Performance evaluation of a wave-front displacement system for vectorial shearing interferometer

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Optics Communications 281 (2008) 347–355

Performance evaluation of a wave-front displacement systemfor vectorial shearing interferometer

Claudio Ramirez *, Marija Strojnik

Centro de Investigaciones en Optica, Apartado Postal 1-948, 37000 Leon, Gto., Mexico

Received 14 November 2006; received in revised form 15 April 2007; accepted 30 July 2007

Abstract

We describe a wave-front displacement system using a novel configuration of a pair of wedge prisms. The wave-front propagation,through the displacement system, is analyzed using the exact ray trace. The main advantage, of this configuration, is that the shearing(direction and magnitude) is constant from the displacement system to the image plane. The direction of the shearing depends on therelative orientation of the prisms. The magnitude of the shearing is proportional to the distance between prisms. The proportionalityconstant depends on the wedge angle and the material of fabrication of the prisms, and it gives the sensitivity to the displacement ofthe system. The deviation of the shearing due to the difference between the wedge angles is larger than that the produced by the obliqueincidence of the wave-front. The experimental results confirm the theoretical data.� 2007 Elsevier B.V. All rights reserved.

Keywords: Displacement system; Wedge prism; Exact ray trace; Vectorial shearing interferometer

1. Introduction

Optical components without rotational symmetry,including segmented and diluted mirrors, are currentlybeing used in large modern optical devices [1–4] and spacetelescopes of remote sensing [5–10]. The fabrication andevaluation of these components imply the utilization ofhigh quality, and thus expensive, reference surfaces. Onthe other hand, optical devices with one aspherical surfacereplace several spherical ones. In this case, null correctorsare used for evaluation of the aspheric surfaces [11,12].The shearing interferometry is an alternative techniquefor assessment and measurement of wave-front error [13–15]. This technique is also useful to analyze local changesin the refraction index due to variations in the medium den-sity [16,17]. It is possible to retrieve the topography or pro-file of a surface knowing the shape of the fringe pattern[18–20]. It is also used to measure the distance between

0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2007.07.043

* Corresponding author. Tel.: +52 477 4414200.E-mail address: [email protected] (C. Ramirez).

objects [21,22]. The shearing interferometry does notrequire a reference surface, because the wave-front ampli-tude is divided in two parts and compared with itself. Sometechniques do not require a reference wave-front; theyinduce a perturbation in the material under test, and thenmeasure and compare the effects [23,24]. Fig. 1 shows thatthe wave-fronts follow a similar path in a vectorial shearinginterferometer, but they undergo a (small) relativedisplacement.

The vector D~q describes the displacement of the wave-front Wd(xd,yd) with respect to the original one W(x,y).The superposition of the wave-fronts gives rise to the inter-ference pattern. This pattern carries the information of thepath difference (DW), and it can be expressed as [25]:

DW ¼ W ðxþ Dx; y þ DyÞ � W ðx; yÞ: ð1ÞIn order to shear a wave-front, prisms are used as displace-ment system to modify the direction of propagation of awave-front [26,27]. Previously, a pair of wedge prisms hasbeen used as displacement system for a vectorial shearinginterferometer [28,29]. The prisms were located in onearm of a Mach–Zehnder interferometer. Two identical

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Fig. 1. The fringe pattern is generated at the superposition region of thewave-fronts.

348 C. Ramirez, M. Strojnik / Optics Communications 281 (2008) 347–355

wedge prisms were placed in the other arm to compensatethe optical path in both arms. They used a pair of prismsinstead of a single one, to make the displacements moreprecise [30,31]. Then, the direction and the magnitude ofthe wave-front displacement depended on the rotation an-gle of each prism, the difference between them, and the dis-tance from the displacement system to the detection plane[32,33]. The sensitivity of the interferometer was chosen toobtain an adequate quantity of fringes, in the selecteddirection [34,35].

Recently, we use the vectorial shearing interferometer inorder to characterize the parts of a flame [36].

