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Liu et al. Vol. 23, No. 10 /October 2006 /J. Opt. Soc. Am. B 2223

Time–frequency uncertainty relationship of apartially temporally coherent pulse train

Haitao Liu, Guoguang Mu, and Lie Lin

Key Laboratory of Opto-electronic Information Science and Technology, Ministry of Education, Institute of ModernOptics, Nankai University, Tianjin 300071, China

Received April 26, 2006; accepted June 23, 2006; posted July 10, 2006 (Doc. ID 70236)

On the basis of the recently introduced definition of energy spectrum of a partially temporally coherent pulsetrain [Opt. Lett. 29, 394 (2004)], we show that the well-known time–frequency uncertainty relationship appli-cable to a fully temporally coherent pulse train is still correct for a partially temporally coherent pulse train.The condition under which the uncertainty relationship reaches equality is derived. An example is presented toillustrate the validity of the uncertainty relationship. © 2006 Optical Society of America

OCIS codes: 030.0030, 320.5550.

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. INTRODUCTIONt is well known that an optical pulse has a time–requency uncertainty relationship of1

��t�2����2 � � 1

4��2

, �1�

here �t and �� are the rms width (or the uncertainty) ofime and the rms width (or the uncertainty) of frequency,espectively. The uncertainty relationship has not onlyheoretical merit but also application significance. For ex-mple, the measurement of the temporal pulse width �ts difficult and needs complex techniques andnstruments,2,3 while the spectrum width �� can be easily

easured with a spectrometer. By using the uncertaintyelationship and measuring ��, one can obtain an estima-ion of �t without direct measurement.

Inequality (1) can be applied to a fully temporally co-erent pulse train (FT), which is under the assumptionhat all the pulses in the train have the same temporalhape. However, this assumption is not always true be-ause of unavoidable random factors in a pulse train. Aandom pulse train that does not satisfy the assumptions called a partially temporally coherent pulse train (PT),hich has to be treated as a nonstationary randomnsemble.4,5 In this paper, we show that the uncertaintyf inequality (1) is still true for a PT, based on the intro-uced definition4 of energy spectrum of a PT. The condi-ion under which the uncertainty relationship reachesquality is also obtained.

In Section 2, the uncertainty relationship of a PT is de-ived. In Section 3, the condition for the equality of thencertainty relationship is obtained. In Section 4, an ex-mple is provided. In Section 5, conclusions are summa-ized.

. DERIVATION OF THE UNCERTAINTYELATIONSHIPor a PT, the temporal intensity I�t� and the energy spec-rum S��� are defined as4

0740-3224/06/102223-4/$15.00 © 2

I�t� = ��e�t��2�, S��� = ��E����2�, �2�

here e�t� is the analytic signal of the random electriceld of the pulses in the train, t is time, E����−�

� e�t�exp�i2��t�dt is the presentation of the pulse inhe spectral domain, � is frequency, and � � means the av-rage over the ensemble of pulses in the train. Then thencertainty �t of time and the uncertainty �� of fre-uency are defined by1

��t�2 =

−�

�

I�t��t − t̄�2dt

−�

�

I�t�dt

, ����2 =

−�

�

S����� − �̄�2d�

−�

�

S���d�

,

�3�

here

t̄ =−�

�

I�t�tdt/−�

�

I�t�dt

nd

�̄ =−�

�

S����d�/−�

�

S���d�

re the central time and the central frequency, respec-ively.

For a PT, a mutual coherence function is defined as�t1 , t2�= �e�t1�e*�t2�� and a cross-spectral density function

s defined as W��1 ,�2�= �E��1�E*��2��, which satisfy5

W��1,�2� = −�

�

��t1,t2�expi2���1t1 − �2t2��dt1dt2.

�4�

t is obvious that I�t�=��t , t� and S���=W�� ,��. Because�t1 , t2� corresponds to a Hermitian and nonnegative defi-ite operator in the Hilbert space, it has a coherent modeecomposition of5,6

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2224 J. Opt. Soc. Am. B/Vol. 23, No. 10 /October 2006 Liu et al.

