quasi-stationary plane-wave optical pulses and the van cittert-zernike theorem in time
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2530 J. Opt. Soc. Am. A/Vol. 23, No. 10 /October 2006 Lajunen et al.
Quasi-stationary plane-wave optical pulses andthe van Cittert–Zernike theorem in time
Hanna Lajunen* and Ari T. Friberg
Royal Institute of Technology (KTH), School of Information and Communication Technology, Electrum 229,SE-164 40 Kista, Sweden
Petter Östlund
Sony Ericsson Mobile Communications AB, SE-221 88 Lund, Sweden
Received April 21, 2006; accepted May 16, 2006; posted May 19, 2006 (Doc. ID 70151)
We study the properties of quasi-stationary, partially coherent, plane-wave optical pulses in the space–timeand space–frequency domains. A generalized van Cittert–Zernike theorem in time is derived to describe thepropagation of the coherence function of quasi-stationary pulses. The theory is applied to rectangular pulseschopped from a stationary light source, and the evolution characteristics of such pulse trains with differentstates of coherence are discussed and illustrated with numerical examples. © 2006 Optical Society of America
OCIS codes: 030.1640, 320.5550, 030.6600, 060.5530.
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. INTRODUCTIONn wave optics, light is often modeled as deterministicarmonic waves, but all real fields in nature contain someandom fluctuations. The well-established formalism ofptical coherence theory is routinely used to account forhe random features of statistically stationary fields.1,2
ut also nonstationary fields, such as light pulses andransient optical waves, may exhibit statistical variationsn their temporal and spectral characteristics. Severalew techniques of statistical coherence theory have re-ently been developed and have been applied for the stud-es of nonstationary optical phenomena.3–9
The concept of quasi-homogeneous sources, based on aathematical model initially introduced for nonstation-
ry random processes,10 is often used for describing thepatial coherence properties of stationary fields.1,11 Inhat case it is assumed that the spatial intensity profile islowly varying compared with the effective coherencerea of the field. The far-zone behavior of such quasi-omogeneous fields is governed by a generalized form ofhe classic van Cittert–Zernike theorem,2,11,12 which inhe special case of incoherent source fields states that theegree of coherence at a sufficient distance is proportionalo the Fourier transform of the intensity distributioncross the source. Following from the well-known dualityetween the paraxial diffraction of beam fields and theropagation of pulses in linearly dispersive media,13
nalogous definitions and laws can be applied as well forartially coherent plane-wave pulses. The mathematicalodel of quasi-stationary pulses10 has been briefly men-
ioned even in this context,7,14 and a form of the vanittert–Zernike theorem for a time-gated, temporally
ully incoherent source has been formulated.14
In this paper we provide a more detailed analysis of theemporal and spectral coherence properties of quasi-tationary pulses, including a discussion on the limiting
1084-7529/06/102530-8/$15.00 © 2
ases of temporally incoherent pulses and stationaryelds. The accurate propagation formulas for such pulsesre presented, and an approximative form valid for suffi-iently long propagations distances, corresponding to aime-domain generalized van Cittert–Zernike theorem, iserived. While the properties of Gaussian Schell-modelulses6 in dispersive media have already been studied byarious authors,13,15,16 we consider in this paper the alter-ative, practically important example of partially coher-nt rectangular plane-wave pulses chopped from a sta-ionary light source.
The paper is organized as follows. In Section 2 we recallhe basic definitions for partially coherent plane-waveulses and present the general laws that govern theirropagation in linearly dispersive media. The concept ofuasi-stationary pulses is introduced in Section 3, andheir properties are discussed both in the space–time andpace–frequency domains. The propagation law for quasi-tationary pulses in linearly dispersive media, taking theorm of the van Cittert–Zernike theorem in time, is de-ived in Section 4. In Section 5 the presented theory is ap-lied to partially coherent chopped pulses to illustrate theropagation characteristics of such fields and to elucidatehe limits of the quasi-stationary approximation. Finally,he main results and conclusions are summarized in Sec-ion 6.
