Axial resonance of periodic patterns by usinga Fresnel biprism

Ana Doblas,* Genaro Saavedra, Manuel Martinez-Corral, Juan C. Barreiro,Emilio Sanchez-Ortiga, and Anabel Llavador

Department of Optics, University of Valencia, Burjassot E-46100, Spain*Corresponding author: a.isabel.doblas@uv.es

Received August 7, 2012; revised October 19, 2012; accepted December 9, 2012;posted December 10, 2012 (Doc. ID 172859); published January 1, 2013

This paper proposes a method for the generation of high-contrast localized sinusoidal fringes with spatiallynoncoherent illumination and relatively high light throughput. The method, somehow similar to the classicalLau effect, is based on the use of a Fresnel biprism. It has some advantages over previous methods for thenoncoherent production of interference fringes. One is the flexibility of the method, which allows the controlof the fringe period by means of a simple axial shift of the biprism. Second is the rapid axial fall-off in visibilityaround the high-contrast fringe planes. And third is the possibility of creating fringes with increasing or withconstant period as the light beam propagates. Experimental verifications of the theoretical statements are alsoprovided. © 2013 Optical Society of America

OCIS codes: 050.1950, 050.1960, 110.6760, 260.3160.

1. INTRODUCTIONThe formation of self-images, both in the coherent (Talbot ef-fect) and in the incoherent (Lau effect) version, has attractedthe attention of scientists since its discovery [1]. This phenom-enon has deserved the interest of the scientific communityas a basic property of certain wave fields [2–9], but it alsohas been fruitfully exploited in many other applications[1,10–14]. The Talbot effect appears when a certain class ofdiffracting objects, Montgomery objects, are illuminated by amonochromatic plane wave. Montgomery objects have theability to produce wave fields that are periodic along the di-rection of propagation of the wave. Then at axial distancesthat are integer multiples of the so-called Talbot distance, theamplitude distribution of the field is the same as the one emer-ging from the original diffracting object. It is worth mentioningthat although it was originally associated with the amplitudeor the irradiance of light fields, the self-imaging phenomenonalso occurs to other magnitudes as the mutual intensity.This function exhibits axial periodicities for partially coherentfields when some conditions, which generalize thosedefining a Montgomery object for coherent illumination, arefulfilled [15,16].

The Lau effect, described first in 1948, is the spatially inco-herent counterpart of the Talbot effect. The Lau effect is ob-tained when one allows the superposition in consonance ofTalbot fringes generated by a series of mutually incoherentquasi-monochromatic sources. The most common practicalimplementation of the Lau effect is achieved when two 1Dgratings, oriented parallel to each other, are illuminated bythe light proceeding from a quasi-monochromatic spatially in-coherent planar source. After imaging the planar source ontothe first grating by a large-aperture lens, this plays the role ofan incoherent source encoded linearly. (However, in general,imaging an incoherent quasi-monochromatic source results inpartially coherent illumination; if the aperture of the imaging

system is large enough, then the residual spatial coherencecan be neglected (critical illumination scheme) [17].) Thesecond grating acts as a diffracting object. The Lau effectis achieved only in Talbot planes (self-imaging planes) andresults in the production of high-contrast nonncoherent fringepatterns. Similar approaches have been used to produce loca-lized and nonlocalized interference patterns in a wide set ofgrating interferometers [18], including Talbot and moirédevices [19,20].

On the other hand, the Fresnel biprism is an optical elementformed by two thin equal prisms joined at the base [21,22].When this element is illuminated with the light from a mono-chromatic point source, it generates two equal-intensity vir-tual sources. Since these are the images that the biprismgenerates from the real source, they are mutually coherentand, therefore, generate behind the biprism an axiallyextended sinusoidal interference pattern, similar to that ob-tained in a Young experiment. The period of this pattern in-creases proportionally to the distance between the pointsource and the observation plane. The high contrast obtainedis directly related to the degree of spatial and temporal coher-ence of the real source. A change in any of these conditionsproduces a dramatic decay in the visibility of the fringes. Thisdevice was designed by Augustin Fresnel to help silence theobjections raised by Young’s experiment, among the detrac-tors of the wave theory of light. Contrary to what happensin the double-slit experiment, with the biprism a pure interfer-ence pattern is obtained without relying upon diffraction onapertures. The Fresnel biprism has recently found applicationin its ability to greatly simplify a commonly used technique formeasuring ultrashort laser pulses [23], to develop a type oftransmission phase-shifting laser microscope [24], to illustratethe wave-particle behavior of light in the single-photon regime[25,26], or, in the case of its x-ray or electron-beam ana-logue, to implement a biprism interferometer for coherence

140 J. Opt. Soc. Am. A / Vol. 30, No. 1 / January 2013 Doblas et al.

1084-7529/13/010140-09$15.00/0 © 2013 Optical Society of America

measurement of x-ray sources [27,28], as well as for use ininterference electron microscopy [29].

