Optics Communications 284 (2011) 1517–1525

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Optics Communications

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Discussion

Estimation of the degree of asphericity of a glass sphere using a vectorialshearing interferometer

Claudio Ramirez ⁎, Marija StrojnikCentro de Investigaciones en Optica, Apartado Postal 1-948, 37000 Leon, Gto., Mexico

⁎ Corresponding author.E-mail address: cramirez@cio.mx (C. Ramirez).

0030-4018/$ – see front matter © 2010 Elsevier B.V. Aldoi:10.1016/j.optcom.2010.10.017

a b s t r a c t

a r t i c l e i n f o

Article history:Received 1 June 2010Received in revised form 29 September 2010Accepted 4 October 2010

Keywords:Vectorial shearing interferometerSensitivityGlass sphereAsphericityAberrated wave front

The degree of asphericity is estimated by determining the average radius of curvature in different sections, atvarious points on the surface of a sphere, and the deviation from it. We employ the vectorial shearinginterferometer (VSI) as the instrument to determine the radius of curvature from two subapertures of thetransparent glass sphere. We incorporate the sphere as a thick lens into the interferometric setup,illuminating it with an expanded beam. The spherical aberration, introduced by the sphere in the wave front,depends on the local sphere radius, on the refraction index of the glass, and on the cone angle of the source.The wave front aberrated by the sphere impinges on the VSI. Here, the wave front is divided in two inamplitude, it is sheared vectorially, and it is superimposed with itself. The fringe pattern is formed in theintersection of the wave fronts. The shape of the resulting fringe pattern is directly related to sphericalaberration. We estimate qualitatively the degree of asphericity, comparing the phase gradients in differentsections of the sphere. Here, we report on the experimental setup to test the asphericity, the results withdifferent vectorial shearing (magnitude and direction). Finally, we perform a comparison with the theoreticalpredictions.

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© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Aspherical surfaces are of great interest in optics due to the abilityof some such surfaces to form a perfect point image of a point object,excluding the diffraction effects. Additionally, in the last 40 years, thedevelopment of increasingly more sophisticated software resulted indiffraction limited designs incorporating small number of asphericaloptical elements.

1.1. Glass sphere

We are investigating the degree of asphericity of a glass sphere. Itis used as a primary density standard by metrology laboratories [1].Density measurement is useful both for the industrial production andin the scientific work. In the petroleum industry, for example, themeasurement of the density of the produced oils is crucial for theassessment of the quality control. In scientific applications, glassspheres are used to determine a precise value of the Avogadroconstant [2].

The density (D) may be calculated by applying its definition,D=m/V, where m and V are the mass and the volume of the sphere,respectively. The value of the mass is obtained by comparisons withthe mass standard and the volume is determined in terms of

dimensional measurements [3]. The primary density standard iskept under controlled conditions of temperature and humidity, until itis used for creating secondary standards. It is then compared to theprimary reference in weight and volume.

1.2. Optical characteristics of a glass sphere

The spherical form of the primary density standard is selectedbecause it is much less susceptible to damage than a cube or acylinder, with their sharp and clearly defined edges. Furthermore, thevolume of a sphere with an excellent sphericity may be determinedwith a high degree of certainty using the mean of diameters overmany directions and at many points. We studied the degree ofasphericity of a sphere fabricated of BK7 glass with high homogeneityand high mechanical and thermal resistance.

The glass sphere is believed to have been finished optically with ahigh precision; however no specific values are provided due to theabsence of established procedures and adequate tools to measuresphericity of a sphere. This experimental determination is in generalconsidered challenging because its relatively small radius of curva-ture, over a large sphere segment produces a high fringe densityrelative to those of traditional optical surfaces. Thus, a detector with avery high resolution is necessary to resolve individual fringes. Finally,an instrument is needed that may be adjusted for variable surfacequality, such as a vectorial shearing interferometer that allows precisecontrol over measurement sensitivity.

Fig. 1. Original and displaced wave fronts. The Δx and Δy values are obtained fromvector Δ→ρ and the angle θ.