. PARTIALLY COHERENT PULSES ANDHEIR PROPAGATION IN DISPERSIVEEDIA
et us consider (nonstationary) scalar plane-wave opticalulses that are described in complex field notation by�z , t�, where z is the axial coordinate in the propagationirection and t denotes the time. The mutual coherenceunction for such fields is defined as
006 Optical Society of America
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��z1,z2,t1,t2� = �U*�z1,t1�U�z2,t2��, �1�
here the asterisk denotes complex conjugation and thengle brackets indicate ensemble averaging over all pos-ible field realizations. In practice, the ensemble may con-ist of, e.g., a train of pulses emitted by some nonstation-ry source, in which case each pulse in the trainorresponds to one member of the ensemble.9,15 The mu-ual coherence function considered in a single plane z andt one instant of time t gives the average intensity distri-ution of the field realizations I�z , t�=��z ,z , t , t���U�z , t��2�. The normalized form of the mutual coherence
unction
��z1,z2,t1,t2� =��z1,z2,t1,t2�
�I�z1,t1�I�z2,t2�. �2�
s the complex degree of coherence.Since we are considering temporally finite pulses, the
eld realizations are square-integrable with respect toime, and they can be represented in the space–frequencyomain by their Fourier transforms
U�z,�� =1
2��
−�
�
U�z,t�exp�i�t�dt, �3�
here � denotes the angular frequency. We note that forptical pulses in complex notation U�z ,���0 for ��0.he correlations between different frequency componentsf the field realizations in two different planes z1 and z2re measured by the cross-spectral density
W�z1,z2,�1,�2� = �U*�z1,�1�U�z2,�2��, �4�
hich at a single frequency � and in a given plane z re-uces to the average spectral density S�z ,��W�z ,z ,� ,��= ��U�z ,���2�. In analogy to Eq. (2), the com-lex degree of spectral coherence is defined as
��z1,z2,�1,�2� =W�z1,z2,�1,�2�
�S�z1,�1�S�z2,�2�, �5�
.e., as the normalized form of the cross-spectral density.The coherence functions in the space–time and space–
requency domains are connected by the generalizediener–Khintchine relations. Using definitions (1), (3),
nd (4) we get the cross-spectral density
W�z1,z2,�1,�2� =1
�2��2 � �−�
�
��z1,z2,t1,t2�
�exp− i��1t1 − �2t2�dt1dt2, �6�
nd the mutual coherence function is obtained by the in-erse Fourier transform
��z1,z2,t1,t2� =� �−�
�
W�z1,z2,�1,�2�
�expi��1t1 − �2t2�d�1d�2. �7�
gain, for random optical pulses in complex field repre-entation, W�z1 ,z2 ,�1 ,�2� is effectively zero for negativerequencies � and � .
1 2The propagation of partially coherent plane-waveulses in linearly dispersive media may be studiedtraightforwardly in the space–frequency domain.15 Allhe frequency components of the field realizations mustatisfy the Helmholtz equation
�2
�z2U�z,�� + 2���U�z,�� = 0, �8�
here ���=n���� /c is the propagation constant; here��� is the refractive index of the material at frequency �,is the speed of light in vacuum, and we recall that onlyositive frequencies � are present. Assuming that theulses are known in some initial plane z=0, the generalolution can be expressed as
U�z,�� = U�0,��A�z,��, �9�
here A�z ,��=expi���z is the frequency-dependentransfer function. We emphasize that since we considerulses in homogeneous dispersive media and assume thato light is incident from infinity at right, only propagation
n the positive z direction appears above in the solution ofhe Helmholtz equation. The propagation law for theross-spectral density in a dispersive medium,
W�z1,z2,�1,�2� = W�0,0,�1,�2�A*�z1,�1�A�z2,�2�,
�10�
s now readily obtained by inserting Eq. (9) into Eq. (4).The representation for the propagated pulses in the
pace–time domain can be obtained by a Fourier trans-orm of Eq. (9). Taking definition (3) in the plane z=0 intoccount we can express the propagation law in the timeomain as
U�z,t� =1
2��
−�
�
U�0,t��A�z,t − t��dt�, �11�
here
A�z,t� =�−�
�
A�z,��exp�− i�t�d�. �12�
nserting Eq. (11) into Eq. (1) gives
��z1,z2,t1,t2� =1
�2��2 � �−�
�
��0,0,t1�,t2��
�A*�z1,t1 − t1��A�z2,t2 − t2��dt1�dt2� , �13�
hich constitutes the general propagation law for the mu-ual coherence function in the space–time domain, a coun-erpart to Eq. (10).
It is often convenient to express the propagation con-tant ��� as a Taylor expansion about the central fre-uency �0 of the pulses. If the refractive index can be ap-roximated by a linear function over the whole frequencyange covered by the random pulses, the spectral transferunction is then expressible as
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2532 J. Opt. Soc. Am. A/Vol. 23, No. 10 /October 2006 Lajunen et al.
