Effect of coherence and polarization on frequencyresolution in optical Fourier transforming system

Andrey S. Ostrovsky,* Miguel Á. Olvera-Santamaría, and Paulo C. Romero-SoríaFacultad de Ciencias Físico Matemáticas, Universidad Autónoma de Puebla, Puebla 72000, México

*Corresponding author: andreyo@fcfm.buap.mx

Received September 23, 2011; revised October 26, 2011; accepted October 28, 2011;posted October 28, 2011 (Doc. ID 155015); published December 1, 2011

Using an example of vector Gaussian Schell-model beam, we demonstrate and analyze the dependence of the spatialfrequency resolution in optical Fourier transforming system on the intrinsic coherence-polarization structure ofillumination. © 2011 Optical Society of AmericaOCIS codes: 030.1640, 260.5430.

For more than ten years much prominence has beengiven to the vector coherence theory [1–8] (an extendedlist of references on this topic can be found in review[9]). Recently we have reported on the effect of coher-ence and polarization of illumination on resolution ofoptical imaging system [10]. It has been shown thatthe minimum resolvable separation between two objectpoints depends on the intrinsic correlation structure ofillumination, increasing with the increase of the correla-tion lengths of orthogonal field components and decreas-ing with the increase of the ratio of their powers. Here wego on with analyzing the effect of statistical properties ofan electromagnetic illumination, considering this timeoptical Fourier transforming system.As well known, the purpose of optical Fourier trans-

forming system is to form in its output plane the intensitydistribution that is proportional to the spatial frequencypower spectrum of an object [11]. For the majority ofpractical applications the quality of the obtained spec-trum can be evaluated by spatial frequency resolution

assumed here to be the minimum difference betweenspatial frequencies of spectrum components which areregistered just separately. Besides, it is customary to con-sider that an object is illuminated by completely coherentand linearly polarized field (usually gas laser radiation),when the spatial frequency resolution reaches its lowerlimit. But, in some applications an object is illuminatedby partially coherent and partially polarized light of nat-ural or artificial origin. At the same time, to the best ofour knowledge, the effect of coherence and polarizationof vector electromagnetic illumination on the spectralfrequency resolution of optical Fourier transforming sys-tem has not been yet reported.Let us consider the process of generating the spatial

frequency power spectrum of an object placed just infront of a thin spherical lens as it is shown in Fig. 1.We assume that the object represents a semitransparentthin screen characterized at any position point x byamplitude transmittance tðxÞ. We assume also that theobject is illuminated by partially coherent and partiallypolarized field characterized by the cross-spectral den-sity matrix (for brevity we omit the explicit dependenceof the considered quantities on frequency ν) [2]

Winðx1; x2Þ ¼�W in

xxðx1; x2Þ W inxyðx1; x2Þ

W inyxðx1; x2Þ W in

yyðx1; x2Þ�; ð1Þ

where Wijðx1; x2Þ ¼ hE�i ðx1ÞEjðx2Þi with Ei and Ej being

the orthogonal components of the electric field vector,asterisk denoting the complex conjugate, and the anglebrackets denoting the average over the statistical ensem-ble. Then, using the paraxial approximation, it may beshown (see, e.g. [12]), that the cross-spectral densitymatrix of the field in the back focal plane of the lenshas the form

Woutðx01; x02Þ ¼1

ðλf Þ2 exp�iπλf ðx

021 − x022 Þ

� ZZ∞

−∞

t�ðx1Þtðx2Þ

×Winðx1; x2Þ

× exp

�−i

2πλf ðx

01 · x1 − x02 · x2Þ

�dx1dx2: ð2Þ

Correspondingly, the power spectrum of the field at theoutput of the system, expressed in the terms of the spatialfrequency vector p ¼ x0=λf , is given by

SoutðpÞ ¼ TrWoutðp; pÞ

¼ Tr1

ðλf Þ2ZZ

∞

−∞

t�ðx1Þtðx2Þ

×Winðx1; x2Þ exp½−i2πp · ðx1 − x2Þ�dx1dx2; ð3Þ

where Tr stands for the trace.To evaluate the spatial frequency resolution of the

system, we chose the one-dimensional model object ofthe form

tðxÞ ¼ cosð2πp0xÞ þ cos½2πðp0 þΔpÞx�; ð4Þ

where p0 is some fixed spatial frequency and Δp isthe spectral distance between two neighborimg spatial

Fig. 1. Optical Fourier transforming system.

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harmonics. Substituting from Eq. (4) into one-dimensional version of Eq. (3) and introducing thetwo-dimensional Fourier transform

~Winðp1; p2Þ ¼ZZ

∞

−∞

Winðx1; x2Þ

× exp½−i2πðp1x1 þ p2x2Þ�dx1dx2; ð5Þ

after straightforward calculation one finds that, for largeenough value of p0 and p > 0, the output power spectrumSoutðpÞ takes the form

SoutðpÞjp>0 ¼ Tr1

4ðλf Þ2 ½~Winðp − p0;−pþ p0Þ

þ ~Winðp − p0 −Δp;−pþ p0 þΔpÞ�: ð6Þ

The typical shape of the spectrum given by Eq. (6) issketched in Fig. 2. Observing this figure, we come tothe conclusion that spatial frequency resolution of theconsidered system is determined by the effective widthof function ~Winðp;−pÞ, i.e.,