In this paper, it is proposed for the first time (to ourknowledge) that the Fresnel biprism be used to producehigh-contrast fringes with spatially incoherent light. The meth-od for producing such fringes is somehow based on the Lauexperiment. Specifically, we propose the use of an array oflinear, spatially incoherent sources for the illumination of theFresnel biprism. This arrangement allows the production ofsets of axially extended, mutually incoherent, laterally shiftedsinusoidal patterns. Depending on the parameters of the set-up, there are some planes where the patterns superposeresonantly, giving rise to fringe patterns of high visibility.These high-visibility, or resonant, patterns are surrounded bypatterns with very low visibility.

The paper is organized as follows. In Section 2, we explainthe basic parageometrical theory [30,31] behind this interfer-ential phenomenon, and we derive the equations that describethe fringe patterns that can be obtained when an array of lin-ear sources illuminates the biprism. In Section 3, we propose anew interferential architecture for the production of resonantfringes whose period does not change with the field propaga-tion. Section 4 is devoted to studying the effect of the diffrac-tion at the edge of the biprism and to deriving the formula thattakes into account its influence on the structure of the inter-ference patterns under issue. Finally, in Section 5, we showthe corresponding experimental verifications, and in Section 6we summarize the main achievements of our research.

It is worth noting that the analysis of the Lau effect has beensuccessfully approached by some authors in the spatial fre-quency domain by using the optical transfer function formal-ism and linked to coherence theory [32]. However, here weuse a more direct approach in the spatial domain, which ismore convenient for the purpose of this paper.

2. BASIC THEORYLet us start by recalling the case in which a quasi-monochromatic light beam proceeding from a spatially inco-herent planar source illuminates a thin object. We assume thatthe object and source planes are parallel to each other, so thatwe can define an optical axis, as is done in Fig. 1. If we nameas η the distance between the source and the object, then wecan write the irradiance distribution on an arbitrary planeplaced at a distance z beyond the object as [12]

I�x⃗; z; η� � 1M2

S

IS

�x⃗MS

�⊗2 I0�x⃗; z; η�; (1)

where ⊗2 represents the 2D convolution product perfor-med over the transverse coordinates, x⃗ � �x; y�. In the above

equation, IS�x⃗� denotes the normalized irradiance distributionof the planar source, I�x⃗; z; η� is the irradiance distributiongenerated at the observation plane by a point source placedat the intersection between the optical axis and the planarsource, and MS � −z∕η is a magnification factor betweenthe source and the observation planes.

This formula has been successfully used in a number of in-teresting applications, for example, the measurement of thedegree of coherence of a light beam [11] or the implementa-tion of lensless correlators [12]. In our case, we aim to design amethod for the production of high-visibility periodic patternswith incoherent light.

Thus, we start by studying the patterns produced when aFresnel biprism is used in our architecture. At this point itis important to remember that when a spherical wavefrontproceeding from a point source illuminates a Fresnel biprism,the exiting wavefront is split into two spherical waves thatvirtually proceed from two virtual point sources placed atthe plane of the source and separated by the distance a �2η�n − 1�δ [21], where n is the refracting index and δ the re-fringence angle of the biprism. Note that we are assuming athin biprism, so that the angle δ is considered to be small. Thevirtual point sources are mutually coherent, thus producingan interference pattern beyond the Fresnel biprism whoseirradiance distribution can be written as

I0�x⃗; z; η� � 1� cos�2πxp

�: (2)

In this equation we considered the biprism central edge tobe aligned with the Cartesian OY direction, and the period ofthe interference pattern is given by

p � M2u0

; (3)

whereM � �η� z�∕η is the magnification factor that accountsfor the increase of the period along the axial coordinate andu0 � �n − 1�δ∕λ. It is worth mentioning that Eq. (2) gives aparageometrical description of the irradiance pattern underissue, thus neglecting the effect of the diffraction at the edgeof the biprism and considering only the region of the geome-trical superposition of rays coming from the virtual sourcesthrough the biprism. A more accurate study will be providedin Section 4.

