analytic relation for recovering the mutual intensity by means of intensity information

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Analytic relation for recovering the mutual intensity by means of intensity information Jinhong Tu and Shinichi Tamura Department of Informatics and Mathematical Science, Osaka University, Osaka, Japan Received February 24, 1997; revised manuscript received May 12, 1997; accepted September 2, 1997 An analytic relation for recovering the mutual intensity by means of intensity information under the condition of the fractional Fourier transform is derived. The results may simplify the reconstruction of the mutual in- tensity in comparison with the Wigner tomographic method and can be regarded as an inverse transform for- mula that expresses output intensity in terms of input mutual intensity under partially coherent illumination. © 1998 Optical Society of America [S0740-3232(98)02201-7] OCIS codes: 030.1640, 070.2590, 110.4980, 350.7420. 1. INTRODUCTION Wave-field determination is of some importance in many areas of science. The reconstruction of a complex wave field with a partially coherent state by measurements of intensity only has drawn the attention of many authors 19 in recent years. Nugent suggested that three- dimensional intensity information is sufficient to com- pletely characterize the second-order statistics of a gen- eral quasi-monochromatic wave field in the paraxial approximation, although the relation for recovering the cross-spectral-density function (CSDF) given by Nugent was proved by Hazak to be incomplete. 1 Later Raymer et al., 3 and McAlister et al. 4 presented a method using phase-space tomography from intensity measurements in a refractive optical system for reconstructing the full Wigner distribution function (WDF). The method is based on a rotation of the WDF 10 in phase space and a de- termination of the projected function for a sufficient set of values of the rotation angle, because the projected WDF of various rotation angles is related to the intensity by ad- justment of configuration parameters corresponding to different fractional Fourier transform orders. 1012 As a result, the WDF is normally reconstructed by using the filtered backprojection algorithm for the inverse Radon transform, and then the mutual intensity, i.e., the CSDF, can be obtained through a Fourier transform of the recon- structed WDF. Another recent method proposed by Gase et al. 5 uses a diffractive element to measure integrals of Page distribution, and these can be differentiated to yield the Page distribution itself, which, in turn, is inverted to reconstruct the CSDF. A more direct way to obtain the same information was more recently offered by a self- referencing interferometric method given by Iaconis and Walmsley. 6 In their experiment the real and the image parts of the CSDF were simply obtained through the in- tensity of the image field with different shear. The tomographic method, apart from its complicated mathematical treatment, is also simple: it uses only the lens, and less field power is lost; furthermore, the method may be straightforwardly extended to the other classical or equivalent classical wave fields such as tomography of atom beams 7 or chronocyclic tomography. 13 In addition, the method has been explained within the framework of the generalized Fresnel transform, and its application to obtaining the parameters of the refractive medium has been introduced. 8 The complicated mathematics of the tomographic technique are simplified by our recently pro- posed method, 9 which uses the ambiguity function (AF) theory. 14 In the theory, 9 by performing one-dimensional inverse Fourier transforms of intensities with some ad- justment in various longitudinal optical system param- eters, one obtains the corresponding AF values along the lines at different angles in the AF phase space, and there- fore one can reconstruct the mutual intensity function by performing a one-dimensional Fourier transform of the reconstructed AF values. The paper 9 dealt mainly with the Cartesian coordinate system and thus failed to lead to an analytic relation for recovering the mutual intensity. Although the AF method 9 for reconstructing the classi- cal wave field was naturally inspired by the direct Fourier transform algorithm known in the theory of computerized tomography, it is interesting to note that a similar idea that avoided the detour via the WDF was adopted by Kuhn et al. 15 in the quantum-mechanical context, using completely different algebra. In fact, since Smithey et al. 16 first succeeded in determining the density matrix from the reconstructed Wigner function following a pro- posal later named quantum tomography by Vogel and Risken, 17 the experimental work stimulated a series of theoretical studies. 15,18,19 In particular, a simple ana- lytic relation that connects the density operator of the electromagnetic field with the tomographic homodyne probabilities by means of a sampling function has been provided by D’Ariano et al. 18 As there exists a math- ematically analogous description between a scalar elec- tromagnetic field and the quantum-mechanical wave function of a matter wave, the theory of wave determina- tion with use of a refractive optical system is closely re- lated to the theory of wave determination through homo- dyne detection in the context of quantum optics. 3,9 In particular, the CSDF in optics may be represented as the density matrix in the coordinate representation. 202 J. Opt. Soc. Am. A / Vol. 15, No. 1 / January 1998 J. Tu and S. Tamura 0740-3232/98/010202-05$10.00 © 1998 Optical Society of America