In this paper, we propose a novel configuration of a dis-placement system using two wedge prisms. The wedge sur-faces are face to face and they are always parallel. It hastwo principal advantages. The first one is that the shearingis constant from the displacement system to the detectionplane. The second one is that there is no tilt introducedin the wave-front. In the next section, we present the dis-placement system, its components, the degrees of freedom,and constrains of movement. The system is placed in one of

Fig. 2. The vectorial shearing interferometer is based on a Mach–Zehnderconfiguration. BS1 and BS2 are the beam-splitters. M1 and M2 are themirrors. The wave-front displacement system is placed in one arm of theinterferometer. Here d is the distance between prisms and x is the rotationangle of both.

the arms of a Mach–Zehnder interferometer to shear later-ally the wave-front. In the other arm, the wave-front con-tinues its path without changes, as shown in Fig. 2. InSection 3, we introduce the equations of refraction andpropagation of a ray (considering its geometrical propertyof being normal to the wave front). We use the ray traceequations, in Section 4, to examine the wave-front propa-gation through the displacement system, in order to deter-mine the shearing at the detection plane. In Section 5, weanalyze the shearing deviations induced when the dimen-sions of the prisms are different from their design valuesor when they are misaligned. In Section 6, we compareexperimental results with the theoretical prediction basedon exact ray trace.

2. Novel wave-front displacement system

The displacement system consists of a pair of identicalwedge prisms (see Fig. 3). The geometrical centers of thesurfaces (denoted S1, S2, S3, and S4, respectively) of theprisms are aligned with the rotation axis of both prisms,as indicated in Fig. 4. The rotation axis is perpendicularto the surfaces S1 and S4ðbN 1 ¼ �bN 4Þ. The wedge surfaces(S2 and S3) are always parallel and face each other,ðbN 2 ¼ �bN 3Þ.

There are 2 degrees of freedom available for implement-ing a shearing: the angle of rotation of both prisms, x, andthe distance d between prisms. The prism rotations are exe-cuted about the same axis.

3. Exact ray trace

We determine the wave-front propagation throughthe displacement system, using the method of exactray trace. This method has been used formerly toanalyze the wave-front propagation across other configu-

Fig. 3. The prisms are made of optical glass with a refractive index (n).They have cylindrical shape with thickness (a) and wedge angle a.

Fig. 4. In the wave-front displacement system bN 1; bN 2; bN 3 and bN 4 are thenormal vectors to the surfaces S1, S2, S3, and S4, respectively.

Fig. 5. A ray impinges perpendicular to the surface S1 and it crosses theprisms at r1, r2, r3, and r4.

C. Ramirez, M. Strojnik / Optics Communications 281 (2008) 347–355 349

rations of wedge prisms. We use the refraction and prop-agation equations in order to calculate the wave-frontshearing.

The refraction equation is:bP i ¼ libP i�1

þ �liðbP i�1 � bN iÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� l2

i 1� ðbP i�1 � bN iÞ2h ir� �bN i;

i ¼ 1; 2 . . . 5 ð2Þ

l1 ¼ l3 ¼n1

n2

; l2 ¼ l4 ¼n2

n1

:

The propagation equation is:

~ri ¼~ri�1 þðtik �~ri�1Þ � bN ibP i�1 � bN i

" #bP i�1 ðmmÞ: ð3Þ

The normal to any surface in terms of the wedge a and therotation angle x is:bN ¼ ðsin a cos xÞiþ ðsin a sin xÞjþ cos ak: ð4ÞHere, n1 and n2 are the refraction indices for the mediumand for the prism material, respectively; t is the distancefrom the origin to each surface, measured along the opticalaxis z. i, j and k are the unitary vectors in the direction x, y

and z, respectively. Eq. (2) expresses the direction bP i of therefracted ray by the ith surface. Eq. (3) gives the positionri(xi,yi,zi), where the ray crosses the ith surface.

4. Wave-front shearing induced by the displacement system

Fig. 5 illustrates the ray propagation from the objectplane S0 to the image plane S5. The sheared ray crossesthe image plane at r5.