1518 C. Ramirez, M. Strojnik / Optics Communications 284 (2011) 1517–1525

A small number of optical techniques to measure the radius ofcurvature of a surface exist, such as a cross-section contour orprofilometer [4,5]. Traditional interferometric methods are also usedto certify the sphericity of the glass spheres. Currently, the spheres aretested in specialized standard laboratories, the Physikalisch-TechnischeBundesanstalt (PTB) in Germany or the National Institute of Standardsand Technology (NIST) in USA [6,7].

PTB employs a spherical interferometer to determine the diameterof the sphere. This interferometer consists of a spherical etalonformed by the spherical reference faces of two Fizeau lenses; thephase stepping is performed by wavelength tuning [8]. The NISTdeveloped an interferometric instrument called XCALIBIR, also basedon a spherical Fizeau interferometer. A variant of the radius benchmethod is used to measure the radius of curvature [9]. These methodsrequire acquisition and fabrication of high quality reference surfaces[10,11]. These, in turn, have to be tested to demonstrate their quality.

We propose the vectorial shearing interferometer (VSI) as analternative instrument to determine the degree of asphericity of aglass sphere. The advantage of using this instrument is that it requiresno high quality reference surfaces, has adjustable sensitivity, and it iseconomical. Other configurations of shearing interferometers havealso been studied with varying degree of success [12,13].

We use the VSI to analyze the aberrations introduced by the spherein an expanded laser beam. Local sphere radius is calculated from thevalue of spherical aberration coefficient. Thus, we use the VSI todetermine how the radius of curvature changes from reference sub-aperture at a number of locations on the sphere surface.

In the next section, we describe the vectorial shearing interfer-ometer that we developed in our laboratory and optimized for thisapplication. In Section 3, we use the glass sphere as an opticalcomponent. Furthermore, we analyze the aberrations introduced bythe sphere on the wave front of a laser. We describe sub-aperturetesting of the sphere cross-section. In Section 4, we compare the fringepatterns generated by different sections of the sphere in order toqualitatively determine the degree of sphericity. Section 5 is dedicatedto conclusions and future work. In the next section, we briefly reviewthe features of VSI and elaborate on specific changes needed tocharacterize the shape of the glass sphere.

2. Vectorial shearing interferometer (VSI)

In a VSI the wave-front under test is compared with itself. Twoidentical wave fronts follow paths nearly, but one of them undergoes asmall displacement relative to the other, as shown in Fig. 1. Theshearing vector Δ→ρ describes the displacement of the wave front Wd

(xd,yd) with respect to the original one W(x,y). The superposition ofthe wave fronts gives rise to the fringe pattern. At the interferenceplane, the incidence IT(x,y) of the fringe pattern is [14]:

IT x; yð Þ = Ib x; yð Þ + Im x; yð Þ cosΔW x; yð Þ: ð1Þ

Here, Ib is the background offset, and Im is the modulatedincidence.

This pattern carries the information of the optical path difference(OPD) in the sheared direction, and may be expressed as:

ΔW = Wd xd; ydð Þ−W x; yð Þ;ΔW = W x + Δx; y + Δyð Þ−W x; yð Þ: ð2Þ

From the fringe pattern, the path difference may be obtained as:

ΔW = mλ: ð3Þ

Here m is the order of the interference fringe and λ is thewavelength.

For small displacements, the distances Δx and Δy may beconsidered infinitesimal quantities (dx, dy). Then, the patternrepresents the total differential of the wave-front function in theshearing direction [15]:

∂W x; yð Þ∂x dx +

∂W x; yð Þ∂y dy = mλ: ð4Þ

The shape, and density of the fringe pattern may be controlled bythe direction and magnitude of the shearing vector Δ→ρ. The ability ofthe operator to control the number of fringes is particularly usefulwhen asymmetrical components are tested. ΔW is the OPD (opticalpath difference) between wave fronts, given by Eq. (2). Thereconstructed wave front may be found by direct integration of itsgradient, avoiding the evaluation of the arctangent function and thecomplex phase reconstruction methods [16–20].