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Page 1: Analytic relation for recovering the mutual intensity by means of intensity information

202 J. Opt. Soc. Am. A/Vol. 15, No. 1 /January 1998 J. Tu and S. Tamura

Analytic relation for recovering the mutualintensity by means of intensity information

Jinhong Tu and Shinichi Tamura

Department of Informatics and Mathematical Science, Osaka University, Osaka, Japan

Received February 24, 1997; revised manuscript received May 12, 1997; accepted September 2, 1997

An analytic relation for recovering the mutual intensity by means of intensity information under the conditionof the fractional Fourier transform is derived. The results may simplify the reconstruction of the mutual in-tensity in comparison with the Wigner tomographic method and can be regarded as an inverse transform for-mula that expresses output intensity in terms of input mutual intensity under partially coherent illumination.© 1998 Optical Society of America [S0740-3232(98)02201-7]

OCIS codes: 030.1640, 070.2590, 110.4980, 350.7420.

1. INTRODUCTIONWave-field determination is of some importance in manyareas of science. The reconstruction of a complex wavefield with a partially coherent state by measurements ofintensity only has drawn the attention of manyauthors1–9 in recent years. Nugent suggested that three-dimensional intensity information is sufficient to com-pletely characterize the second-order statistics of a gen-eral quasi-monochromatic wave field in the paraxialapproximation, although the relation for recovering thecross-spectral-density function (CSDF) given by Nugentwas proved by Hazak to be incomplete.1 Later Raymeret al.,3 and McAlister et al.4 presented a method usingphase-space tomography from intensity measurements ina refractive optical system for reconstructing the fullWigner distribution function (WDF). The method isbased on a rotation of the WDF10 in phase space and a de-termination of the projected function for a sufficient set ofvalues of the rotation angle, because the projected WDF ofvarious rotation angles is related to the intensity by ad-justment of configuration parameters corresponding todifferent fractional Fourier transform orders.10–12 As aresult, the WDF is normally reconstructed by using thefiltered backprojection algorithm for the inverse Radontransform, and then the mutual intensity, i.e., the CSDF,can be obtained through a Fourier transform of the recon-structed WDF. Another recent method proposed by Gaseet al.5 uses a diffractive element to measure integrals ofPage distribution, and these can be differentiated to yieldthe Page distribution itself, which, in turn, is inverted toreconstruct the CSDF. A more direct way to obtain thesame information was more recently offered by a self-referencing interferometric method given by Iaconis andWalmsley.6 In their experiment the real and the imageparts of the CSDF were simply obtained through the in-tensity of the image field with different shear.

The tomographic method, apart from its complicatedmathematical treatment, is also simple: it uses only thelens, and less field power is lost; furthermore, the methodmay be straightforwardly extended to the other classicalor equivalent classical wave fields such as tomography of

0740-3232/98/010202-05$10.00 ©

atom beams7 or chronocyclic tomography.13 In addition,the method has been explained within the framework ofthe generalized Fresnel transform, and its application toobtaining the parameters of the refractive medium hasbeen introduced.8 The complicated mathematics of thetomographic technique are simplified by our recently pro-posed method,9 which uses the ambiguity function (AF)theory.14 In the theory,9 by performing one-dimensionalinverse Fourier transforms of intensities with some ad-justment in various longitudinal optical system param-eters, one obtains the corresponding AF values along thelines at different angles in the AF phase space, and there-fore one can reconstruct the mutual intensity function byperforming a one-dimensional Fourier transform of thereconstructed AF values. The paper9 dealt mainly withthe Cartesian coordinate system and thus failed to lead toan analytic relation for recovering the mutual intensity.