��t1,t2� = �n

�nun�t1�un*�t2�, �5�

here �n�0, �−�� um

* �t�un�t�dt=�mn, and �mn is the Kro-ecker delta symbol. The insertion of Eq. (5) into Eq. (4)an yield the coherent mode decomposition of W��1 ,�2� as

W��1,�2� = �n

�nUn��1�Un*��2�, �6�

here Un���=�−�� un�t�exp�i2��t�dt and �−�

� Um* ���Un���d�

�mn. With Eqs. (5) and (6) inserted, Eqs. (3) become

��t�2 =�

n�n��tn�2

�n

�n

, ����2 =�

n�n���n�2

�n

�n

, �7�

here

��tn�2 =−�

�

�un�t��2�t − t̄�2dt

nd

���n�2 =−�

�

�Un����2�� − �̄�2d�.

ecause the normalized functions un�t� and Un��� are con-ected by the Fourier transform, un�t� and Un��� can bereated as a wave function in the position representationn quantum mechanics7 and the corresponding wave func-ion in the momentum representation, respectively, and�tn�2 and ���n�2 are the uncertainty of position and theorresponding uncertainty of momentum, respectively. Bysing the uncertainty relationship between the positionnd the momentum in quantum mechanics,7 we can ob-ain

��tn�2���n�2 � � 1

4��2

, �8�

here the equality is reached only if

un�t� = un�0�exp t̄ 2 − �t − t̄�2

4��tn�2 �exp�− i2��̄t�, un�0�

=

exp −t̄ 2

4��tn�2�exp�in�

2���tn�2�1/4 , �9�

here n can be any real number. Inequality (8) has beenentioned as inequality (1). Now we have

��t�2����2 =�

n�n��tn�2

�n

�n

�n

�n���n�2

�n

�n

�

�n

��n��tn�2��n���n�2�2

��n

�n�2 � � 1

4��2

,

�10�

hich yields an uncertainty relationship given by in-quality (1).

In Eq. (10), the Cauchy–Schwarz inequality8 is used forhe first inequality, and inequality (8) is applied for theecond inequality. Inequality (1) is just the time–requency uncertainty relationship satisfied by a PT,hich is the same as that satisfied by a FT.

. CONDITION FOR THE EQUALITY OF THENCERTAINTY RELATIONSHIPy the Cauchy–Schwarz inequality,8 the first inequality

n inequality (10) can become equality only if

��n��tn�2

��n���n�2= C, �11�

here C is a constant independent of n. The second in-quality in inequality (10) can become equality only if�tn�2���n�2= �1/4��2 for any n, which with Eqs. (11) and7) can determine ��tn�2=C /4�= ��t�2. So the equality ofhe uncertainty relationship of inequality (1) can beeached only if

un�t� = un�0��t�, un�0� =

exp −t̄ 2

4��t�2�exp�in�

2���t�2�1/4 ,

�t� = exp t̄ 2 − �t − t̄�2

4��t�2 �exp�− i2��̄t�. �12�

nder the condition of Eqs. (12), the mutual coherenceunction is

��t1,t2� = A�t1�*�t2�, A =

exp −t̄ 2

2��t�2��n

�n

2���t�2�1/2 ,

�13�

nd the degree of temporal coherence4,5 is

��t1,t2� =��t1,t2�

��t1,t1���t2,t2��1/2 =�t1�*�t2�

��t1��t2��. �14�

quation (14) satisfies ���t1 , t2� � =1, which with theauchy–Schwarz inequality8 can yield e�t1�=c�t1 , t2�e�t2�,r, equivalently,

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Liu et al. Vol. 23, No. 10 /October 2006 /J. Opt. Soc. Am. B 2225

e�t� = c�t,0�e�0�, �15�

here c�t1 , t2� is a deterministic function of t1 and t2. Withq. (15), ��t1 , t2�= �e�t1�e*�t2�� can be rewritten as

��t1,t2� = ��e�0��2�c�t1,0�c*�t2,0�. �16�

hrough the comparison between Eqs. (13) and (16), onean obtain

c�t,0� = c0�t�, �17�

here c0 is a constant complex number independent of t.ith Eq. (17) inserted, Eq. (15) can become

e�t� = c0�t�e�0� = �t�e�0�, �18�

here c0=1 is determined by �0�=1. So a PT reachinghe equality of the uncertainty relationship of inequality1) must be a FT, in which all the pulses have the sameaussian temporal shape described by �t�.

. EXAMPLEn this section, the Gaussian Schell-model (GSM) pulserain5 will be taken as an example to illustrate the uncer-ainty relationship of a PT. The GSM pulse train has aemporal mutual coherence function of

��t1,t2� = �0 exp −t12 + t2

2

2T2 −�t1 − t2�2

2Tc2 − i2��0�t1 − t2�� ,

�19�

here �0=��0,0�, T describes the temporal width, Tcharacterizes the temporal coherence, and �0 describeshe central frequency, as explained in the following. Byq. (4), the cross-spectral density function of the GSMulse train can be determined as

W��1,�2� = W0 exp −��1 − �0�2 + ��2 − �0�2

2M2 −��1 − �2�2

2Mc2 � ,

�20�

ith

M =1

�� 1

4T2 +1

2Tc2�1/2

, W0 =�0T

M, Mc =

TcM

T, �21�

here M describes the spectrum width and Mc character-zes the spectrum coherence, as explained in the follow-ng. The temporal intensity I�t� and the energy spectrum��� are

I�t� = �0 exp�−t2

T2�, S��� = W0 exp −�� − �0�2

M2 � .