In previous configurations, in one arm of a Mach Zehnderinterferometer the sheared wave front was displaced in a specificdirection according to the relative angle between the wedge prismsand their separation, but this configuration introduces tilt in theshearedwave front. A compensator systemwas necessary in the otherarm of the interferometer in order to minimize the tilt deviation[21,22].

Currently, our implementation of the VSI uses a configuration ofwedge prisms that do not introduce tilt in the sheared wave front anda compensator system is not needed, as illustrated in Fig. 2. The wavefront under test impinges on the first beam splitter, BS1. Here, theamplitude is divided into two. In order to shear one wave front, weuse a pair of wedge prisms P1 and P2 as a displacement system in onearm of the interferometer. In the other arm the wave front continuesits path without being modified. Both wave fronts, partially displacedwith respect to each other, are superimposed after the second beamsplitter, BS2. The fringe pattern is produced in the space behind thesecond beam splitter. A high resolution CCD captures the digitalimage. The camera is interfaced to a computer, where the images arestored and further analyzed.

In the current application, it is particularly important to be able tocontrol separately the magnitude and direction of the shear, in orderto examine an area on the sphere from different directions. Thedisplacement system introduces no tilt in the wave front and nochanges in the image orientation. The degrees of freedom of thedisplacement system, the angle ω of rotation of both prisms and thedistance d between them, allow the operator the requisite control todisplace the wave front vectorially (on any direction and anymagnitude). Additionally, the magnitude of the displacement deter-mines the sensitivity of the VSI [23].

In the sheared wave front (see Fig. 1), the angle θ depends on theangle of rotation ω of both prisms. The shearing magnitude jΔ→ρ j is

Fig. 2. The schematic depicts the upper view of the system. The wave front under test is from the sphere. The elements of the Mach Zehnder interferometer are: the beam splittersBS1 and BS2, the mirrors M1 and M2. The shearing system, in one arm of the Mach Zehnder, is formed by a pair of wedge prisms P1 and P2. The fringe pattern is registered by a CCDwith 4096×4096 pixels and sent via a USB port to the computer.

Fig. 3. Ray tracing through the sphere as a thick lens. The width and the radius of thesphere are t and r, respectively. The incidence angle of the ray on the sphere surface is I1,on the other hand the transmission angle is I1′. A ray parallel to the optical axis (u0=0),with height y0, is deviated by the sphere to the focal point f.

1519C. Ramirez, M. Strojnik / Optics Communications 284 (2011) 1517–1525

proportional to the distance d between them. The angle and theshearing magnitude are described by Eqs. (5) and (6) [24]:

θ = ω; ð5Þ

jΔ→ρ j = βd; ð6Þ

where β is the proportionality factor given by:

β n1;n2;αð Þ = 1−n2n1cosαffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1− n2n1sinα

� �2r2664

3775 sinα cosα: ð7Þ

The value β represents the sensitivity of the displacement of thesystem. It depends on the refractive index of the material n2, on therefractive index of the medium n1 (air in our case, n1=1), and on thewedge angle of the prisms (α).

3. Aberrations introduced by a glass sphere

We use the sphere under test as a thick lens, with radius ofcurvature r=r1=−r2. In this case, the principal planes pass andcoincide at the center of the sphere, as indicated in Fig. 3. The paraxialfocal points, the effective focal length (efl) and back focal length (bfl)are [25]:

ef l = f = n1−n0ð Þ 1r1

− 1r2

+n1−n0ð Þtn1r1r2

� �� �−1; ð8Þ

bf l = f− tf n1−n0ð Þn1r2

: ð9Þ

Employing exact ray trace, we calculate the back focal length as afunction of ray height (y0),

bf ly = − y01 + cos I1

+t sinu′

1

cosu′1 + cos I′1

" #cos 2u′

1 + cos I1sin 2u′

1: ð10Þ

The term sinI1 may be identified from Fig. 3 as follows,

sin I1 =y0r1

= y0c1: ð11Þ

From Snell's law, we compute sinI′1,

sin I′1 =n0

n1sin I1 =

n0y0n1r1

=n0y0c1n1

: ð12Þ

The terms sinu′1 and cosu′1 are calculated next,

sinu′1 = sin I′1−I1

� �=

y0c1n1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin20− n0y0c1ð Þ2

q−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin21− n0y0c1ð Þ2

q� �;