Although the AF method9 for reconstructing the classi-cal wave field was naturally inspired by the direct Fouriertransform algorithm known in the theory of computerizedtomography, it is interesting to note that a similar ideathat avoided the detour via the WDF was adopted byKuhn et al.15 in the quantum-mechanical context, usingcompletely different algebra. In fact, since Smitheyet al.16 first succeeded in determining the density matrixfrom the reconstructed Wigner function following a pro-posal later named quantum tomography by Vogel andRisken,17 the experimental work stimulated a series oftheoretical studies.15,18,19 In particular, a simple ana-lytic relation that connects the density operator of theelectromagnetic field with the tomographic homodyneprobabilities by means of a sampling function has beenprovided by D’Ariano et al.18 As there exists a math-ematically analogous description between a scalar elec-tromagnetic field and the quantum-mechanical wavefunction of a matter wave, the theory of wave determina-tion with use of a refractive optical system is closely re-lated to the theory of wave determination through homo-dyne detection in the context of quantum optics.3,9 Inparticular, the CSDF in optics may be represented as thedensity matrix in the coordinate representation.

1998 Optical Society of America

Page 2: Analytic relation for recovering the mutual intensity by means of intensity information

J. Tu and S. Tamura Vol. 15, No. 1 /January 1998 /J. Opt. Soc. Am. A 203

For the reader’s convenience, in Section 2 we give asimple review of the important relation between the AF inthe input plane and the intensity in a set of outputplanes, which is equivalent to the results given in Ref. 9.In Section 3, inspired by the result given in the context ofquantum optics,18 we derive an analytic relation for re-covering the mutual intensity by means of intensity infor-mation in classical optics by considering a fractional Fou-rier transform system, which is often used in thetomographic method.3,4 Because of the analogy betweenthe classical optics and quantum optics, it is not surpris-ing that this analytic relation takes the same mathemati-cal forms as those derived in Ref. 18 in the context ofquantum optics with use of operator algebra. Since theinterpretation of the mathematical equation and thephysical concept are different in the two fields, the modi-fication based on the limitation of the classical optical ex-periment is discussed so that the unbounded problem ofthe integral kernel is avoided.

2. RELATION BETWEEN THE AMBIGUITYFUNCTION IN THE INPUT PLANEAND THE INTENSITY IN THE OUTPUTPLANESAs is known, a temporally stationary light source is usu-ally described by the mutual coherence functionG(r1 , r2 , t) defined by the ensemble average,

G~r1 , r2 , t! 5 ^C~r1 , t !C* ~r2 , t 1 t!&, (1)

where C(r, t) is the analytic signal associated with onetransverse component of the electric field vector and r5 (x, y). The Fourier transform of G with respect to thedelay t is called the CSDF and is given by

W ~r1 , r2 , v! 5 E G~r1 , r2 , t!exp~2ivt!dt. (2)

For convenience, we neglect the time and temporal fre-quency dependence by restricting our discussions toquasi-monochromatic illumination, so that the frequencycoordinate v is omitted. This simplified quantity iscalled the mutual intensity and is given by14

J ~r1 , r2 , z ! 5 ^F~r1 , z !F* ~r2 , z !&, (3)

where z denotes the longitudinal location of the consid-ered xy plane. For simplicity we take only onetransverse-dimensional case into consideration. Two-dimensional case can be extended according to Refs. 3 and8. Also, we omit some unimportant constants before theintegrals in the following deductions, which do not affectthe result. Therefore one-dimensional mutual intensitycan be expressed in terms of the center and the differencecoordinates x and Dx, respectively:

J~x, Dx, z ! as

J~x, Dx, z ! 5 J ~x1 , x2 , z !, (4)

with

x 5 ~x1 1 x2!/2, (5)

Dx 5 x1 2 x2 . (6)

The mutual intensity J(x, Dx, z) at z can be trans-formed into the light intensity distribution functionI(x, z) at z by letting Dx 5 0, as in the following equa-tion:

I~x, z ! 5 J~x, Dx 5 0, z !. (7)

In an actual experiment, the intensity distributionI(x, z) is measured by an optical intensity sensor such asa CCD camera located at a distance z from the inputplane, the sensor element has finite sampling width (i.e.,resolution), and the output plane may have a valid extentsuch as the finite size of the lens aperture or the totalnumber of sensor elements. However, in many theoreti-cal treatments, the sampling width is assumed to be zeroand the measuring extent x is assumed to be infinity.For simplicity, we first accept these assumptions to con-sider the ideal case, and then we discuss the problem ofthe actual case.

