference profiles with multiple spherical waves : general case

As is well known, the analysis of the interferential processes in different optical structures must include a solution of the geometrical aspect that relates the fringe localization in the interferential pattern with the intensity distribution across it. In the case of multiple-reflection interferometers, such as the Fabry-Perot interferometer, several kinds of experimental plications.1The geometry and the intensity profile of the interference pattern produced by such optical structures are well Jmown for the cases in which mirrors are at normal alignment con­ ditions. Recently, Brossel4 and Aebischer5 researched the geometry and the intensity profile of the interference pattern produced by a misaligned Fabry-Perot interferometer, and Rogers6 developed a program to compute the profile of multiple-beam Fizeau fringes. Our paper follows the methodology of Refs. 4duces the more general conditions for the calculation of the optical-path difference, the impulse response, and the in­ tensityThese general conditions are as follows: (1) Three different refractive indices in the entire optical structure are considered. (2) The light source can be placed near the interferometer, servation plane can also be placed near the interferometer. These conditions correspond to the generalization of the twoa compact device for interferometric holography.8 In Section 3 the theoretical results obtained in Section 2 ported, ensuring the consistency of the methodology employed and of the results. Section 4 describes the experimental test of the theoretical results. .


INTRO DUCTIO N
As is well known, the analysis of the interferential processes in different optical structures must include a solution of the geometrical aspect that relates the fringe localization in the interferential pattern with the intensity distribution across it.
In the case of multiple-reflection interferometers, such as the Fabry-Perot interferometer, several kinds of experimental -plications.1- The geometry and the intensity profile of the interference pattern produced by such optical structures are well Jmown for the cases in which mirrors are at normal alignment con ditions.
Recently, Brossel4 and Aebischer5 researched the geometry and the intensity profile of the interference pattern produced by a misaligned Fabry-Perot interferometer, and Rogers6 developed a program to compute the profile of multiple-beam Fizeau fringes.
Our paper follows the methodology of Refs.4-duces the more general conditions for the calculation of the optical-path difference, the impulse response, and the in tensity-These general conditions are as follows: (1) Three different refractive indices in the entire optical structure are considered.(2) The light source can be placed near the interferometer, servation plane can also be placed near the interferometer.These conditions correspond to the generalization of the twoa compact device for interferometric holography.8 In Section 3 the theoretical results obtained in Section 2 ported, ensuring the consistency of the methodology employed and of the results.
Section 4 describes the experimental test of the theoretical results.
. We consider a wedge with plane partial reflecting surfaces S i and S 2 forming an angle a. Figure 1 shows a cross section This circumference is in the plane containing the normal to the surface Si and the point source 0*, and it is.normal to the edge of the wedge (in Fig. 1, the x -y plane).7 On the other hand, as 0 \ is an image of O'0, we can write Besides, as is shown in Fig. 2, the segment 0*1 is parallel to the normal to surface Si, and we may write for any light ray leaving the source 0* at an angle < t > 0*S sin < f > = S I = O'qS sin < $> . (4) Combining Eq. ( 4) with Snell's law, ri\ sin 4> = ri2 sin <£', produces the following result: Then, in general, we may write ( 5) Going back to Fig. 1 and using Eq. ( 5) for point C (the in tersection with surface Si), Eq. ( 3) for a ray that is reflected q times inside the wedge, and Eq. ( 6) for point T (the inter section with the surface S 2), we find that the length of the optical path [q] is given by . Tj-5 Ta -O0c 4" 0 q T 4 n 30 qP , ( 7) of the dihedral angle between S i and S 2, where is the vertex of a, 0* is a light point source, and Mi, n 2, and n 3 are the re fractive indices of the media.We take into account the multiple reflections of rays in the medium characterized by «2 placed between S i and S 2.
There are rays (such as 0 ) that arise from O* and travel through the wedge without suffering reflections, and there are others (such as q) that are reflected q times on each surface before leaving the cavity.
