Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qi.phys.msu.ru/papers/2011-jetp(e)-112-20.pdf Äàòà èçìåíåíèÿ: Tue Sep 6 08:03:49 2011 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 19:51:31 2012 Êîäèðîâêà:

ISSN 1063 7761, Journal of Experimental and Theoretical Physics, 2011, Vol. 112, No. 1, pp. 20­37. © Pleiades Publishing, Inc., 2011. Original Russian Text © K.G. Katamadze, S.P. Kulik, 2011, published in Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki, 2011, Vol. 139, No. 1, pp. 26­45.

ATOMS, MOLECULES, OPTICS

Control of the Spectrum of the Biphoton Field
K. G. Katamadze and S. P. Kulik
Moscow State University, Moscow, 119992 Russia e mail: katamadze@inbox.ru, Sergei.Kulik@gmail.com
Received June 11, 2010

Abstract--The main methods for controlling the biphoton field, as well as the problems for which the width and the shape of the spectrum of the biphoton field are of decisive importance, are discussed. The method for controlling the spectrum of the spontaneous parametric downconversion of light based on the spatial modu lation of the refractive indices of a nonlinear crystal in which the generation of biphotons has been analyzed. Modulation is due to the thermo optic and electro optic effects. DOI: 10.1134/S1063776110061111

1. INTRODUCTION One of the main aims of experimental quantum optics and quantum communication is the prepara tion of light in a given quantum state, when the param eters of the state are known a priori and/or can be con trolled in the process of the experiment. The state of the biphoton field is specified by the spatial, spectral, and polarization parameters. In this work, we analyze the methods for preparing biphoton fields with various frequency properties. Taking into account the spectral expansion, the state of two pho ton light has the form [1, 2] | = |vac +
s

where p is the pump frequency. Then, it is convenient to represent the frequencies in the form of mismatchs s = s0 + , i = i0 ­ , (2)

where s0 and i0 are the central frequencies of the spectra of the signal and idle fields (s0 + i0 = p), and represent Eq. (1) in the form | = |vac + d F ( ) a s ( s0 + ) a i ( i0 ­ ) |vac . (3)









d s d

i

Here, the generally complex function F(), which is usually called the spectral amplitude of the biphoton, describes the frequency spectrum of the biphoton field. Below, the spectrum of the biphoton field is prima rily treated as its spectral amplitude and the width of 2 the distribution of the function F ( ) is considered as the width of the spectrum, . Note that the spec tral width of the biphoton in the case of collinear degenerate type I phase matching is defined in [5, 6] as the length of the localization region of the ampli tude F(s, i) in the direction of the axis s ­ i = 0, which differs from the spectral width of the signal and idle photons only by a factor 2 . The paper consists of two parts. The first part (Sec tions 2­4) reviews the main aims and methods for controlling the spectrum of the biphoton field. The second part (Sections 5­7) describes in detail the method for controlling the spectrum by means of the spatial modulation of the refractive indices of the non linear crystal. The new experimental results are also presented in the second part, where the method under investigation is compared with the methods reviewed in the first part.
20

(1)

â F ( s, i ) a ( s ) a ( i ) |vac , where s and i are the frequencies of the signal and idle photons, respectively; and a s and a i are the cre ation operators of photons in fixed signal and idle spa tially polarized modes, respectively. Such a field is usu ally obtained using spontaneous parametric downcon version [3]. In the case of a narrowband pump, the frequencies s and i are related as s + i = p,
1


i

1

Hereinafter, we consider the case of the Fourier limited pump for which the spectral width p and pulse duration are related as p ~ 1/. In this case, the pump can be treated as narrowband (i.e., can be approximated by the delta function) under the condition 2( v
( s, i ) (p) g

­v

( s, i ) g

)/L > 1 [4], where v

(p) g

and v g are the group velocities of the pump and parametric radiations, respectively; and L is the length of the nonlinear crystal.


CONTROL OF THE SPECTRUM OF THE BIPHOTON FIELD (a) i Pump
(2)

21 M2

M

(b) Pump M1 i s



(2)

M1 BS

s

BS M2 D

D
Fig. 1. Scheme of the measurement of the first order correlation function using a Michelson interferometer. Radiation generated in a crystal with a nonlinear susceptibility (2) is guided to the interferometer consisting of beam splitter BS and two mirrors M1 and M2; after that, the radiation intensity is measured by detector D. Interference can be observed by displacing mirror M2. The width of the function G(1)() can be estimated from a change in the visibility of the interference pattern: (a) measurement of

G g ( ) in signal mode; (b) measurement of G ( ) in both modes.