Considering the one-dimensional Fourier transforms ofthe mutual intensity J(x, Dx), we arrive at the AF andthe WDF of the wave field as

A~Dn, Dx, z ! 5 E J~x, Dx, z !exp~2i2pDnx !dx, (8)

W~x, n, z ! 5 E J~x, Dx, z !exp~2i2pDxn!dDx,

(9)

related by

A~Dn, Dx, z ! 5 EE W~x, n, z !

3 exp@i2p~Dxn 2 xDn!#dxdn,

(10)

where Dn is the difference of the spatial frequency n. Inthe above and in the following deductions the integralsare from 2` to ` without declaration.

Transforming the AF in Eq. (10) into the polar coordi-nate system with

j 5 ~Dn2 1 Dx2!1/2, (11)

Dn 5 j cos u, (12)

Dx 5 j sin u, (13)

yields

A~j cos u, j sin u, z !

5 EE W~x, n, z !

3 exp@2i2pj~x cos u 2 n sin u!#dxdn. (14)

In the Fresnel approximation, as is well known, thepropagation of the AF and the WDF in a first-order sys-tem (i.e., through a distance or a transparent inhomoge-neous medium, such as a thin lens or a quadratic graded-index medium) can be described by a certain transportmatrix M(z), the so-called A B C D matrix, as

M~z ! 5 FA B

C DG . (15)

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204 J. Opt. Soc. Am. A/Vol. 15, No. 1 /January 1998 J. Tu and S. Tamura

Hence, the WDF in the output plane at z is related to theWDF in the input plane at z 5 0 by

W~x, n, z ! 5 W~A x 1 B n, C x 1 D n, 0!. (16)

Let I(h8, z) denote the Fourier transform of the lightintensity distribution at z as

I~h8, z ! 5 E I~x8, z !exp~2i2ph8x8!dx8. (17)

As the light intensity distribution function at z can beobtained in terms of the WDF at z by10

I~x8, z ! 5 E W~x8, n, z !dn, (18)

taking the Fourier transforms in both sides of the Eq. (18)and then taking the transformation of the variants of x8,n8, and h8 in terms of x, n and h, respectively, accordingto the following definition,

S x8n8 D 5 Fa b

c dG S xn D , (19)

h8 5 h/Aa2 1 b2, (20)

where a, b, c, and d are the components of the inversematrix of the A B C D matrix given by9

Fa b

c dG 5 F D 2B

2C AG , (21)

and using relation (15), we obtain

I~h/Aa2 1 b2, z ! 5 EE W~x, n, z 5 0 !

3 exp@2i2ph~xa/Aa2 1 b2

1 nb/Aa2 1 b2!#dxdn. (22)

Comparing Eq. (22) with Eq. (14), we obtain an importantrelation,

I~h/Aa2 1 b2, z ! 5 A~ha/Aa2 1 b2, 2hb/Aa2 1 b2, 0!,(23)

which is equivalent to Eq. (22) of Ref. 9.

3. DIRECT AND ANALYTIC SOLUTION TOTHE PROBLEMThe algorithm proposed in Ref. 9 is, in principle, not con-fined to the fractional Fourier transform system; however,the explicit analytic relation between the mutual inten-sity in the object plane and the intensity in a set of outputplanes is not given in Ref. 9 that is, we cannot obtain theanalytic solution to this problem by directly substitutingEq. (23) or its counterpart, Eq. (22) in Ref. 9, into the fol-lowing relation as

J~x, Dx, 0! 5 E A~Dn, Dx, 0!exp~i2pDnx !dDn.

(24)

It is interesting that both of the equivalentsolutions15,18 to the density matrix in the quadrature rep-resentation (in the context of quantum optics) are in ex-plicit forms. Thus one may ask whether the solution in

classical optics is analogous, and the answer is yes: wefind that a relation for recovering the mutual intensity inclassical optics that is completely analogous to that for re-covering the density matrix in quantum optics by usingphase-space tomography requires that the fractional Fou-rier transform configuration be used.