Next, we substitute Eq. (2) into Eq. (1) to straightforwardlyobtain

I�x⃗; z; η� �~IS�0⃗�M2

S

�1� V�z� cos

�2πxp

��z���

; (4)

where

~IS�u⃗� � j~IS�u⃗�j exp�jΦ�u⃗��; (5)

which stands for the Fourier transform of the irradiancedistribution of the source.

The visibility of the sinusoidal pattern in Eq. (4) is given by

V�z� �

���~IS�MSp ; 0

����~IS�0⃗�

�

���~IS�−2u0

zz�η ; 0

����~IS�0⃗�

: (6)

x0

y0xS

yS

y

xη

z

yxS

yS

η

Fresnelbiprism

ExtendedsourceObservation plane

Fig. 1. (Color online) Geometric scheme to study the diffractedfield by a Fresnel biprism illuminated by a spatially incoherentquasi-monochromatic planar source.

Doblas et al. Vol. 30, No. 1 / January 2013 / J. Opt. Soc. Am. A 141

Thus, the spatially incoherent planar source produces afringe pattern whose visibility is given by the Fourier trans-form of the irradiance variations of the source along theOX direction (perpendicular to the biprism edge). Note thatthe visibility depends on the propagation distance z, i.e., itis different in each observation plane.

The above general study can be particularized to the case ofspecial interest in which the quasi-monochromatic source iscomposed by an array of mutually incoherent, equidistantpoint sources arranged perpendicular to the biprism edge.For simplicity, we assume N sources that have the same irra-diance, IP , and are distributed symmetrically to the opticalaxis. Then, the irradiance distribution of the source is

IS�x⃗� �XNI�1

IPδ�x − x1; y�; (7)

where

x1 ��N � 1

2− I

�x0 I � 1;…; N; (8)

x0 being the separation between neighbor point sources. Byusing Eq. (1), the irradiance distribution in this case isobtained to be as follows:

I�x⃗; z; η� � IPXNI�1

I0�x −MSx1; y; z; η�: (9)

According to Eq. (6), the visibility of the irradiance patternat any observation plane can be written in this case as

V�z� �sin

�πN MSx0

p

�

N sin�π MSx0

p

� : (10)

Equation (10) is a periodic function that takes significantvalues only in a discrete set of planes on which the argumentof both sin functions achieve an integer multiple of π. Thiscondition determines the planes of maximum visibility, andthese planes will be named hereafter as the planes of axialresonance. Therefore, the planes of axial resonance appearat distances

zm � mλη

2x0�n − 1�δ −mλ�11�

from the biprism plane, where m is an integer number. Sincethe relation between zm and m is not linear, the maximum-visibility planes are not equidistant. Note, however, that theposition of such planes does not depend on the number ofpoint sources N . Furthermore, Eq. (10) indicates that thegreater the number of slits, the higher the axial localizationof the planes of resonance.

It is worth noting that, as stated above, the structure of theincoherent source along the y direction does not influence thevisibility of the fringes but only their maximum irradiance.Thus, the same resonances could be obtained if instead ofusing an array of point sources one used an array of linearsources. It is interesting to note that the result in Eq. (11)can be also obtained by realizing that Eq. (9) represents a

superposition of laterally shifted versions of the periodic pat-tern I0�x⃗; z; η�. In planes for which these lateral shifts equal aninteger multiple of the period, a high-visibility irradiance dis-tribution is obtained.

On the other hand, in a realistic experimental situation, thewidth Δ of the sources along the x direction would not beinfinitesimal. In that case, the irradiance distribution of thesource can be written as the convolution between IS�x⃗� inEq. (7) and a rectangle of width Δ. Thus, in such a realisticcase the visibility at any observation plane is given by

V�z� �sin

�πN MSx0

p

�

N sin�π MSx0

p

� sinc�ΔMS

p

�: (12)

The main difference between Eqs. (12) and (10) is the sincfunction, which appears because of the finite size of the inco-herent sources. This factor causes the visibility of the planesof axial resonance to decrease along the axial direction, asshown in Fig. 2. Note that the lower themodulation (fill factoror duty cycle), Δ∕x0, of the set of slit sources, the smootherthe variation of this function. Note also that the positions ofthe planes of axial resonance (maximum, although not unitvisibility) are still given by Eq. (11) as far as this variationis smooth. It is also interesting to mention that Eq. (12) de-scribes, in a different context, the Fraunhoffer pattern of abinary diffraction grating.