(1)

(1)

2. RELATION OF THE SPECTRUM OF THE BIPHOTON FIELD WITH THE CORRELATION CHARACTERISTICS In most cases where the biphoton field is used, its correlation characteristics are of primary importance. For this reason, the control of the spectrum of the biphoton field is interesting in the context of the first and second order correlation functions. Note that the function F() is not measured directly in an experi ment. However, the spectral intensities of the field in the signal and idle modes can be measured: S s F ( = ­ s0 ) ,
2

The state of the biphoton can be described not only by the spectral amplitude, but also by the time ampli tude i ~ (9) F ( ) = d e F ( ) ,



whose absolute value squared provides the second order correlation function [1]: G ()
(2)



F ( ) cos ( ) d .

2

(10)

S i F ( = ­ i0 ) . (4)

2

According to the Wiener­Khinchin theorem, the first order correlation function for the signal and/or idle modes (i.e., for the single photon field) is given by the expression G ()
(1)



F ( ) cos ( ) d . = = p / 2

2

(5)

Under the conditions F ( ­ ) = F ( ) ,
s0 i0

(6)

the total spectral intensity of the field, S = Ss + Si, has the form S ( ) F ( = ­ p / 2 ) .
2

(7)

In this case, the correlation function of the biphoton field is expressed similar to the single photon correla tion function given by Eq. (5) [1]. We emphasize that the width of the spectrum S(), which is related only to the absolute value of the amplitude F ( ) , com pletely determines the width of the correlation func tion G(1)(): 1 / = 1 / .
(1)

(8)

This expression does not require condition (6). An important difference of Eq. (10) from Eq. (5) is that the width of the second order correlation function (2) is determined not only by the absolute value, but also the phase of the amplitude of F(). The relation (2) min ~ 1/ = 1/ is satisfied only for the mini mum possible value of the width of (2), which is reached when the phase of F() depends only on the frequency. In this case, the broad spectrum of the biphoton field is a necessary, but not sufficient condi tion for the smallness of the time (2). At the same time, the biphoton field with a narrow spectrum always has a wide second order correlation function. The correlation functions G(1)() and G(2)(), as well as the spectral intensity, can also be measured in an experiment. The first order correlation function is manifested in interference experiments. For example, if the signal and idle photons are in different spatial modes, the correlation function G(1)() can be mea sured with a Michelson interferometer placed in either signal or idle mode (see Fig. 1a). In the case of the col linear degenerate regime and the same polarization states of photons, the correlation function G(1)() of the biphoton field can be measured with the same interferometer (see Fig. 1b) [7]. One of the remarkable effects that are conse quences of the quantum properties of the biphoton
Vol. 112 No. 1 2011

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


22 M i Pump
(2)

KATAMADZE, KULIK

D1

M M M

BS s D2 CC

P
Fig. 2. Scheme of the measurement of the first order cor relation function by measuring the Hong­Ou­Mandel dip. The biphoton radiation is generated in the crystal with the nonlinear susceptibility (2) in the noncollinear regime degenerate in the frequency and polarization. Using mir rors M, the signal (s) and idle (i) modes are mixed on a 50% beam splitter BS. Then, radiation is guided to detec tors D1 and D2, which are connected through the coinci dence scheme CC. The length of the signal channel can be changed by displacing prism P. When the optical paths of the signal and idle photons are the same, they are shared with a probability of 100% to one of two outputs of the beam splitter and the coinciding photocounts will exhibit a dip whose width corresponds to the half width of the cor relation function G(1)() given by Eq. (12).

field is the so called Hong­Ou­Mandel dip: if both photons of a pair arrive simultaneously on both inputs of the 50% beam splitter, they come to the same arm in the output if they are absolutely indistinguishable. This indistinguishability includes indistinguishability in time. spontaneous parametric downconversion is accompanied by the generation of photons correlated in the creation time and, for they to remain correlated to the time of their arrival at the beam splitter, the lengths of their optical paths should coincide with each other with an accuracy of the inverse spectral width. The effect is experimentally observed in coin cidences of photocounts of the detectors placed in the output modes of the beam splitter (see Fig. 2). The length of one of the optical paths is varied and, when the lengths of both paths are the same, a dip appears in coincidences because both photons are always guided to one detector. It is known that, if the window of the coincidence scheme is sufficiently wide, the shape and width of this dip are related to the first order correla tion function as [9, 10] Rc 1 ­ g ( 2 ) ,
(2) (1)