ðΔpÞmin ¼ TrR∞

−∞

~Winðp;−pÞdpTr ~Winð0; 0Þ : ð7Þ

To make possible the analysis of the obtained result inclosed form, we assume that the object is illuminated bythe Gaussian Schell-model beam whose cross-spectraldensity matrix has the form [13]

Winðx1; x2Þ ¼�Wxxðx1; x2Þ 0

0 Wyyðx1; x2Þ�; ð8Þ

with elements

Wiiðx1; x2Þ ¼ S0i exp

�−

x21 þ x224σ2

�exp

�−

ðx1 − x2Þ22σ2i

�;

ð9Þwhere S0i is the power of the field component at theorigin of the object plane, σ is the effective width of thebeam, and σi is the parameter characterizing the trans-verse correlation length of the field component. As wellknown [14], the ratio γi ¼ σi=σ represents a measure ofthe degree of global (spatial) coherence of the field com-ponent with the cross-spectral density given by Eq. (9).Besides, as has been shown in [15], the ratio β ¼ S0x=S0y

(we assume that S0y ≠ 0) is associated with the degree ofpolarization P according to equality P ¼ j1 − βj=ð1þ βÞ.Hence, the set of values β, γx, and γy may be regardedas the intrinsic coherence-polarization structure of thevector Gaussian Schell-model beam defined by Eqs. (8)and (9).

On making use of Eqs. (5) and (9), we find that for thechosen illumination

~W iniiðp;−pÞ ¼ S0i

ZZ∞

−∞

exp

�−

x21 þ x224σ2

�exp

�−

ðx1 − x2Þ22σ2i

�

× exp½−i2πpðx1 − x2Þ�dx1dx2: ð10Þ

Introducing the new variables

ξ ¼ x1 − x2; η ¼ x1 þ x22

; ð11Þ

and using the integrals [16]

Z∞

0expð−a2u2Þdu ¼

ffiffiffiπp2a

ða > 0Þ; ð12Þ

Z∞

0expð−a2u2Þ expð−iuvÞdu ¼

ffiffiffiπpjaj exp

�−

v2

4a2

�; ð13Þ

we obtain

~W iniiðp;−pÞ ¼

4πσ2S0iγið4þ γ2i Þ1=2

exp

�−

8π2σ2γ2i4þ γ2i

p2�: ð14Þ

Finally, substituting from Eq. (14) into Eq. (7) and usingagain the integral (12), we find the following expressionfor the spatial frequency resolution of optical Fouriertransforming system:

ðΔpÞmin ¼�

1

2ffiffiffiffiffi2π

pσ

�1þ β

γxð4þγ2xÞ1=2 þ β γy

ð4þγ2yÞ1=2: ð15Þ

Expression (15) is the main result of the presentLetter which reveals the effect of the intrinsic coherence-polarization structure of illumination on the spatialfrequency resolution in optical Fourier transforming sys-tem. Indeed, as can be seen from Fig. 3(a) correspondingto the case of equal degrees of global coherence of ortho-gonal illumination components, i.e. γx ¼ γy ¼ γ, the spa-tial frequency resolution ðΔpÞmin decreases with theincrease of γ irrespective of the degree of polarizationstarting from infinity for γ ≪ 1 (completely incoherent il-lumination) and tending asymptotically to its lower limit1=2

ffiffiffiffiffi2π

pσ for γ ≫ 1 (completely coherent illumination).

On the other hand, as can be seen from Fig. 3(b), for γx ≠

γy (for definiteness we chose γx ¼ 1 and γy ¼ 0:1) thespatial frequency resolution ðΔpÞmin increases with theincrease of β starting from

ffiffiffi5

p=2

ffiffiffiffiffi2π

pσ for β ¼ 0 and tend-

ing asymptotically to its higher limitffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4=γ2y þ 1

q=2

ffiffiffiffiffi2π

pσ

for β → ∞.Fig. 2. The typical shape of the spectrum given by Eq. (6).

4720 OPTICS LETTERS / Vol. 36, No. 23 / December 1, 2011

The analysis given above shows that when designingan optical Fourier transforming system, to provide thebest spatial frequency resolution, one must give prefer-ence to the perfectly coherent illumination with any de-gree of polarization. When for some reasons it is notpossible, the partially coherent illumination may be used.In this case it is advisable to filter beforehand the mostcoherent in the global sense polarization component ofillumination; this can be done, e.g., employing the appro-priate polarization beam splitter. It is not out of place tonote here that in some applications it is not necessary togain the best frequency resolution. This situation occurs,e.g., in the spectral analysis of random objects, whenlow frequency resolution can be used advantageouslyto smooth the so-called “raw” spectrum with purposeof improving the statistical reliability of the obtained

spectral estimate [17,18]. In this case the function~Winðp;−pÞ plays the role of the so-called “spectral win-dow” whose effective width determines the degree ofstatistical smoothing [19]. Concluding, we mention thatthe illumination given by Eqs. (8) and (9) can be easilygenerated in physical experiment [20–23]. This allowsus to suppose that the results obtained in the presentLetter can find some useful applications in practice.

This work has been supported by the BeneméritaUniversidad Autónoma de Puebla under project VIEPOSA-EXC-11 G.

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Fig. 3. Dependence of the normalized spatial frequency reso-lution on the degree of global coherence (a) and the ratio of thepowers of polarization components (b) of the illumination field.

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