A�z,�� = exp�i��0� + �� − �0�/vg + a�� − �0�2z�,
�14�
here vg is the group velocity and a is a parameter rep-esenting the dispersion of the group velocity.15 Using thisssumption, the propagated cross-spectral density is theneadily obtained by inserting Eq. (14) into Eq. (10). In thepace–time domain, on the other hand, we first find17
rom Eqs. (12) and (14)
A�z,t� =� i�
azexp�i��0�z − �0t�exp − i
�t − z/vg�2
4az � ,
�15�
nd the propagation law defined by Eq. (13) takes on theorm
��z1,z2,t1,t2� =exp�− i��0��z1 − z2� − �0�t1 − t2��
4�a�z1z2�1/2
�� �−�
�
��0,0,t1�,t2��exp− i�0�t1� − t2��
� exp�it1� − �t1 − z1/vg�2
4az1
− it2� − �t2 − z2/vg�2
4az2�dt1�dt2� . �16�
his is the basic formula for the propagation of the mu-ual coherence function in a linearly dispersive medium.etting z1=z2=z it can be used to study the temporal co-erence properties of the propagated pulses in any givenlane z.However, before proceeding, let us consider some limit-
ng cases of the material medium. If the group velocityispersion a=0, but otherwise Eq. (14) remains valid, werst find that A�z , t�=2� exp�i��0�−�0 /vgz��t−z /vg�,here is the Dirac delta function, and Eq. (13) gives
��z1,z2,t1,t2� = exp�− i��0� − �0/vg�z1 − z2��
���0,0,t1 − z1/vg,t2 − z2/vg�. �17�
f the medium is vacuum, ���=� /c, and the group veloc-ty vg=c. In this case A�z , t�=2��t−z /c� and�z1 ,z2 , t1 , t2�=��0,0, t1−z1 /c , t2−z2 /c�, indicating nondis-ersive pulse propagation at speed c.
. QUASI-STATIONARY PULSESn this section we introduce the concept of quasi-tationary pulses and discuss their properties in thepace–time and space–frequency domains. The propertiesf such plane-wave optical pulses will be considered in aiven plane z1=z2=z, and thus, for simplicity, the spatialependence is not explicitly shown in the following ex-ressions.Let us assume that the complex degree of coherence of
he light pulses is solely a function of the time difference
� = t2 − t1, �18�
nd that the temporal width of the pulses is considerablyarger than the coherence time. Further, the intensity dis-ribution is assumed to be slowly varying compared withhe effective range of the function describing the degree ofemporal coherence. In such a situation we can use theemporal equivalent of the quasi-homogeneous approxi-ation for the mutual coherence function18
��t1,t2� = �I�t1�I�t2���t1,t2� � I�t����� = ��t,��, �19�
here
t = �t1 + t2�/2 �20�
s the average time instant. In other words, the intensityistribution is assumed to remain approximately constantver the time span during which the degree of coherences appreciably different from zero.
In the space–frequency domain we introduce the newariables
� = ��1 + �2�/2, �21�
� = �2 − �1, �22�
hich correspond to t and �. Using these definitions,q. (6) can be expressed in the general form
W��1,�2� = W��,�� =1
�2��2 � �−�
�
��t,��
�expi��t + ���dtd�, �23�
hich gives the cross-spectral density as a function of theverage frequency � and the frequency difference �. Us-ng the quasi-stationary approximation defined byq. (19), the cross-spectral density takes on the form
W��,�� = W1���W2���, �24�
here
W1��� =1
2��
−�
�
����exp�i���d�, �25�
W2��� =1
2��
−�
�
I�t�exp�i�t�d�. �26�
hus it is seen that the cross-spectral density of quasi-tationary pulses can be expressed as a product of twoeparate functions of � and � that are obtained by Fou-ier transforms of the complex degree of coherence andhe intensity distribution, respectively.