To conclude this section, we next discuss briefly whetherthe results obtained with this technique can be extended toother devices which are an alternative to Young’s experimentin producing two coherent sources. Although all of them arecapable of generating interference fringes with monochro-matic point source illumination, their behavior when usinga quasi-mochromatic extended light source is quite different.

Let us consider first the classical Young’s double-slit experi-ment. It is straightforward to find that in this case the visibilityof the fringes is also given by the function in the middle termof Eq. (6), but with a ratio MS∕p equal to −a∕λη, a being theseparation between the two slits and η the distance from thesource to the slits. Since this ratio is independent of the ob-servation distance z, the visibility of the fringes is the same forany observation plane, in contrast to the results presented inthis section for the Fresnel biprism setup. In other words, no

0.20 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Normalized axial coordinate: z/η

Vis

ibil

ity:

|V|

N = 1N = 3N = 5N = 10

Fig. 2. (Color online) Absolute value of the visibility of the interfer-ence patterns obtained in planes behind the biprism. For the calcula-tion of these curves we assumed a specific setup in which n � 1.5,δ � 0.51°, λ � 0.634 μm, x0 � 400 μm, and Δ � 60 μm.

142 J. Opt. Soc. Am. A / Vol. 30, No. 1 / January 2013 Doblas et al.

axial modulation of the contrast is obtained in the double-slitconfiguration.

However, some other alternatives to Young’s interferom-eter share the axial modulation for the visibility of the inter-ference fringes that we have presented here when usingincoherent quasi-monochromatic sources. In fact, any interfe-rometer fulfilling the condition stated in Eq. (1) and showingsome variation in the ratio MS∕p with the propagation dis-tance z may present some axial modulation for the contrastof the interference fringes, as stated in Eq. (6). This is, amongothers, the case of a Wollaston prism. But in this case, the de-vice cannot work with unpolarized light unless the prism islocated between two crossed (or parallel) linear polarizers,thus providing a lower light gathering power compared withthe Fresnel biprism.

Other devices that do not allow us to obtain the axial reso-nance of sinusoidal patterns by the proposed technique areLloyd’s mirror and the Kösters prism. In Lloyd’s mirror, a shiftof the point source normal to the mirror causes the same dis-placement of its image, but in the opposite direction, so thatthe separation between the two sources generated by the de-vice changes. Therefore, a shift of the source makes a changein the period of the interference pattern instead of just a lat-eral displacement of the fringes. In other words, for Lloyd’smirror the condition stated in Eq. (1) is not fulfilled and, con-sequently, Lloyd’s mirror does not show the capability ofgenerating resonant superposition planes with sinusoidal irra-diance variations. The same argument applies to the Köstersprism. This prism is also capable of generating two virtualsources, but due to the fact that one of the sources is gener-ated by a reflection in the semi-reflecting plane surface of theprism, it exhibits the same behavior as Lloyd’s mirror. Somedetails of this discussion can be found in [33].

3. AXIAL RESONANCE OF FAR-FIELDPERIODIC PATTERNSOur aim here is to obtain periodic fringes whose period doesnot change with the axial distance, so that the period is thesame in all the planes of resonance. Since the period in suchplanes corresponds to the one provided by the biprism underpoint source illumination [see Eq. (3)], the only way of achiev-ing this goal is to set the distance η to infinity. In other words,we have to separate infinitely the source from the biprism.Experimentally, this can be performed by simply inserting aconverging lens, with focal length f , between the sourcesand the biprism, as shown in Fig. 3, in such a way that thesource plane coincides with the front focal plane of the lens.

Following the same reasoning as in the previous section,the irradiance distribution obtained in this case at planesbeyond the biprism is given by

I�x⃗; z� ∝ 1� V 0�z� cos�2πxp0

�; (13)

where now the period of the fringe pattern,

p0 � 12u0

�14�

does not change with z. The visibility is given in this con-figuration by

V 0�z� �sin

�πN

M 0Sx0p0

�

N sin�π

M 0Sx0p0

� sinc�ΔM 0

S

p0

�; (15)

where M 0S � −z∕f .