ð13Þ

cosu′1 = cos I′1−I′1

� �=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1− n2

0 + n21

y0c1n1

� �2+

n0

n1

� �2y0c1ð Þ4

s

+n0

n1y0c1ð Þ2:

ð14Þ

Here c1=1/r1. Other terms are obtained using trigonometricidentities. In Fig. 4, we graph bfly as a function of ray height y0. Thefocal length depends on the aperture for nonparaxial rays (sphericalaberration). The aberrations introduced by the sphere, on anexpanded wave front, are smallest at distance of bfl between thesphere and the illumination source.

Fig. 4. Back focal length bfly of a sphere decreases when ray height y0 increases. Wemaydetermine the radius (y0) of transmitted beam by the sphere as a function of its bfly.

Fig. 6. a) R-coefficients are minimum at focal length (x0=bfly) and they may beneglected, except R4, b) the spherical aberration term R4 ρ4 changes slowly. The R-coefficients are calculated for a sphere of glass BK7 (n=1.515) and radius r=36.2 mm.

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The transmitted beam carries the wave front under test, asindicated in Fig. 5. We may describe a general wave front in polarcoordinates as a Zernike polynomial [26]

W ρð Þ = C0 + C3 2ρ2−1� �

+ C8 6ρ4−6ρ2 + 1� �

+ C15 20ρ6−30ρ4 + 12ρ2−1� �

+ C24 70ρ8−140ρ6 + 90ρ4−20ρ2 + 1� �

:

ð15Þ

The coefficients Ci of ith term corresponds to different aberrationsof the wave front: C0 is piston, C3 is focus, C8 is spherical and focus.When the glass sphere is perfectly round (it has the same radius atevery point on its surface), homogeneous, and centered on the opticalaxis at the focal distance (bfly) from beam expander, only thespherical aberration is present in the emerging measured wave front[27]. We may rewrite Eq. (15) as follows:

W ρð Þ = C0−C3 + C8−C15 + C24ð Þρ0 + 2C3−6C8 + 12C15−20C24ð Þρ2

+ 6C8−30C15 + 90C24ð Þρ4 + 20C15−140C24ð Þρ6 + 70C24ρ8;

W ρð Þ = R0ρ0 + R2ρ

2 + R4ρ4 + R6ρ

6 + R8ρ8:

ð16Þ

In Fig. 6 we illustrate how the values of the R terms depend ondistance x0. At the back focal distance (x0=bfly) all coefficients arezero except for R4ρ4 that corresponds to the spherical aberration.

Fig. 5.We are testing surfaces S0, S1 and inhomogeneities in the dashed region. Here h0is the input pupil radius and h1 is the radius of the collimated wave front. The distancebetween the beam expander and the sphere surface S0 is x0=blfy.

When the distance between beam expander and sphere is not equal tothe focal distance, defocus is additionally introduced in thewave front.

The interferometer measures a single value of radius of curvaturefor each position and orientation of the sphere. For completeevaluation, the sphere is rotated around a vertical axis in order toexamine different subapertures, as indicated in Fig. 7. The differenceson the fringe patterns between subapertures are attributed to thedeviations in radius in curvature. Actually, each beam samples two

Fig. 7. The sphere is rotated an angle γ, around an axis through its center andperpendicular to the direction of beam propagation. We allow nearly 50% apertureoverlap to permit complete surface covering.