The second order correlation function is mani fested in correlation between the counts in the detec tors detected signal and idle photons. When the win dow of the coincidence scheme is much narrower than (2), the function G(2)() can be measured with a Han bury Brown­Twiss interferometer (see Fig. 3a) by varying the time delay between pulses arriving the coincidence scheme and measuring the count rate of the coincidences of photocounts Rc() ~ G(2)(). How ever, (2) is in most cases much smaller than the win dow of the coincidence scheme; in these cases, such measurements are impossible. The function G(2)() is also manifested in two photon interactions with mat ter, as well as in parametric processes, e.g., in the gen eration of sum frequency radiation. The probability of such process is proportional to the second order cor relation function. The authors of [11] considered an experiment (see Fig. 3b) in which light from a para metric amplifier operating in the collinear degenerate regime was split by a beam splitter into two channels; after that, a controlled delay was introduced to one of the channels and the beams were reunited on a nonlin ear crystal in which the second harmonic generation occurred in the noncollinear regime. The intensity of the double frequency radiation was measured as a function of the delay, I2() ~ G(2)(). 3. PROBLEMS IN WHICH THE INCLUSION OF THE SPECTRUM OF BIPHOTON FIELD IS IMPORTANT In a number of applications, a source of the bipho ton field with a narrow spectrum is required. First, to increase the efficiency of the single photon interac tions of light with individual atoms and to implement quantum memory [12, 13], the frequency of photons should be in resonance with energy levels. Since one of the basic elements of the scheme (heralded scheme) for generating single photon states is the biphoton field [14­17], its typical spectral width for these prob lems should be no more than 1­10 MHz. Second, in the case of the transmission of quantum information through optical fibers, the arrival time of single pho ton packages is smeared in view of chromatic disper sion; for this reason, the spectral composition of the packages should be minimized [18]. In addition, the biphoton field with a narrow spectrum (and, there fore, with a large correlation time (2) is necessary for measuring the time characteristics of single photon detectors [19], which can be determined in the Han bury Brown­Twiss scheme (see Fig. 3a). The biphoton field with a broad spectrum is used in another group of applications. Here, we primarily point to the problem of increasing the entanglement degree of the biphoton field, which is of current importance for quantum information encoding. An
3

2

3

(11)

where the function Rc() describes the dependence of the coincidence count rate on the time delay introduced for one of the photons and g(1)() is the normalized first order correlation function for a pair of photons.
2

Note that there are a number works in the field of quantum optics concerning the so called postponed compensation in which the manifestation of two photon interference in the Hong­Ou­Mandel scheme is not associated with the simulta neous arrival of two photons at the beam splitter [8].

Hereinafter, numerical values are presented for the frequency of the light field rather than for the angular frequency . Vol. 112 No. 1 2011

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


CONTROL OF THE SPECTRUM OF THE BIPHOTON FIELD (a) i Pump
(2)

23 BS M M

(b) Pump

PA

i s

CC M P

s



(2)

D
Fig. 3. Scheme of the measurement of the second order correlation function. (a) Scheme of the measurement of the correlation function G(2)() in a Hanbury Brown­Twiss interferometer. The signal s and idle i photons obtained in a nonlinear crystal in the presence of spontaneous parametric downconversion are guided in different spatial modes and are detected by the detectors. The function G(2)() can be measured by varying the delay of signals propagating from one of the detectors to coincidence scheme CC (if the window of the coincidence scheme is much narrower than the width (2)()). (b) Scheme of the measurement of the correlation function G(2)() due to the generation of the sum frequency. Radiation from parametric amplifier PA is separated into two channels by beam splitter BS; after that, mirror M guides both beam into a nonlinear crystal in which the second harmonic generation occurs in the noncollinear regime. Radiation at doubled frequency is registered by detector D. The second order cor relation function can be measured by varying the length of one of the optical paths through the displacement of prism P. In this case, the nonlinear crystal serves as a precision coincidence scheme.