Contrary to what happened in the results of Section 2, nowthe argument of the functions in Eq. (15) depends linearly onthe axial coordinate, z. Thus, the planes of resonance are equi-distant and the fringe patterns have the same period, p0. Theaxial period for this resonant superposition is given by

T � f2u0x0

: (16)

From Fig. 4 this equidistance is clear, as is, interestingly,that the decrease of the visibility around a resonant planeis symmetric in the axial direction. Additionally, the axialextent to which the visibility remains above a threshold value(in a sense, the depth of focus of the high-visibility planes) isuniform for every plane of resonance.

4. DIFFRACTION EFFECTS AT THECENTRAL EDGE OF THE BIPRISMAs stated above, the formalism developed in the previous sec-tions gives an approximate description of the irradiance pat-terns generated by the Fresnel biprism only in the interferenceregion defined by the parageometrical approximation.Furthermore, the effect of the edge diffraction on the biprismhas been neglected in previous sections. In this section, wepresent a more accurate description of the field diffractedby the biprism. As we will show, the full description of the

xS

yS

x0

y0

x

z

f

Fresnel biprism

N incoherent sources Converging lens

Observation plane

y

Fig. 3. (Color online) Scheme for the production of far-field periodicresonant patterns.

Vis

ibili

ty: |

V|

0

0.2

0.4

0.6

0.8

1

2.5210 1.50.5Normalized axial coordinate: z/T

N = 1N = 3N = 5N = 10

Fig. 4. (Color online) Visibility of fringe patterns behind the biprismwhen the array of linear sources is placed at infinity by using aconverging lens. For the calculation of these curves we assumed aparticular case: n � 1.5, δ � 0.51°, λ � 0.634 μm, f � 200 mm,x0 � 400 μm, and Δ � 60 μm.

Doblas et al. Vol. 30, No. 1 / January 2013 / J. Opt. Soc. Am. A 143

irradiance pattern at every transverse plane can be expressedas a modulated version of the parageometrical model predic-tions. In this way, the results presented in the previoussections are accurate enough that this envelope function isvery smooth as compared with the sinusoidal pattern. Thisapproximation gives rise to equations that predict very accu-rately the axial position of the maxima and minima of visibilityin Eq. (4) or (13), and it gives the qualitative behavior of V�z�,but it is not accurate enough for a precise calculation of itsvalues.

Accordingly, the fringes obtained with the parageometricalmodel are modified by diffraction effects, due to the fact thatthe waves, which diverge from the two virtual sources, are notcomplete but are abruptly cut off at the arista of the biprism[21,22], which acts as a straight edge. A reliable evaluationof the fringes’ profile and their visibility requires the exactcalculation of the diffraction pattern produced by the biprism.To obtain the amplitude distribution behind the Fresnel bipr-ism when it is illuminated by a spherical wave, one uses theFresnel–Kirchoff integral [34], assuming the usual approxi-mation of a thin object. We have modeled the biprism as aslim object whose transmission function is given byt0�x⃗� � �t�x⃗� � t�−x⃗��, where t�x⃗� can be written as

t�x⃗� � exp�−j2πu0x�step�x⃗�; (17)

step�x� being the unit step function [35].After straightforward mathematical manipulations, the

expression of the amplitude distribution beyond the biprism,for the case of a monochromatic point (or linear) sourceplaced at a distance η from it, can be obtained as

U 00 �

1��������2M

p exp�−j

π

λzM�x� λzu0�2

�

×1� j2

� Fres� ����������

2λzM

r�x� λzu0�

�

� 1��������2M

p exp�−j

π

λzM�x − λzu0�2

�

×1� j2

− Fres� ����������

2λzM

r�x − λzu0�

�; (18)

where

Fres�α� �Z

α

0exp

�jπ

2x2�dx � C�α� � jS�α�: (19)

In this definition, C�x� and S�x� are the Fresnel integralsdefined as the criterion of Abramowitz and Stegun [36]. Notethat this wave model only takes into account the central edgeof the biprism, neglecting the external ones. However, theusual practical configurations justify this approach.