Comparing expressions (19) and (24), it would beempting to assume that functions W1��� and W2���ould directly equal the spectral density and the complexegree of spectral coherence, respectively, of quasi-tationary pulses. However, this interpretation is not ex-ctly correct. According to the definition, the spectrum isbtained from the cross-spectral density when �1=�2 inhich case �=0. Thus we find from Eq. (24)
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Lajunen et al. Vol. 23, No. 10 /October 2006 /J. Opt. Soc. Am. A 2533
S��� = W��,0� = W1���W2�0�, �27�
hich shows that the spectral density is proportional tohe function W1��� multiplied by a constant,
W2�0� =1
2��
−�
�
I�t�dt, �28�
hich equals the time-integrated average intensity of theulses.Furthermore, from definition (5) and Eq. (27) we obtain
���1,�2� =W1���
�W1��1�W1��2�
W2���
W2�0�=
S���
�S��1�S��2�
W2���
W2�0�,
�29�
hich shows that generally the complex degree of spectraloherence depends on both functions W1��� and W2���.owever, taking into account the assumptions that I�t� isslowly varying function of time and ���� correspondinglyas a narrow effective width, it follows from the knownroperties of Fourier transforms that W1��� may be con-idered to be slowly varying compared with W2���. Thus,rovided the functions are sufficiently smooth and wellehaved, we may usually use the approximation S����S��1�S��2� in Eq. (29). In that case the complex degree
f coherence takes on the form
���1,�2� � ���� =W2���
W2�0�, �30�
nd we see that it is directly proportional to the normal-zed Fourier transform of the intensity distribution of theuasi-stationary pulses. The cross-correlation functionan now be expressed in the form
W��,�� � S�������, �31�
orresponding to Eq. (19), where S��� and ���� are de-ned by Eqs. (27) and (30).As a limiting case we approach the situation in which
he pulses are temporally completely incoherent (whiteoise pulses). The mutual coherence function can then, inractice,19 be approximated in the form ��t1 , t1�=I�t1��t2t1�, where is the Dirac delta function. Thus we may
ormally write ����=��� which, when inserted in Eq. (25),eads to the result that the spectral density is constantver all the frequencies. The degree of spectral coherencetill depends on the intensity distribution, so temporallyncoherent pulses obtained from random white-noise ra-iation are generally not also spectrally incoherent.On the other hand, if the spectral correlations are as-
umed to vanish, we approach the familiar concept of sta-ionary fields that can be considered as a special case ofhe quasi-stationary approximation. If the average inten-ity is taken to have a constant value over time, the cross-pectral density is according to Eqs. (24)–(26) of the formˆ �� ,��=W1������. Here the function W1��� effectivelyepresents the spectrum of the stationary field that is ob-ainable from the Fourier transform of the complex de-ree of coherence, complying with the established rules ofhe coherence theory of stationary fields. While the exactnalysis of temporally infinite fields requires the use of
ophisticated mathematical theories, such complicationsre easily avoided with the help of the quasi-stationarypproximation for fields with finite extent, which also bet-er corresponds to the actual physical situation.
. VAN CITTERT–ZERNIKE THEOREM INIME
n the following we examine the propagation of quasi-tationary pulses in linearly dispersive media. For thaturpose, the assumptions and definitions of the previousection will remain the same, but now the spatial depen-ence is added explicitly into the functions. However, inrder to concentrate on the changes in the temporal co-erence properties of quasi-stationary pulses, we restricturselves to the study of the evolution of the coherenceunction only between constant planes, so that z1=z2=z.
Let us consider the propagation law for the mutual co-erence function in linearly dispersive media given by Eq.16). If we change the variables t1 and t2 into � and t de-ned by Eqs. (18) and (20), respectively, the propagationormula can be expressed after some manipulations as
��z,z,t,�� =1
4�az � �−�
�
��0,0,t�,���
�exp i��0 −1
2avg���� − ���
�exp −i
2az�t� + t��� − t�� − t����dt�d��,
�32�
hich provides an alternative, compact form of Eq. (16) inconstant plane z1=z2=z. If a=0, simply ��z ,z , t ,��
��0,0, t−z /vg ,��, and in vacuum further vg=c.Now the mutual coherence function of quasi-stationary
ulses in the plane z1=z2=0 is taken to be of the form
��0,0,t1,t2� = ��0,0,t,�� = I�0,t���0,0,��, �33�
n agreement with the theory presented in the previousection. Further, we assume that t�r��2az, so that wean approximate the exponential term as exp�it��� /2az��1 in the integral of Eq. (32). This assumption
s justified after sufficient propagation distances (fareld), approximately when z�Ttc /a, where T is the effec-ive length and tc is the coherence time of the pulses. Inhat case, inserting Eq. (33) into Eq. (32) gives the for-ula
��z,z,t,�� =1
4�azexp − i��0 +
t − z/vg
2az �����
−�
�
I�0,t��exp� i�t�
2az�dt�
��−�
�
��0,0,���exp i��0 +t − z/vg
2az ����d��
�34�
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2534 J. Opt. Soc. Am. A/Vol. 23, No. 10 /October 2006 Lajunen et al.