-path lengths of rays such as [q] and [0] when both arrive at the same observation point P(x , y , z), interfering with each other.The optical path r q for the [qj ray is and, in the same way, we obtain for the optical path [0] Let 0* = ( | , 0,0) and 0'¡ = (jc-| 0) be the coordinates of the point light source and their images given by refraction and reflection, respectively, and 0¿ = ( i ñ I = ( i ¡ , -y 'i, 0).Then, from Fig. 1 and using Eq. ( 6), we obtain Defining the coordinates T = = z q ), we find it possible to calculate?
where h is the width of the wedge in the x -z plane (y = 0).Points P, T, and Q are not necessarily in the x -y plane be cause the last expressions are valid for any angle at which light impinges upon the wedge.Now, in order to calculate both optical paths [q] and [O] and their difference, we must know
(2) fraction at Si, and let 0[ be the successive image positions of Oq, O'i.........Oi-i generated by multiple reflections at the surfaces.Also, let O-tion at S 2. The images 0Ó,..., 0\ of the source 0 * , which can be interpreted as virtual sources in the medium of index n 2, lie on a circumference of radius l, whose center is fi, and they are separated from one another by an angular distance 2a.
(1) The coordinates of the image sources O'i and 0¿ as functions of the wedge angle a and the position of 0*.
(2) The coordinates of points Q and T as functions of the positions of the observation point P(x, y, z) and 0 ¿.
Let h be the width of the wedge in the x -z plane and y be the distance between Q, and the origin of the coordinates, as illustrated in Fig. 2; then h = y tan a.
As we already know that the coordinates of the source are O* = ( | , 0,0), easy calculations give Before using this last expression for searching the opticalpath difference, we can make several approximations taking into account the different orders of magnitude that play a significant role and then subtract rq from ro.
The order of magnitude of the main parameters involved is as follows: The width h can measure several millimeters.-The angle a of the wedge is small.It is of the order of 10-4y'q = y -Cv " Coicos 2qa + x'0 sin 2q a ► z q = 0 i for O'q = (-Xg, -y'q, 0),  7) and (8) as functions of OqP, obtaining the general expression that is valid for q = y = h/tan a ~ h i a « 100 m. q, the number of reflections, may be several times 10.2qa « 2 X 10 X 10"3 « 0 (10~2).
As we want to obtain a general formula for the optical path that is valid both when the source and the record are at infinity and when the source and the record are near the interferom eter, we will consider the possibility that p ranges from 10-1 m to infinity.Then we can replace sin a with a and cos a with 1 and neglect h 2 compared with p2 and also a 2 compared with 1; then Eq. ( 7) is reduced to and ro is obtained by replacing q with zero.
where After finding the difference between r 9 and r 0 and rear ranging terms, we obtain the following expression for the optical path: The phase difference corresponding to each optical path r q is < f> q = (2x A ) r 9, where A is the wavelength in free space.
Suppose that at plane x* (see Fig. 3) there is a plane object th at is illuminated by monochromatic light.Let 1/(0*) be the complex field amplitude of the object at plane x*, and let U (P ) be the complex field amplitude at point x that contains observation point P.
Using the linearity property, we can express the complex amplitude U as function of U*, that is, where -path deference be tween two rays arriving at the same observation point P, one mitted after q reflections inside the wedge, when we make the approximations h 2 « p2 and a 2 « 1 .

Impulse Response ,
In the preceding section we calculated the optical path rq for light coming from source 0 *, having q reflections inside the cavity and reaching observation point P.
where h{P, 0*) is the amplitude at P coordinates in response to a point-source object at 0* and h is the impulse response of the optical system.We calculate the complex amplitude of the field at point P in plane x (refractive index 713) when the interferometer is illuminated by monochromatic light coming from a point source placed at 0 * ( | , 0, 0) in plane x* (refractive index ni).
As we have already said, the incident field amplitude U¡ at point P / of plane x / of the system can be written as where ro/ is the optical path from 0* to P /(0, £, tj) without reflections inside the instrument and ro/ is the geometric distance from 0* (the image of 0* calculated at 713) to P j (see Fig. 4).
Let if(£, 77) be the pupil function a t plane x/.The field amplitude lf¡ at the output of the system (and also at plane x /) will be

* <7=o rqi
where rqi is rq of Eq. ( 14) for point P / = -Fresnel principle, we find that the field amplitude at point P of x is and rqi is given from Eq. ( 14) for x = 0, y = £, and z = q.