From Eq. (18), the irradiance distribution can be written as

I 00�x⃗; z; η� �env�x⃗; z; η�

2M

�1� V�x⃗; z; η� cos

�2π

xp

��; (20)

where the expression of env�x⃗; z; η� is

env�x⃗;z;η��1�C� ����������

2λzM

r�x�λzu0�

�2�S

� ����������2

λzM

r�x�λzu0�

�2

�C� ����������

2λzM

r�x�λzu0�

��S

� ����������2

λzM

r�x�λzu0�

�

�C� ����������

2λzM

r�x−λzu0�

�2�S

� ����������2

λzM

r�x−λzu0�

�2

−C� ����������

2λzM

r�x−λzu0�

�−S

� ����������2

λzM

r�x−λzu0�

�;

(21)

and V�x⃗; z; η� is given by

V�x⃗; z; η� � 1env�x⃗; z; η�

×1� 2C

� ����������2

λzM

r�x� λzu0�

�C� ����������

2λzM

r�x − λzu0�

�

− 2S� ����������

2λzM

r�x� λzu0�

�S� ����������

2λzM

r�x − λzu0�

�

� C� ����������

2λzM

r�x� λzu0�

�� S

� ����������2

λzM

r�x� λzu0�

�

− C� ����������

2λzM

r�x − λzu0�

�− S

� ����������2

λzM

r�x − λzu0�

�:

(22)

The irradiance diffraction patterns behind the Fresnel bipr-ism are completely described by Eq. (20). The interferenceterms of Eqs. (2) and (20) have the same period. Thus, follow-ing a similar reasoning to that in Section 2, the position ofmaximum-visibility planes, given by Eq. (11), remains valid.In order to show the deviations of the parageometrical modelwith respect to the more rigorous prediction presented inEq. (20), we present in Fig. 5 a representation of the envelopefunctions, env�x⃗; z; η� and V�x⃗; z; η�, for two different trans-verse planes. As can be seen, in this figure, within the regionof superposition of the light coming from the virtual sources inthe parageometrical approximation, V�x⃗; z; η� attains an al-most constant unity value, as predicted by the simpler model.However, the effect of the modulating function env�x⃗; z; η�may be of certain importance to the accurate description ofthe interference patterns. Note that this effect is less signifi-cant at large distances from the biprism.

For the case of an incoherent source composed by N point(or linear) sources [such as the source that is described in theEq. (7)], the irradiance distribution, given by the Eq. (1), canbe calculated by using Eq. (9). As an example, in Fig. 6, weshow the numerically evaluated fringe patterns for the caseof N � 1, 2, and 5 point (or linear) incoherent sources placedat a distance η � 207.8 mm from the biprism. As predicted inSection 2, the positions of the planes of maxima visibility donot depend on the number of sources. Nevertheless, the axialmodulation of the visibility increases with the number ofslits in the sources. The planes of resonance are markedwith dashed lines. Next, in Fig. 7 we show the profile of thefringes in a plane of maximum visibility and one of minimumvisibility.

144 J. Opt. Soc. Am. A / Vol. 30, No. 1 / January 2013 Doblas et al.

When the incoherent source is located at infinity (or in thefront focal plane of a converging lens), Eqs. (20)–(22) are stillvalid. The only change is that for this particular case theparameter M takes the unit value. Also in this case we havecalculated the incoherent fringes for the case of a differentnumber of point sources, as shown in Fig. 8.

5. EXPERIMENTAL VERIFICATIONSTo demonstrate the capability of a Fresnel biprism for produ-cing the resonant superposition of interference fringes, weprepared the experimental setup shown in Fig. 9. For the im-plementation of the quasi-monochromatic mutually incoher-ent linear sources, we illuminated a 1D binary grating ofperiod x0 � 400 μm and aperture Δ � 60 μm, with the lightproceeding from an ultrabright red LED whose emission cen-tral wavelength was 634 nm with 14 nm spectral FWHM. TheFresnel biprism (BK7 glass) had a refringence angle δ and arefractive index n such that �n − 1�δ � �0.25� 0.02�°. To cap-ture the interference patterns we used a 2.5× objective lens(NA � 0.075) and a CCD composed of 765 × 578 square pixels11 μm on each side. Note that although it was not strictlymonochromatic, the light source that we employed had a verynarrow spectral bandwidth, and no chromatic aberration ef-fects were expected to occur. In fact, the relative variationin the refractive index n of the biprism used in the spectral

0

0.4

0.8

1.2

1.6

-0.4

0

0.4

0.8

Transverse coordinate: x (a.u.)