It is found that under certain conditions the first-ordersystem can be managed to be the fractional Fourier trans-form system.3,4,10,12 The transport matrix M(z) associ-ated with the fractional Fourier transform of order f isthe rotation matrix12

Mf~z ! 5 F cos f sin f

2sin f cos fG , (25)

in which f is also viewed as rotation angle in the phasespace of the WDF. From Eq. (10) we may write an in-verse form of the relation between the WDF and the AFas

W~x, n, z ! 5 EE A~Dn, Dx, z !

3 exp@2i2p~Dxn 2 xDn!#dDxdDn.

(26)

By transforming the integral of Eq. (26) into the polar co-ordinate system through Eqs. (11)–(13), we can write theWDF at z 5 0 in the form of the AF as

W~x, n, 0! 5 E0

pE2`

1`

A~j cos u, j sin u, 0!

3 exp@2i2pj~n sin u 2 x cos u!#ujudjdu.

(27)

It is obvious9 that the rotation angle f in WDF space10

is equal to the one in AF space, denoted u in the system offractional Fourier transform.

Putting Eqs. (17) and (23) into Eq. (27) and consideringthe configuration of the fractional Fourier transform of adifferent fractional order f according to Eq. (25), we ob-tain the algorithm for recovering the WDF, called the in-verse Radon transformation:

W~x, n, 0! 5 E0

pE2`

1`E2`

1`

If~x8, z !

3 exp@2i2pj~x8 1 n sin f

2 x cos f!#dx8ujudjdf, (28)

which may be rewritten as

W~x, n, 0! 5 E0

pE2`

1`

If~x8, z !g~x8 1 n sin f

2 x cos f!dx8df, (29)

where

g~x ! 5 E2`

1`

ujuexp~2i2pjx !dj. (30)

As is well known in the theory of computerizedtomography,7 the function g(x) is unbounded, and a cutoff

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J. Tu and S. Tamura Vol. 15, No. 1 /January 1998 /J. Opt. Soc. Am. A 205

frequency jm needs to be introduced to act as a filter func-tion in the numerical calculation:

g~x ! ' gm~x !,

5 E2jm

1jm

ujuexp~2i2pjx !dj

5 2F21

x2 1cos~jmx !

x2 1jm sin~jmx !

x G . (31)

To obtain the mutual intensity, recall the relation be-tween mutual intensity and the WDF given by

J ~x1 , x2 , 0! 5 E WS x1 1 x2

2,n, z 5 0 D

3 exp@i2p~x1 2 x2!n#dn. (32)

Putting Eq. (28) into Eq. (32) and exchanging the inte-grals over x8 and f with respect to the outer integral overj, we obtain an analytic relation for recovering the mutualintensity with x1 Þ x2 by means of the intensity measure-ment under the condition of the fractional Fourier trans-form system by

J ~x1 , x2 , 0! 5 E0

p

dfE2`

1`

dx8If~x8, z !K~x1 , x2 , f, x8!,

(33)

where

K~x1 , x2 , f, x8!

5 E f~j sin f 1 x2 2 x1!

3 expF2i2pjS x8 2x1 1 x2

2cos f D G ujudj, (34)

in which

f~x ! 5 E exp~2i2pnx !dn 5 d~x !. (35)

By using properties of the d function and assuming fnot too close to np (n 5 0, 61, 62, ...), we can obtain ananalytic form of the integral kernel K (x1 , x2 , f, x8) as18

K~x1 , x2 , f, x8! 5ux1 2 x2u

sin2 fexpH 2i2p

x1 2 x2

sin f

3 Fx8 21

2~x1 1 x2!cos fG J .

(36)

As indicated in Refs. 18 and 20 the integral kernel K(x1 , x2 , f, x8) will be unbounded and diverges as f5 np (n 5 0, 61, 62, ...), so that it is not possible tosample the density matrix in quadrature representationsfor its perfect reconstruction in quantum optics; however,in the case of classical optics, as noted in Section 1, thesampling intervals and the extent depend on the sensorand/or the size of illumination. That is, as in the treat-ment of the filtered backprojection algorithm,7 the cutoffspatial frequency nm , which is dependent on the resolu-tion of the sensor, and the cutoff frequency of the polar co-

ordinate jm , which is dependent on the valid extent of thesensor in the x coordinate, can be naturally introduced.As a result, the error introduced in recovering the mutualintensity might be due not only to the theoretical approxi-mations but also to the experimental limitations. There-fore, in the numerical evaluation, the integral kernel K(x1 , x2 , f, x8) can be modified as

K~x1 , x2 , f, x8!