Fig. 8. a) R-coefficients as a function of changes in sphere radius, b) the spherical andfocus aberration terms are modified quickly when the radius sphere changes. The R-coefficients are calculated for a sphere of glass BK7 (n=1.515) and expected radiusr=36.2 mm.

Fig. 10. B-coefficients as a function of changes in sphere radius. These coefficients arecalculated for a sphere of BK7 (n=1.515) glass and expected radius r=36.2 mm.

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radii of curvature, one upon entering and the other one upon exitingthe sphere. Therefore, each measurement of the transmitted wavefront may be represented by:

Wmeasured x; yð Þ = WSphereError x; yð Þ + WSphericalAberration x; yð Þ: ð17Þ

Here WSphereError is the contribution to the measured error due tothe errors in the spherical geometry, and WSphericalAberration is theintrinsic spherical aberration.

The measured values of the R term as a function of sphere radius rare depicted in Fig. 8. If the radius changes when the sphere is rotated,the values of focus R2 and spherical R4 aberrations are affected and thefringe patterns are correspondingly modified, as illustrated in Fig. 9.The coefficient corresponding to the spherical aberration R4 depends

Fig. 9. Simulated fringe pattern obtained with VSI jΔ→ρ j = 1:633mm. The simulated s

on the input pupil height h0, the output pupil height h1, the nominalsphere radius r, and the refractive index n1. The equation to obtain thespherical aberration coefficient of surface S0 of the sphere is [15]

B0 =12h40 n2

01r0

− 1x0

� �2 1n1x

′0− 1

n0x0

!" #: ð18Þ

Here x0′ is the image distance, given as follows

n1

x′0=

n1−n0

r0+

n0

x0: ð19Þ

Similarly, on surface S1 the equation of spherical aberration is

B1 =12h41 n2

11r1

− 1x1

� �2 1n0x

′1− 1

n1x1

!" #: ð20Þ

The unknown x1′ is given as

n0

x′1=

n0−n1

r1+

n1

x1: ð21Þ

The total spherical aberration introduced by the sphere is the sumof the contributions from surfaces S0 (Eq. (18)) and S1 (Eq. (20))

B = B0 + B1: ð22Þ

The values of the B term as a function of sphere radius r arepresented in Fig. 10.

The spherical aberration may be determined experimentally foreach sub-aperture. Therefore, the amount of sphericity error on theprobed wave front may be calculated from the measured wave front:

WSphereError ρ; θð Þ = Wmeasured ρ; θð Þ−R4ρ4: ð23Þ

phere is of BK7 (n=1.515) glass, with deviation radius Δr of its expected value r.

Fig. 11. Experimental setup implemented at the laboratory. The beam splitters of theMach Zehnder interferometer aremade of optical glass (BK7) of high quality, with a thickness of6 mm. The beam splitters surfaces have a peak-valley relation of λ/5 at 80% of surface. The mirrors are made of Pyrex glass with a thin film of aluminum of high reflectance. A secondthin film of silica covers the aluminum in order to avoid its oxidation. The mirrors surfaces have a peak-valley relation of λ/5 at 80% of surface. In the detection plane we locate a CCD.The fringe pattern is recorded by the CCD and is sent it to a computer.

1522 C. Ramirez, M. Strojnik / Optics Communications 284 (2011) 1517–1525

In the next section, we use the vectorial shearing interferometer todetect changes on the phase gradients in different subapertures of theglass sphere.

4. Experimental

We implement the experimental setup as in Fig. 11, in order todemonstrate the feasibility of the technique. A laser beam passesthrough a spatial filter, incorporated inside the beam expander. Theglass sphere is illuminated with the laser beam expanded. Thetransmitted wave front impinges on the VSI. The superposition of thewave fronts (original and sheared) is observed at the detection planeand recorded by a computer-interfaced CCD.