Intensity-Profile Distribution
In order to corroborate Eq. ( 15), we must calculate the transm itted amplitude and intensity through the wedge and the recorded intensity at point P.
Taking into account that observation point P is placed a finite distance from the wedge and supposing that the pupil is large, we can write the transmitted vibrations as These results correspond to the well-known concentric rings, whose theoretical intensity profile, calculated using Eqs.( 18) and ( 19), is shown in Fig. 5(a If the observation is made in a position near the axis (sin2 0 « 1 ), the optical-path difference has the expression With that intensity equation it is possible to plot the theo retical intensity profile for comparison with the experimental microdensitometric record of the interference pattern.

ANALYSIS OF LIM ITING CASES
Equation (15) reproduces the theoretical expression of ô for the limiting cases already known: (1) When a = 0 and n i~n 2 = n3 = n, the optical-path difference given by Eq. ( 15) take the form We can also neglect terms in p cos 0 /(x 2 + y 2 + z 2)1/2, but, in the case of offaypf(x2 + y 2 + z 2)1/2, obtaining (28) which corresponds to interference patterns whose intensity profiles have symmetry only with respect to the z axis, as is shown in Fig. 5(b), which is the theoretical profile for z = 0.
We can add a comment for the case of close axis observation.If y « 0, Eq. ( 28) coincides with Eq. ( 22), which means th at a small interferometer misalignment does not modify the structure of the central rings.This is also true if y ~ z « 0 and n = «i = n 3 zi2 = n' in Eq. ( 15).In this case, we again ob tain Eq. ( 22).

EXPERIM ENTAL RESULTS
The theoretical intensity profiles computed from Eq. ( 19) were plotted using an IBM/360 computer with a 2250 visual unity.Such plots were compared with the experimental microdensitometric records obtained for typical values of the pa rameters involved.
Figure 6 is a schematic of the experimental setup used.For the simulation of the light point source O* we used a cw 1.5-mW He-Ne laser and a beam collimator.Those cases with n\ = were tested with a glass fixed interferometer of 7.6-mm width, and those cases with n\ = = -13, with an air-scanning Fabry-Perot interferometer.For recording the interference pattern we used Ilford PAN-F film (50 ASA).
The intensity densitometric profiles of the photographic records were scanned with a Grant automatic microdensi tometer comparator.
We have observed an adequate coincidence between the theoretical and the experimental profiles.Figures 7(a

ACKNOW LEDGM ENTS
The invaluable support provided by the Subsecretaría de Ciencia y Tecnología, Argentina, is gratefully acknowl edged.
0740-3232/84/050495-07$02.00 -PATH DIFFERENCE, THE IM PULSE RESPONSE, A N D --path difference as well as the impulse ferentikl pattern produced by the most general case of a misaligned Fabry-Optical-Path Difference -path difference between rays passing through a misaligned Fabry-essary to characterize the optical structure under consider ation.

Fig. 1 .Fig. 2 .
Fig. 1.Cross section of the wedge, showing the 0* real point source and its virtual images.
Q are defined by the intersections between the y -z plane (x = 0) and the segments OqP and OqP, respectively; such intersections give and xq = 0 Combining Eqs.(11) and (12) and introducing the result into Eqs.(10) and using Eqs.(9), we can Eqs.(

Fig. 3 .
Fig. 3. Diagram showing the positions of the object and observation planes.
Figure 8(d) shows the experimental profile when x = 300 mm and