We assume that the constraints of the displacement sys-tem are:bN 2 ¼ �bN 3;bN 1 ¼ �bN 4:

ð5Þ

Furthermore, we consider that the initial conditions of theray (wave-front) are:

~r0 ¼ 0iþ 0jþ 0k;bP 0 ¼ �bN 1 ¼ k:ð6Þ

We substitute the expressions in Eqs. (5) and (6) in exactray trace equations for the surfaces S1 to S5. After somealgebraic operations, the coordinates of the ray crossingthe image plane S5 are:

x5 ¼ bd cos x;

y5 ¼ bd sin x;

z5 ¼ t5:

ð7Þ

Here, b is defined as

bðn1; n2; aÞ ¼ 1�n2

n1cos affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� n2

n1

� �2

sin2 a

r0BB@

1CCA sin a cos a: ð8Þ

The magnitude and the angle of the shearing vector D~q, interms of the coordinates of r5(x5,y5,z5), are:

jD~qj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx5 � x0Þ2 þ ðy5 � y0Þ

2q

ðmmÞ; ð9Þh ¼ xþ p ðradÞ: ð10Þ

We substitute Eq. (7) in (9) in order to obtain the shearingmagnitude:

jD~qj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

5 þ y25

q¼ bd

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðcos xÞ2 þ ðsin xÞ2

qðmmÞ; ð11Þ

jD~qj ¼ bd ðmmÞ: ð12Þ

Eq. (12) denotes that the shearing magnitude of the ray isproportional to the distance d between prisms, and it isindependent of the rotation angle x. The proportionalityconstant b depends on the refractive index of the materialn2, on the medium refractive index n1, and on the wedge an-gle a of the prisms. During mathematical development wefound that bP 0 ¼ bP 4. Therefore, no tilt is introduced inthe sheared wave-front.

Fig. 8. The shearing amount increases with the refractive index of thewedge prisms.


The magnitude jD~qj and the angle h of the shearingdepend on the distance d and on the angle of rotation xof both prisms, respectively.

A specific performance for a practical prism is shown inFig. 6. The value b represents the sensitivity to the displace-ment of the system. In this case, the value b gives theamount of lateral shearing when the prisms are separated.We can modify the sensitivity changing the values of therefractive index of the material and/or the wedge angle ofthe prisms.

The shear magnitude jD~qj as a function of the distance d,for different values of wedge angle, is illustrated in Fig. 7.The rate of change of jD~qj (shearing sensitivity) increaseswith the angle a. In Fig. 8, we exhibited the change in theshearing sensitivity when the material of the prismschanges. In this case, the shearing sensitivity increases withthe refractive index.

Finally, Fig. 9 presents the sensitivity of the displace-ment system as a function of the wedge angles, with therefractive index of several materials as parameter. Thisgraph is very useful when the displacement system isdesigned for a specific sensitivity. We can choose the sys-tem sensitivity for any prism material, varying the wedgeangle.

Fig. 6. The value of b is the slope of the straight line, and it represents thesensitivity to the displacement of the system.

Fig. 7. The shearing amount increases with the a angle of the wedgeprisms.

Fig. 9. The sensitivity to the displacement as a function of the wedge angle(a), and prism material (n).

5. Analysis of the deviation of the wave-front shearing

The two primary deviation sources of the sheared wave-fronts are: deviations by differences between the wedgeangles and oblique incidence of the wave-front on theshearing system.

5.1. Deviations by differences between the wedge angles

The dimensions and angles of any prism are bound bytolerance margins. These mainly depend on severalaspects: the kind of material, the prism shape, the technol-ogy, and fabrication process. The literature reports thatthe typical tolerances of prism angles are ±5 00. Fig. 10shows the ray when it is propagating through prisms withdifferent wedge angles. The ray impinges perpendicularlyon the prism A ðbP 0 ¼ �bN 1Þ. When the ray crosses fromthe prism A to the prism B, it is not deviated becausethe prisms are in contact (d = 0mm). The difference onwedge angles (aA 5 aB), produces a deviation of the nor-mal to the surface S4ðbN 4 6¼ bN 1Þ. The ray impinges on S4

with an angle hi, equal to the difference Da between thewedge angles

hi ¼ Da ¼ aA � aB: ð13Þ

Fig. 12. Deviation angle cP4 as a function of the difference Da betweenwedge angles. The direction of original ray and displaced ray are differentðbP 1 6¼ bP 4Þ. This add tilt to the wave-front.