or the mutual coherence function of quasi-stationaryulses in any plane z in the far field.Equation (34), which is one of the main results of this
aper, corresponds to the generalized form of the spatialan Cittert–Zernike theorem2,11,12 in the time domain.irst, we see that the intensity in the plane z, obtainedhen �=0, is given by
I�z,t� =I0
4�az�−�
�
��0,0,���exp i��0 +t − z/vg
2az ����d��,
�35�
here we have used notation
I0 =�−�
�
I�0,t��dt� �36�
or the average time-integrated intensity of the pulses.hus the intensity distribution in the far field is propor-ional to the Fourier transform of the complex degree ofoherence of the quasi-stationary pulses in the inputlane z=0.Using the above definitions, the propagated mutual co-
erence function given by Eq. (32) can also be expresseds
��z,z,t,�� =I�z,t�
I0exp − i��0 +
t − z/vg
2az �����
−�
�
I�0,t��exp� i�t�
2az�dt�. �37�
his result shows that the far-zone temporal coherenceunction of quasi-stationary pulses ��z ,z , t ,�� does not ex-ctly (owing to the phase term) factor in the two variablesand �. However, it is proportional to the Fourier trans-
orm of the average intensity distribution in the plane z0, modulated by the far-field intensity. Further, because�0,0,�� is assumed to be a narrow function in compari-on with I�0, t�, the intensity factor I�z , t� is slowly vary-ng compared with the �-dependent part of the coherenceunction.
From Eq. (37) we in fact obtain, in view of definition (2)nd using the approximation I�z , t−� /2��I�z , t+� /2�I�z , t�, for the complex degree of coherence of the quasi-
tationary pulses in the far field
��z,z,t,�� = exp − i��0 +t − z/vg
2az �����
−�
� I�0,t��
I0exp� i�t�
2az�dt�, �38�
hich implies that ��z ,z , t ,0�=��z ,z , t , t�=1 as it should.ence the far-zone complex degree coherence, i.e., the ab-
olute value ���z ,z , t ,��� in Eq. (38), depends in time onlyn the difference variable �.
As an example, we may consider the limiting case ofemporally incoherent pulses. As discussed in the previ-us section, we may formally write ��0,0,��=���, whichhen inserted in Eq. (35) gives a constant value I�z , t�I /4�az for the average temporal intensity in the far
0eld. Thus the absolute value of the far-zone complex de-ree of coherence assumes the form of a time-domain vanittert–Zernike theorem
���z,z,t1,t2�� ��−�
�
I0�0,t��exp�it2 − t1
2azt��dt�� , �39�
s discussed in Ref. 14.Finally, we comment on the far-zone condition
�Ttc /a, which implies that the distance z is inverselyroportional to a and in the absence of group-velocity dis-ersion �a=0� z becomes infinite. Indeed, our results be-ow Eq. (32) indicate that when a=0, the equal-plane co-erence function propagates at velocity vg without anyistortion, and so the far-field condition is in this caseever reached. However, all media (except vacuum) con-ain some amount of group-velocity dispersion.
. CHOPPED PULSESn this section we apply the theory presented above to anxample of rectangular plane-wave pulses chopped from atationary source with a Gaussian spectrum. First, thepectral coherence properties of such pulses are discussedoth in the quasi-stationary limit and in general cases.hen, the propagation of quasi-stationary chopped pulses
n dispersive media is studied analytically by using theeneralized van Cittert–Zernike theorem and is comparedith numerical results obtained for general partially co-erent chopped pulses.Let us start by considering the coherence properties of
ulses obtained by the temporal modulation of a station-ry source.20 If the stationary field U0�t� is modulated by
function M�t�, the pulses obtained are of the form�t�U0�t� in the plane z=0. The mutual coherence func-
ion of the resulting pulses then is by definition
��0,0,t1,t2� = �M*�t1�U0*�t1�M�t2�U0�t2�� = M*�t1�M�t2��s���,
�40�
here �s���= �U0*�t1�U0�t2�� is the coherence function of
he stationary source. We can see directly from this ex-ression that the intensity distribution of the pulses is de-ermined by the modulation function, and their degree ofemporal coherence remains the same as that of the sta-ionary source. Furthermore, the mutual coherence func-ion of the source is obtained by the Wiener–Khintchineheorem, which in the stationary limit is of the form
�s��� =�−�
�
Ss���exp�− i���d�, �41�
here Ss��� is the spectral density of the source.Let us now assume that the stationary source has a
aussian frequency spectrum
Ss��� = S0exp −�� − �0�2
�02 � , �42�
n which case its mutual coherence function is, accordingo Eq. (41),
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Lajunen et al. Vol. 23, No. 10 /October 2006 /J. Opt. Soc. Am. A 2535
�s��� = �0exp�−�2
Tc2�exp�− i�0��, �43�
here �0=S0�0�� and Tc=2/�0. Further, the modulationunction is assumed to be simply of the form
M�t� = �1 when 0 � t � T,
0 otherwise.�44�
epeating the modulation, we generate a statistical en-emble of finite, temporally separated pulses, for each ofhich the time coordinate is normalized so that t=0 at
he beginning of the pulse. The mutual coherence func-ion of the obtained pulses in the plane z1=z2=0 then is
��0,0,t1,t2� = �0exp −�t2 − t1�2
Tc2 �exp− i�0�t2 − t1�,
�45�
hen 0� t1�T and 0� t2�T, and 0 otherwise.Considering the pulses defined above, we see that the
uasi-stationary approximation does not hold in generalt time instants near the beginning and the end of theulses. However, if we assume that Tc�T, i.e., the tem-oral coherence time is considerably shorter than theength of the pulses, the influences of the edge regions cane neglected, in analogy with the spatial quasi-omogeneous approximation for finite sources.11 Then theutual coherence function of the chopped pulses in the
lane z=0 can be expressed as ��t ,��=I�t�����, where
I�t� = ��0 when 0 � t � T,
0 otherwise,�46�
���� = exp�−�2
Tc2�exp�− i�0��, �47�
ith the variables � and t defined as in Eqs. (18) and (20),espectively.