(x,z

;η)

(a)

-0.4

0.4

0.8

0

0

0.4

0.8

1.2

(b)Transverse coordinate: x (a.u.) Transverse coordinate: x (a.u.)

Transverse coordinate: x (a.u.)

env

(x,z

; η)

env

(x,z

;η)

(x,z

;η)

Fig. 5. (Color online) Numerically evaluated envelope functions V�x⃗; z; η� and env�x⃗; z; η� for (a) z � 116 mm and (b) z � 261 mm. The dashedlines define the region of interference superposition in the parageometrical approximation. For the calculation we set n � 1.5, δ � 0.51°,λ � 0.634 μm, and η � 207.8 mm.

40 80 120 160 200z (mm)

x (a

.u.)

40 80 120 160 200z (mm)

x (a

.u.)

40 80 120 160 200z (mm)

x (a

.u.)

N=5

N=2

N=1

Fig. 6. Numerically evaluated fringe patterns located behind thebiprism for different numbers of incoherent point sources. For thecalculation we set n � 1.5, δ � 0.51°, λ � 0.634 μm, η � 207.8 mm,x0 � 400 μm, and Δ � 60 μm.

(a)

(b)

x (a.u.)

x (a.u.)

Fig. 7. (Color online) Numerically evaluated fringe profile in theplanes (a) z � 105 mm (maximum visibility) and (b) z � 160 mm(minimum visibility), for the case N � 5 of point sources.

Doblas et al. Vol. 30, No. 1 / January 2013 / J. Opt. Soc. Am. A 145

band of the source is lower than 0.05%. Other chromaticallyinduced effects that might affect the matching of the experi-mental results to the quasi-monochromatic predictions, suchas the coherence loss in the periphery of the interference field,are mostly negligible in the situations considered here.

In our experiment we placed the biprism at a distance η �207.8 mm from the grating. Experimental patterns obtained atdifferent axial positions behind the biprism are shown inFig. 10. In the left column we show patterns of very low vis-ibility, and in the right one we show some planes of resonance.Clearly, the planes of maximum visibility are not equidistant,and the fringe period varies with the axial coordinate, as pre-dicted in Section 2. In Fig. 11 we show the irradiance distribu-tion behind the biprism in the meridian plane (XZ) for variousvalues of N (N � 1, 2, and 5). We can compare these patternswith the ones evaluated numerically from our formula (seeFig. 6). The similarity between the experimental and calcu-lated results is apparent. Next, in Fig. 12 we show the struc-ture of the fringe pattern in a plane of maximum visibility(z � 105 mm) and also in a plane of minimum visibility(z � 160 mm). In this case also the similarity to the calculatedresults (Fig. 7) is apparent.

Finally, we calculated, plane by plane, the visibility of thefringes. For this calculation we obtained the Fourier trans-form of the irradiance pattern in each plane and evaluatedthe visibility as two times the quotient between the �1 orderand the zero order. In Fig. 13 we show these experimentalvalues as compared with the theoretical ones. The agreementbetween them is high.

To demonstrate the feasibility of the resonance of far-fieldfringes, we arranged the experimental setup schematized inFig. 14. For this experiment we used a source grating com-posed of N � 5 slits. The grating was placed at the front focalplane of a converging lens of focal length f � 200 mm. With

x (a

.u.)

x (a

.u.)

10 20 30 40 50 60 70 80z (mm)

10 20 30 40 50 60 70 80z (mm)

N=5

N=1

Fig. 8. Numerically evaluated fringe patterns located behind thebiprism for different numbers of incoherent point sources locatedat infinity. For the calculation we set n � 1.5, δ � 0.51°,λ � 0.634 μm, f � 200 mm, x0 � 400 μm, and Δ � 60 μm.

ηz

LEDGrating

Objectivelens

CCD

FresnelbiprismObservationplane

Fig. 9. (Color online) Schematic diagram of the experimental setupused for the production of resonant superposition of interferencefringe.

z=22.9 mm z=44.59 mm

z=81 mm z=105 mm

z=160 mm z=228 mm

Fig. 10. Interference patterns obtained for N � 5 at different dis-tances from the Fresnel biprism. The patterns shown in the right col-umn correspond to the first, second, and third planes of resonance,respectively.