5 E2jm

jm

sinc@nm~j sin f 1 x2 2 x1!#

3 expF2i2pjS x8 2x1 1 x2

2cos f D G ujudj, (37)

where sinc(x) 5 sin(px)/px.It is clear that we may arrive at an analytical solution

to the integral kernel that is just like Eq. (36) when fÞ np (n 5 0, 61, 62, ...) if the cutoff frequencies nm andjm in Eq. (37) are large enough to be approximately infin-ity. Therefore Eq. (37) might be seen as a generalizedformula of the integral kernel for the reconstruction of themutual intensity.

Equation (37) might be useful for the numerical calcu-lation; in particular, when f 5 np (n 5 0, 61, 62, ...),the kernel K (x1 , x2 , f 5 np, x8) is well behaved withuse of using Eq. (31) as

K~x1 , x2 , f 5 np, x8!

5 sinc@nm~x2 2 x1!#gmS x8 2x1 1 x2

2 D . (38)

Finally, we obtain the diagonal elements of the mutualintensity according to Eqs. (23) and (24) by setting x15 x2 as

J ~x1 , x1 , 0! 5 E2`

1`

dx8Ib50~x8, z !d~x1 2 x8/a !,

(39)

where parameters a and b show the imaging condition orcorrespond to the case of f 5 np (n 5 0, 61, 62, ...) inthe fractional Fourier transform system.

It can be noted that using the results discussed abovewill simplify the reconstruction of the mutual intensity incomparison with using the WDF tomographic method.3

In the latter, in addition to the complicated reconstruc-tion of the WDF based on Eqs. (28)–(31), another one-dimensional Fourier transform must be performed on thereconstructed two-dimensional WDF to obtain the mutualintensity function, whereas Eq. (33) contains only inte-grals of x8 and f, and the integral kernel does not dependon the specific distribution of the intensity and can be cal-culated beforehand and only once. It is worth notingthat the method proposed in this paper is equivalent tothe method of using the two Fourier transform methodsproposed in Ref. 9 as a special case of the fractional Fou-rier transform system used here. The analytical form ofthe method in Ref. 9 is readily obtained15,20 after somechanging of integration variables from the method pro-posed in this paper. The difference is that the methodproposed here recovers J (x1 , x2 , z), whereas the

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206 J. Opt. Soc. Am. A/Vol. 15, No. 1 /January 1998 J. Tu and S. Tamura

method in Ref. 9 recovers J (x, Dx, z) (density matrix infield-strength base in the context of quantum optics20).

Furthermore, according to a recent study of the propa-gation theory of mutual intensity expressed in terms ofthe fractional Fourier transform,21 it is interesting to notethat Eq. (33) is just the inverse transform formula of thepropagation relation given by Eq. (24) of Ref. 21, wherex1 5 x2 to express output intensity in terms of input mu-tual intensity. Therefore the results in the current papermay provide another view to understand the theory ofpropagation under partially coherent illumination.

4. CONCLUSIONIn this paper, from the relation between the input AF andthe output intensities, we derived an analytic relation forrecovering the mutual intensity by means of intensity in-formation under the condition of the fractional Fouriertransform system, and we discussed the case of actual nu-merical implementation. The results may simplify thereconstruction of the mutual intensity in comparison withusing the WDF tomographic method and may be regardedas an inverse transform formula that expresses output in-tensity with input mutual intensity in terms of the frac-tional Fourier transform under partially coherent illumi-nation. As an application, the extension of this theoryto tomography of atom beams7 and chronocyclictomography13 is straightforward.

ACKNOWLEDGMENTThe authors thank the reviewers for their important com-ments.

The authors are also at Division of Functional Diagnos-tic Imaging, Medical School, Osaka University, Osaka,Japan.

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