In the quasi-stationary case, the cross-spectral densityakes on the form of Eq. (24) where, based on Eqs. (25)nd (26),
W1��� =��Tc
2�exp −
�� − �0�2
4/Tc2 � , �48�
W2��� =�0T
2�sinc��T
2��exp� i�T
2 � . �49�
he sinc function above is defined as sinc�x�sin��x� / ��x�, and � and � are given by Eqs. (21) and
22). The spectrum then is, according to Eq. (27),
S��� =S0T
2�exp −
�� − �0�2
�02 � , �50�
here we have made use of the relations between �0 ,Tc,0, and �0. Thus the spectrum of the chopped pulses is,ccording to the quasi-stationary approximation, theame as that of the original stationary source, only mul-iplied by an energy coefficient that depends on the length
f the pulses. It may also be noted that the dimension ofhe pulse spectrum differs from the one of the initial spec-rum given by Eq. (42), which is due to the different defi-itions of the energy spectrum of nonstationary fields andhe power spectrum of stationary fields. Furthermore,ince we have assumed that the coherence time is short,.e., the spectrum is wide, we can use Eq. (30) to deter-
ine the complex degree of spectral coherence of theulses, which gives
���� = sinc��T
2��exp� i�T
2 � , �51�
o that its absolute value, the degree of coherence, is ainc function that depends on the temporal duration ofhe pulses.
It should be remembered that the above results arealid only if Tc�T. As the opposite case, we can considerompletely coherent pulses that are obtained by a tempo-al modulation of an ideal monochromatic stationaryource. If we express the spectral density of the stationaryource formally as Ss���=S0���−�0�, the pulses obtainedy similar temporal modulation as above have the spec-rum
S��� = S0�� T
4��2
sinc2 T�� − �0�
2�� , �52�
hile their degree of temporal as well as spectral coher-nce remain unity. Thus the spectral content of the pulsess in the coherent case determined entirely by the modu-ation function, which, on the other hand, does not haven influence on the spectrum in the quasi-stationary case.n the intermediate cases both the temporal length of theulses and their temporal coherence time affect the spec-ral density of the obtained pulses.
Finally, let us proceed to consider the propagation ofhe rectangular chopped pulses in linearly dispersive me-ia. Using the quasi-stationary approximation, we mayake the pulses to be defined by Eqs. (46) and (47) in thelane z1=z2=0. If we consider pulses that have propa-ated a sufficiently long distance z�Ttc /a, as discussedn the previous section, their mutual coherence functionan be solved analytically by using the generalized vanittert–Zernike theorem, Eq. (34). This yields the result
��z,z,t,�� = I�z,t�sinc� T�
4�az��exp − i���0 +
t − z/vg
2az−
T
4az�� , �53�
here the average temporal intensity distribution of theulses is
I�z,t� =�0TTc��
4�azexp − � t − z/vg
4az/Tc�2� , �54�
nd its functional form is seen to depend only on the tem-oral coherence length Tc. Further, Eq. (53) shows thathe degree of coherence of the quasi-stationary pulses isow effectively proportional to the sinc function.
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2536 J. Opt. Soc. Am. A/Vol. 23, No. 10 /October 2006 Lajunen et al.