40 80 120 160 200z (mm)

40 80 120 160 200z (mm)

N = 2

40 80 120 160 200z (mm)

N = 1

N = 5

x (a

.u.)

x (a

.u.)

x (a

.u.)

Fig. 11. Experimental irradiance distribution along the axialdirection for different numbers of finite-width sources.

146 J. Opt. Soc. Am. A / Vol. 30, No. 1 / January 2013 Doblas et al.

this setup we again captured patterns in planes perpendicularto the optical axis. The irradiance distribution in the meridianplane is plotted in Fig. 15. Again we can see the greatsimilarities between the experimental results and the onescalculated with our formula (Fig. 8). Within the represented

axial range we could find two planes of resonance (markedwith dashed lines in the figure) at distances z1 � 35.2 mmand z2 � 70 mm. Then we measured the period of the fringesin those planes of resonance and obtained p01 � �69� 4� μmand p02 � �67� 4� μm. As predicted, the period of resonantfringes is kept constant within the error range.

We alsomeasured the visibility of the fringes obtained atdifferent distances from the biprism. These values are repre-sented in Fig. 16, as compared with the theoretical ones. Againthe degree of correlation between theoretical and experimen-tal data is very high.

6. CONCLUSIONSA new system for the generation of localized interferencefringes is presented. The proposal works with noncoherentillumination coming from a properly coded spatially incoher-ent quasi-monochromatic source illuminating a Fresnel bipr-ism. In particular, a set of incoherently illuminated equidistantparallel slits is used to generate resonant overlapping in somesingular planes of the periodic interference patterns gener-ated by the biprism from each point in the source plane. Thisresonance generates in these planes high-contrast highly loca-lized sinusoidal interference patterns. A brief discussion onthe possibility of obtaining such a result with some other“Young’s double-slit”-related elements is also presented,and the necessary conditions for achieving this goal are alsoestablished.

The technique proposed here presents some advantagesover other similar devices. On one hand, the period of the re-sonant fringes can be tuned by axially displacing the biprismwith respect to the source plane, as can be seen easily fromEq. (3). On the other hand, when the number of slits in thegrating source is high, the visibility of the fringes falls off veryrapidly in the vicinity of the planes of resonance, generating a

(a)

(b)

x (a.u.)

x (a.u.)

Fig. 12. (Color online) Experimentally measured fringe patternscorresponding to N � 5 point sources and a plane of (a) maximumvisibility (z � 105 mm) and (b) minimum visibility (z � 160 mm).

40 60 80 100 120

0.1

0.3

0.5

0.7

0.9

|V(z

)|

z (mm)

N=2

40 60 80 100 120

0.1

0.3

0.5

0.7

0.9

|V(z

)|

z (mm)

N=5

Fig. 13. (Color online) Numerically evaluated (triangles) and experi-mentally measured (asterisks) values of the visibility of the fringes fordifferent numbers of incoherent finite-width sources.

z

f

Converging lensLED Grating

Objectivelens

CCD

FresnelbiprismObservationplane

Fig. 14. (Color online) Schematic diagram for generating interfer-ence patterns behind a Fresnel biprism when finite-width sourcesare located at an infinite distance.

10 20 30 40 50 60 70 80z (mm)

x (a

.u.)

Fig. 15. Experimental irradiance distribution along the axial direc-tion N � 5 for sources when the grating is located at the front focalplane of a converging lens.

40 45 50 55 60 65 70 75 80

0.9

0.7

0.5

0.3

0.1

z (mm)

|V(z

)|

Fig. 16. (Color online) Numerically evaluated (triangles) and experi-mentally measured (dots) values of the visibility of the fringes forN � 5.

Doblas et al. Vol. 30, No. 1 / January 2013 / J. Opt. Soc. Am. A 147

highly localized fringe pattern. Finally, we have demonstratedthat it is also possible to modify the experimental architectureso that the period of the resonant fringes remains constant, asstated in Section 4. All these features make this setup veryattractive for some modern applications using patterned light,for instance, in structured illumination microscopy [37–39].

ACKNOWLEDGMENTSWe acknowledge the anonymous reviewers for very usefulcomments and suggestions. This work was supported in partby the Ministerio de Ciencia e Innovación, Spain (GrantFIS2009-9135), and also by Generalitat Valenciana (GrantPROMETEO2009-077). A. Doblas acknowledges funding fromthe University of Valencia through the predoctoral fellowshipprogram “Atracció de Talent.”

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