The wedge prisms of the shearing system are made of optical glassBK7 (n=1.515). The wedge angles of the prisms are about 5° (α1=5°01′ and α2=5° 00′ 40″). The prism distortion may be neglected whenwedgeangle is smaller than7° [28]. The sensitivityβof thedisplacementsystem, calculated with Eqs. (6) and (7), is equal to 0.046. In this way,when theprismsare separated d=1mm, the shearingmagnitude jΔ→ρ jis equal to 46 μm. The distance d, between prisms, is set with themicrometric actuator of the slidingmount, shown in Fig. 12. The rotatingmounts allow the prisms to rotate by the same angle ω.

The sphere material is BK7 glass with refraction index n=1.515 atthe wave length of 632.8 nm. The sphere radius is r=36.2 mm,

Fig. 12. Thewedgeprisms of the shearing systemare placed on amechanical support. Thismechthe prism P1, are coupled using three steel bars to spin both prisms simultaneously an angle ω

measured with a micrometer. The paraxial focal length, calculatedwith Eq. (8) is efl=53.246 mm. We use three motorized mounts inorder to place the sphere on the optical axis of the expanded beam, tocontrol the separation x0 between beam expander and sphere, and torotate the sphere, as illustrated in Fig. 13. The sphere is positioned atthe paraxial focal length from the beam expander where x0=bfl,calculated with Eq. (9) to be bfl=17.046 mm.

In Fig. 14 we depict the measured fringe pattern obtained with theVSI and the simulated pattern assuming the same conditions. We usethe interference equation Eq. (1) in order to simulate the fringepattern. The shearing magnitude is jΔ→ρ j = 1:663mm in the −ydirection (θ=270°), the input pupil radius is h0=2.272 mm, and theexit pupil radius is h1=7.099 mm. Table 1 shows the calculatedZernike coefficients. We substitute the coefficients into the Zernikepolynomial Eq. (15). The pattern corresponds to spherical aberration.In Table 2 we present the values of the R-coefficients calculated fromthe Zernike coefficients with Eq. (16).

We verify the length x0, analyzing the shape of the fringe patterns,shown in Fig. 14. When x0 is greater or smaller than bfl, the number offringes increases, due to its out-of-focus positioning. The values of theR-coefficients are expressed in Table 3. Focus aberration R2 increasesrapidly. The spherical aberration changes slowly and the otheraberrations may be neglected, because they are independent ofangle of rotation of the sphere.

anism is formedby two rotatingmounts and one slidingmount. The rotatingmounts,with. The third mount slides on the three bars and holds the prism P2 at a length d from P1.

Fig. 13. The sphere is illuminated by a spherical wave front from the beam expander.The positioning system of the sphere is formed by two motorized linear stagesperpendicularly coupled, in order to move the sphere in a horizontal plane. The linearstages have a resolution of 0.625 μm at 1/8 of step. A third rotating stage is used to spinthe sphere an angle γ and it has a resolution of 4.5 arcsec at 1/8 of step.

Table 1Coefficient values for the Zernike polynomial.

Coefficient Value

C0 1.2522C3 1.8871C8 0.6408C15 0.0059C24 7.52×10−5

Table 2Coefficient values for the wave front polynomial in terms of R.

Coefficient Value

R0 1.1877×10−6

R2 −3.4774×10−5

R4 3.6717R6 0.1092R8 0.0053

Table 3Coefficient values for the wave front polynomial in terms of R.

Value

x0 16.5 mm 17.04 mm 17.5 mmR0 1.0472×10−6 1.1877×10−6 1.3202×10−6

R2 −7.3202 −3.4774×10−5 6.2788R4 3.4978 3.6717 3.8245R6 0.1015 0.1092 0.1162R8 0.0048 0.0053 0.0057

1523C. Ramirez, M. Strojnik / Optics Communications 284 (2011) 1517–1525

In order to assess the degree of asymmetry of the sphere undertest, we perform the shearing in different directions, exhibited inFig. 15. The amount of displacement is constant jΔ→ρ j = 1:663mm,and the length x0=bfl. The shape of the fringes remains unchanged,independently of the shearing direction.