Table 1Deviation d due to difference between prism angles Da

Difference of wedge angle Da ( 0 0) Deviation d (lm)

5 40010 80020 160030 240060 4810

Fig. 10. The ray deviation depends on the difference between wedge anglesaA 5 aB. (a) aA < aB, when the prisms are separated, the magnitude of thedisplacement vector decreases until zero and increases again. (b) aA > aB,the magnitude of the displacement never is zero, it always increases.


Finally, the deviation angle of the ray is

cP4 ¼ ht � hi: ð14ÞThe deviation angle cP4 is approximately equal to one halfof the difference between prism wedge angles Da, as illus-trated in Fig. 11.

In this case, the wave-front shearing magnitude jD~qjDa6¼0

is calculated evaluating the ray trace equations for everysurface of the displacement system. The deviation d ofthe shearing is calculated using the following expression:

Fig. 11. Deviation d as a function of distance d, for several values of Da.The deviation d is constant for any distance d between prisms.

d ¼ jD~qjDa 6¼0 � jD~qjDa¼0 ðmmÞ: ð15Þ

Here jD~qjDa¼0 is the magnitude of the wave-front shearingobtained with Eq. (12).

Fig. 12 represents the deviation of the shearing for dif-ferent values of Da. Table 1 presents the deviation valued when Da changes.

5.2. Deviations due to oblique incidence of the wave-front on

the displacement system

Another deviation source occurs when the wave-frontimpinges obliquely on the displacement system. The anglebetween the incident ray and the normal to surface S1 iscP0, as illustrated in Fig. 13.

Fig. 13. The ray impinges obliquely to the surface S1 and is propagatedthrough the displacement system.

Table 2The shearing deviation d as a function of the incidence angle cP0

Incidence angle cP0 ( 0 0) Shearing deviation d (lm)

5 �1810 �3520 �7530 �11060 �220


In the ideal wave-front displacement system, the wedgeangles of the prism are the same (aA = aB) and the prismsurfaces are parallel ðbN 1 ¼ �bN 4; bN 2 ¼ �bN 3Þ. The shearingmagnitude jD~qj is the difference between the incidencepoint of displaced ray r5, and the projection of originalray r50 (see Fig. 13). Fig. 14 depicts that the ray deviationd depends also on the propagation directionðbP 0 ¼ P 0xiþ P 0y jþ P 0zkÞ of the ray incident on the dis-placement system.

We summarize the deviation d of the wave-front shear-ing, obtained with ray trace equations, in Table 2.

Tables 1 and 2 demonstrate that the deviation, by obli-que incidence, is approximately 20 times smaller than thedeviation due to the differences between wedge angles.Therefore, we conclude that to know the shearing magni-tude with precision, the wedge prisms should be identicalin dimensions and angles.

Fig. 14. The magnitude of the vector D~q depends on the angle and direction ofsystem with components in x, z axes, (b) the ray has components in �x, z axes, (in x, y, z axes, (e) the ray has components in �x, �y, z axes.

6. Experimental verification

In order to compare our theoretical results with the realshearing of the displacement system, we implement in lab-oratory the experimental setup illustrated in Fig. 15. Theillumination source is a He–Ne laser (k = 633 nm), with a

the incident ray. (a) The ray impinges on the surface S1 of the displacementc) the ray has components in y, z or �y, z axes, (d) the ray has components

Fig. 15. Schematic layout of the experimental setup to measure the wave-front shearing by the displacement system.


coherence length of 20 cm. The laser divergence isdecreased with a positive lens. We use a high resolutioncamera to record the displaced ray with high precision.The CCD camera has an array of 4096 · 4096 pixels andthe pixel size is 9 · 9 lm. This camera may digitize imageswith 16 bits of resolution (gray levels).