�22�

hen for a GSM pulse train, the central time t̄, the centralrequency �̄, the uncertainty �t of time, and the uncer-ainty �� of frequency are

t̄ = 0, �̄ = �0, ��t�2 = T2/2, ����2 = M2/2, �23�

here Eqs. (3) are used. Equations (23) explain the physi-al meanings of � , T, and M. Equations (21) and (23)

0

how that �t and �� are connected by

����2 = � 1

4��2 1

��t�2 +1

4�2Tc2 , �24�

here Tc will be interpreted in the following.For a GSM pulse train, the degree of temporal coher-

nce is4

��t1,t2� =��t1,t2�

�I�t1�I�t2�= exp −

�t1 − t2�2

2Tc2 − i2��0�t1 − t2�� .

�25�

he coherence time �c of the GSM pulse train is4

�c2 =

−�

�

���t1,t2��2�t1 − t2�2dt1dt2

−�

�

���t1,t2��2dt1dt2

=Tc

2

2. �26�

imilar to Eqs. (25) and (26), the degree of spectrum co-erence ��1 ,�2� and the coherence spectrum width �c cane defined as

��1,�2� =W��1,�2�

�S��1�S��2�,

�c2 =

−�

�

� ��1,�2��2��1 − �2�2d�1d�2

−�

�

� ��1,�2��2d�1d�2

. �27�

or a GSM pulse train we have

��1,�2� = exp −��1 − �2�2

2Mc2 �, �c

2 =Mc

2

2. �28�

After the above explanation of Tc and Mc, we come backo the discussion of Eq. (24) that connects �t and ��. Withq. (26) inserted, Eq. (24) can become

����2 = � 1

4��2 1

��t�2 +1

8�2�c2 . �29�

o a GSM pulse train satisfies

����2 � � 1

4��2 1

��t�2 , �30�

here the inequality can become equality only if �c→� orf the GSM pulse train becomes a FT with a Gaussianemporal shape. The above conclusions for the GSM pulserain are consistent with the general conclusions derivedn Sections 2 and 3.

. CONCLUSIONSe show that a partially temporally coherent pulse train

PT) has the same time–frequency uncertainty relation-hip as a fully temporally coherent pulse train (FT), basedn the recently introduced definition of energy spectrum

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2226 J. Opt. Soc. Am. B/Vol. 23, No. 10 /October 2006 Liu et al.

f a PT. The equality of the uncertainty relationship of aT can be reached only if the PT becomes a FT in whichll the pulses have the same Gaussian temporal shape.hese conclusions are confirmed by the example of theaussian Schell-model (GSM) pulse train.

CKNOWLEDGMENTShis research is supported by the Natural Science Foun-ation of Tianjin (under grant 06YFJMJC01500), by thepen Research Fund of the Key Laboratory of Opto-lectronic Information Science and Technology of the Edu-ation Ministry of China (under grant 2005-04), and byhe Fund for the Development Project of Science andechnology of Tianjin (under grant 043103011).H. Liu can be reached by e-mail at

iuht@nankai.edu.cn, by phone at 8622-23506422, and byax at 8622-23502275.

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3. C. Iaconis and I. A. Walmsley, “Self-referencing spectralinterferometry for measuring ultrashort optical pulses,”IEEE J. Quantum Electron. 35, 501–509 (1999).

4. S. A. Ponomarenko, G. P. Agrawal, and E. Wolf, “Energyspectrum of a nonstationary ensemble of pulses,” Opt. Lett.29, 394–396 (2004).

5. H. Lajunen, P. Vahimaa, and J. Tervo, “Theory of spatiallyand spectrally partially coherent pulses,” J. Opt. Soc. Am. A22, 1536–1545 (2005).

6. L. Mandel and E. Wolf, Optical Coherence and QuantumOptics (Cambridge U. Press, 1995).

7. X. Ka, Advanced Quantum Mechanics (AdvancedEducation Press, 1999).

8. T. S. Blyth and E. F. Robertson, Further Linear Algebra(Springer, 2002), Chap. 1.