The propagation of fully coherent chopped pulses inispersive media differs, as one might expect, from the re-ults obtained for quasi-stationary pulses. Using a similarar-field approximation as in Ref. 21, namely that�T /4a, we obtain
I�z,t� =�0T2
4�azsinc2 T�t − z/vg�
4�az � �55�
or the temporal intensity distribution of the propagatedulses chopped from a coherent source. General situa-ions in which pulses are partially coherent but not quasi-tationary can be solved numerically based on Eq. (16).ome examples of the far-field intensity distributions ofulses with different states of coherence propagated in ainearly dispersive medium are shown in Figs. 1 and 2nd discussed below.In Fig. 1 we illustrate the normalized far-field intensity
istributions of pulses with initially equal rectangularemporal shapes, obtained from stationary sources withifferent spectral widths. The pulses with the smallestpectral width, and thus the longest coherence time, cor-espond approximately to the coherent case, and theyave the narrowest sinc-shaped intensity distribution inhe far field. As the coherence time decreases, the inten-ity distributions become wider and approach the Gauss-an form predicted by the quasi-stationary approxima-ion.
In Fig. 2 we similarly show normalized far-field inten-ity distributions for pulses with different states of coher-nce. Now the pulses are assumed to have been obtainedrom the same stationary source with a constant spectralidth, but they have different initial durations. In that
ase, the nearly coherent pulses with the shortest initialuration spread the most during the propagation overong distances. Hence the pulses with the longest initialuration and lowest degree of coherence, corresponding
ig. 1. Normalized average intensity distributions of pulseshopped from different sources after propagation of distance z20 km in a dispersive medium with a=25 km/ps2. The initialuration of all the pulses is T=10 ps, and the temporal coherenceimes Tc=2/�0 are assumed to be 1 ps (dotted curve), 4 psdashed–dotted curve), and 40 ps (dashed curve). The solid curveorresponds to the quasi-stationary approximation in the case of
=1 ps.
cpproximately to the quasi-stationary case, have the nar-owest average intensity distribution in the far field.
It may also be noted that the functional forms of thear-field intensity distributions after propagation in lin-arly dispersive media and the spectra of the pulses areimilar. Thus Figs. 1 and 2 also illustrate qualitativelyhe different spectral shapes of the rectangular pulses ob-ained by temporal modulation of stationary sources withhe different initial conditions.
. CONCLUSIONSe have studied the temporal and spectral coherence
roperties of quasi-stationary plane-wave pulses and de-ived a temporal analog for the generalized van Cittert–ernike theorem, which describes the propagation of suchulses in linearly dispersive media. The theory we intro-uced was illustrated by an example of the spectral coher-nce properties and the propagation characteristics ofartially coherent rectangular pulses chopped from a sta-ionary light source.
In the space–time domain the coherence time of quasi-tationary pulses is assumed to be much shorter than theverage temporal width of the pulses. Physically thiseans that the detailed features of the individual pulses
n the ensemble are relatively random when comparedith other field realizations. In the space–frequency do-ain the spectrum of the pulses is proportional to theourier transform of the complex degree of temporal co-erence, and the complex degree of spectral coherence isorrespondingly obtained from the Fourier transform ofhe average intensity distribution. The spectral width ishus considerably larger than the spectral correlationidth.The temporal coherence properties of quasi-stationary
ulses after sufficiently long propagation distances in lin-
ig. 2. Normalized average intensity distributions of pulseshopped from the same stationary source with different initialurations after propagation of distance z=20 km in a dispersiveedium with a=25 km/ps2. The temporal coherence time of all
ulses is Tc=10 ps, and the initial durations T are assumed to be0 ps (dotted curve), 10 ps (dashed–dotted curve), and 4 psdashed curve). The solid curve corresponds to the quasi-tationary approximation in the case of T=40 ps.
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Lajunen et al. Vol. 23, No. 10 /October 2006 /J. Opt. Soc. Am. A 2537
arly dispersive media can be determined by the general-zed van Cittert–Zernike theorem in time. It states thathe average intensity distribution of the pulses in the fareld is proportional to the Fourier transform of the com-lex degree of coherence in the initial plane. As the ex-mples presented in Section 5 show, the ratio of the coher-nce time to the initial temporal duration cruciallynfluences the far-field intensity distributions of partiallyoherent pulses. Thus it is possible, for instance, thatuasi-stationary pulses experience less temporal spread-ng in linearly dispersive media than fully coherentulses.Since no real field can be temporally infinite and thus
ctually stationary, the concept of quasi-stationary fieldsrovides a more realistic generalized model for tempo-ally partially coherent optical fields. It should also beoted that when the temporal duration of the pulses is as-umed to approach infinity, the temporal coherence lengthay also increase, still remaining within the bounds of
he quasi-stationary approximation. This means thathen considering pulses that are nearly stationary, theresented model is valid for any practical states of tempo-al coherence, and consequently also of spectra, excludingnly the idealized concept of fully temporally coherent,onochromatic sources.