Table 3 shows the measured wedge angles of the prisms.Fig. 16a, presents the displacement system composed by

two wedge prisms and their rotating mounts. The mountsare connected by three metallic bars. The bars force both

Table 3Wedge prisms values

Wedge angle, a Refraction index, n

Prism A 5�1 0 1.517 (BK7)Prism B 5�4000 1.517 (BK7)Da 2000

Fig. 16. Wave-front displacement system. (a) The prism A is placed in a rotawedge. (b) To make the wedge surfaces parallel, we join both prisms and fix t

Fig. 17. The wave-front shearing is recorded with a camer

mounts to rotate simultaneously. The prism A is placedin a rotating mount, in such a way that the laser impingesthe most perpendicular as possible on the surfaceS1ðbP 0 ¼ bN 1Þ. The prism B is collocated in the slidingmount. In Fig. 16b, we display that when the prisms arein contact we assure that the wedge surfaces are parallelðbN 2 ¼ �bN 3Þ.

The wave-front displacement is obtained when theprisms are separated. The distance d between prisms isselected with the micrometric actuator on the slidingmount. We acquire an image of the displaced wave frontwith the camera (see Fig. 17). The images are processedand filtered. Using the Gaussian shape of the wave-frontintensity, it is possible to determine the coordinates ofthe wave-front center. We calculate the value of thewave-front shearing with these coordinates.

The experimental and the theoretical wave-front shear-ing are displayed in Fig. 18. The theoretical shearing iscalculated with the exact ray trace (Eqs. (2)–(4)). Wesubstitute in ray trace equations the values of wedge angle(aA and aB), refractive index (n), distance between prismsd, and distance from displacement system to camera(image plane). In Fig. 18, the ‘‘x’’ marks correspond tothe incidence wave-front on the camera. The discontinu-ous line represents the shearing when the differencebetween wedge angles is Da = 20 arcsec and the incidenceangle is cP0 = 0�.

Table 4 shows the RMS error of the differences betweenexperimental and theoretical values.

ting mount, the laser light impinges perpendicular to the surface withouthe prism B in the sliding mount.

a. The laser wave-front maintains its Gaussian shape.

Fig. 18. Displacement jD~qj of a beam as a function of the distance d between prisms: a) 0 < d < 11 mm, b) 0 < d < 40 mm.

Table 4Differences between the experimental and the theoretical values

Shearing Theoretical(lm)

Experimental(lm)

Difference(lm)

RMS error(lm)

(a)X �340 �336 4.00 7.58Y 0 �5.60 5.60 3.53(b)X �705 �699 6.00 7.57Y 0 �6.69 6.69 6.37


7. Summary

We analyzed and characterized a novel wave-front dis-placement system. The system consists of a pair of identicalwedge prisms. The degrees of freedom, under operator con-trol are: the angle of rotation of both prisms x and the dis-tance between prisms d. We demonstrated that the shearingis constant from the system to the image plane. Therefore,no tilt is introduced in the sheared wave-front.

The shearing direction depends on the relative orienta-tion of the prisms. The shearing magnitude is proportionalto distance between them. The proportionality constantrepresents the sensitivity to the displacement of the system.This constant is given in terms of the wedge angle andrefractive index of the prisms. We control the system sensi-tivity for any material, varying the wedge angle of theprisms.

The deviation of the shearing due to the differencebetween wedge angles is larger than that due to the obliqueincidence of the wave-front. Therefore, accurate wedgeangle values are necessary to calculate the wave-frontshearing with precision.

The experimental data confirm the theoretical results.There is an RMS error of about 7 lm, ascribed to theexperimental error.

Acknowledgements

The authors express their appreciation to the ConsejoNacional de Ciencia y Tecnologıa (Mexico) for fundingthe research described in this paper.

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Performance evaluation of a wave-front displacement system for vectorial shearing interferometer

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