We present the fringe patterns obtained in the testing of the glasssphere with different shearing magnitudes jΔ→ρ j in Fig. 16. It ispossible to improve the sensitivity of the VSI by increasing theshearing magnitude jΔ→ρ j , and consequently the number of fringes inthe detection plane. In order to validate the experimental fringepatterns, we simulate the pattern with the associated shearing. Wesubstitute the values of the aberrations (Table 1) in the Zernikepolynomial Eq. (10), and the latter expression into the phase gradient

Fig. 14. a) Fringe pattern simulated with sphere data and specific shearing,b) experimental fringe pattern obtained with the VSI. The sphere is displaced alongthe optical axis, Δx0=± 0.5 mm near to the paraxial focal length x0=17.046 mm. Theshearing magnitude jΔ→ρ j = 1:663mm in the −y (θ=270°) direction is maintained.At focal length, x0=bfl, the pattern obtained has the smallest quantity of fringes.

equation Eq. (2). Finally, the fringe pattern is obtained with theinterference equation Eq. (1). Then, we compare them with the onesobtained experimentally. The predicted performance of the VSI isverified with the simulated fringe patterns.

Finally, we estimate qualitatively the asphericity of the elementunder test, by comparing the phase gradients in different subaper-tures of the sphere. We use the rotation stage in order to changesphere orientation. In this case, the wave-front displacement systemis set for moderate sensitivity.

In Fig. 17 we show that the shape and number of the fringes aresimilar for the subapertures tested on the equator plane of the sphere.Therefore, we may conclude that the component is qualitativelyspherical along this axis, and that the VSI is optimal for testingasphericity in standard spheres. VSI is found qualitatively feasible as ameans to determine the degree of asymmetry degree of a transparentglass sphere.

Fig. 15. Experimental fringe patterns obtained with VSI. The shearing directions (θ) area) 270°, b) 300°, c) 60°, d) 180°, e) 210°, and f) 240°. The shearing magnitude isjΔ→ρ j = 1:663mm.

Fig. 16. a) Simulated and b) experimental fringe patterns obtained with the VSI and different shearing magnitudes jΔ→ρ j . The shearing direction is −y (θ=270°). The length x0 isconstant and equal to the back focal length (bfl=17.046 mm) of the sphere. When the shearing is small jΔ→ρ jb1mm the number of fringes, and thus the sensitivity, are low. The useof a more sensitive configuration aids in the detection of potential fine defects in the tested patron, we observe more easily variations between fringe patterns.

Fig. 17. Experimental fringe patterns obtained with the VSI. The sphere is rotated at angle γ. The shearingmagnitude is jΔ→ρ j = 1:663mm on the−y (θ=270°) direction. The lengthbetween beam expander and sphere is x0=17.046 mm.

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5. Summary

We have demonstrated the performance of the vectorial shearinginterferometer in order to evaluate an aberrated wave front from aglass sphere. We calculated the Zernike coefficients to estimatethe value of the aberrations introduced by a perfect sphere into thewave front, in order to mimic more closely the complexity ofassessing qualitatively the sphericity in standard spheres. We useZernike polynomials in order to simulate the sheared fringepatterns. Simulated fringe patterns correspond to those obtainedexperimentally.

Vectorial shearing interferometry is found qualitatively feasible asa means to determine the degree of asymmetry degree of atransparent glass sphere. The symmetry of the sphere under testwas estimated qualitatively, by comparing the phase gradients (fringepattern) of different subapertures, of the optical component.

It is possible to improve the sensitivity of the VSI by increasing theshearing magnitude, and consequently the number of fringes in thedetection plane. Sensitivity control is a desirable feature whensophisticated measurements are required. The use of a more sensitiveconfiguration aids in the detection of potential fine defects in thetested sphere.

Future work includes improving the scanning method of thesphere in order to ensure the complete measurement of the sphere.

Acknowledgments

The authors wish to express their appreciation to the ConsejoNacional de Ciencia y Tecnología (Mexico) for funding the researchdescribed in this paper under project CONACYT 2007-I0003-60450.Claudio Ramirez is a recipient of the CONACYT fellowship.

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