CKNOWLEDGMENTShe work of H. Lajunen is supported by the Academy ofinland (project 111701), and she also acknowledges theetwork of Excellence on Micro-Optics (NEMO, http://ww.micro-optics.org) and the grants from the Hel-
ingin Sanomat Centennial Foundation and the Vilho,rjö and Kalle Väisälä Foundation of the Finnish Acad-my of Science and Letters. A. T. Friberg thanks thewedish Foundation for Strategic Research (SSF) for fi-ancial support.Corresponding author H. Lajunen may be reached by
-mail at [email protected].
*Permanent address, University of Joensuu, Depart-ent of Physics, P.O. Box 111, FI-80101 Joensuu, Fin-
and.
EFERENCES AND NOTES1. L. Mandel and E. Wolf, Optical Coherence and Quantum
Optics (Cambridge U. Press, 1995).2. J. W. Goodman, Statistical Optics (Wiley, 1985).
3. B. Cairns and E. Wolf, “The instantaneous cross-spectraldensity of non-stationary wavefields,” Opt. Commun. 62,215–218 (1986).
4. M. Bertolotti, A. Ferrari, and L. Sereda, “Coherenceproperties of nonstationary polychromatic light sources,” J.Opt. Soc. Am. B 12, 341–347 (1995).
5. L. Sereda, M. Bertolotti, and A. Ferrari, “Coherenceproperties of nonstationary light wave fields,” J. Opt. Soc.Am. A 15, 695–705 (1998).
6. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F.Wyrowski, “Partially coherent Gaussian pulses,” Opt.Commun. 204, 53–58 (2002).
7. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energyspectrum of a nonstationary ensemble of pulses,” Opt. Lett.29, 394–396 (2004).
8. H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherenceand coherent-mode expansion of spectrally partiallycoherent pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
9. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatiallyand spectrally partially coherent pulses,” J. Opt. Soc. Am. A22, 1536–1545 (2005).
0. R. A. Silverman, “Locally stationary random processes,”Proc. IRE Trans. Inf. Theory 3, 182–187 (1957).
1. W. H. Carter and E. Wolf, “Coherence and radiometry withquasihomogeneous planar sources,” J. Opt. Soc. Am. 67,785–796 (1977).
2. E. Wolf and W. H. Carter, “A radiometric generalization ofthe van Cittert–Zernike theorem for fields generated bysources of arbitrary state of coherence,” Opt. Commun. 16,297–302 (1976).
3. J. Lancis, V. Torres-Company, E. Silvestre, and P. Andrés,“Space-time analogy for partially coherent plane-wave-typepulses,” Opt. Lett. 30, 2973–2975 (2005).
4. C. Dorrer, “Temporal van Cittert–Zernike theorem and itsapplication to the measurement of chromatic dispersion,” J.Opt. Soc. Am. B 21, 1417–1422 (2004).
5. H. Lajunen, J. Turunen, P. Vahimaa, J. Tervo, and F.Wyrowski, “Spectrally partially coherent pulse trains indispersive media,” Opt. Commun. 255, 12–22 (2005).
6. Q. Lin, L. Wang, and S. Zhu, “Partially coherent light pulseand its propagation,” Opt. Commun. 219, 65–70 (2003).
7. Use can be made of “Siegman’s lemma” �−�� exp�−ax2
+bx�dx=�� /a exp�b2 /4a�, valid for any complex numbers aand b with R�a��0. See A. E. Siegman, Lasers (UniversityScience Books, 1986), p. 783.
8. The hat superscript implies that the function depends intime on t and �, as opposed to t1 and t2, or in frequency on �and �, as opposed to �1 and �2. However, for functions ofonly a single variable t or �, or � or �, the symbol is notused.
9. Since ��t1 , t2� has to remain finite, the Dirac delta function�t2− t1� is to be interpreted as a limiting case of a narrowfunction with a very high but normalizable value.
0. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F.Wyrowski, “Spectral coherence properties of temporallymodulated stationary light sources,” Opt. Express 11,1894–1899 (2003).
1. Q. Lin and S. Wang, “Propagation characteristics ofchopped light pulses through dispersive media,” J. Opt. 26,
247–249 (1995).