increase in the entanglement degree is accompanied by an increase in the effective dimension of the Hilbert space of strongly correlated optical states, which is a promising object of quantum cryptography [20­23] and for testing the fundamentals of quantum theory [24­27]. When the biphoton field is in a pure state of form (3), the parameter R that was introduced by M.V. Fedorov [4, 28] and is defined as the ratio of the spectral width of individual photons (unconditional distribution) to the spectral width of coincidences of photocounts (conditional distribution) [29] is very convenient for the numerical analysis of the entangle ment degree. Since the spectral width of coincidences in spontaneous parametric downconversion for the narrowband pump is determined by the spectral width of the pump, the spectral width of single counts coin 2 cides with the width of the function F ( ) . Thus, the frequency entanglement degree under a given pump can be increased only by the broadening of the spec trum of the biphoton field. As was mentioned above, the broadening of the spectrum of the biphoton field leads to a decrease in (1) the correlation time s, i between photons in each mode; this fact can be used in optical coherent tomog raphy (see Fig. 4a). In optical coherent tomography, an interferometer with an object under investigation in one of the arms is placed in one of the modes of the biphoton field. The object can be scanned at various

depths by varying the length of the second arm of the interferometer. The resolution of optical coherent tomography is determined by the radiation coherence (1) length lres = c/ s, i . An alternative of optical coherent tomography is the quantum optical coherent tomogra phy [30] based on the Hong­Ou­Mandel dip. The object under investigation is placed in one of the chan nels in front of the beam splitter (see Fig. 4b) and the optical length of the other channel is varied. Accord ing to Eq. (11), the resolution of such a scheme is determined by double coherence time, lres = c/2(1). Note that quantum optical coherent tomography requires a source of the biphoton field that satisfies conditions (6) and whose signal and idle modes differ in the spatial or polarization parameters. In addition, the biphoton field with a broad spec trum is necessary for problems requiring the efficient biphoton interaction of light with matter. In the case of short times (2), the biphoton behaves as a united object with the effective wavelength [31] 2c 2c 1 2c 1 (12) eff = = = = , E/ 2 / 2 2 where E is the energy of the biphoton and is the angular frequency of constituent photons (in the fre quency degenerate case). This fact is used to increase the resolution of nonlinear microscopy [32] and quan tum interference optical lithography [33]. The spec troscopy of virtual states with the use of entangled photons is based on two photon absorption [34].
Vol. 112 No. 1 2011

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


24 (a) i D

KATAMADZE, KULIK (b) P M Pump
(2)

BS M M

s BS S

M i CC M

Pump

(2)

s

S

Fig. 4. Optical tomography types. (a) Optical coherent tomography. Radiation was fed to a Michelson interferometer consisting of beam splitter BS and mirror M. Sample S was placed in one of the arms of the interferometer. Light at the output of the inter ferometer is detected by detector D. The sample can be scanned at different depths by varying the position of the mirror. (b) Quan tum optical coherent tomography. A pair of photons is created in a nonlinear crystal. Sample S is placed in signal mode s and the length of the path of idle photon i is varied by prism P. Taking photons on beam splitter BS, images of different layers of the sample can be obtained using the Hong­Ou­Mandel dip effect. The scanning depth can be varied by displacing the prism.

Note the problem of the synchronization of clocks for solving which the use of a pair of photons with small coherence time (2) was proposed [35] when a pair of photons is separated into two spatial modes; then, the photons were guided to detectors triggering clocks that should be synchronized. We emphasize that most applications discussed above impose requirements only on the spectral width of the biphoton field, not on its shape. It is always assumed that the spectral shape is close to Gaussian or rectangular; however, this is not necessarily the fact in reality. In view of this circumstance, the problem of controlling not only the width, but also the shape of the spectrum of biphotons is interesting. 4. METHODS FOR CONTROLLING THE SPECTRUM OF THE BIPHOTON FIELD Two photon light is usually generated owing to spontaneous parametric downconversion. As was mentioned above, the state of the field in the case of a narrowband pump at the output of the nonlinear crys tal can be represented in the form of Eq. (3), where the spectral amplitude F() is given by the expression [2]
L

phase mismatch. In the case of the homogeneous crys tal, k is independent of z and Eq. (13) is simplified as k ( ) L F ( ) L exp ­ i k ( ) L sin c . 2 2 (14)

According to Eq. (14), it is seen that the spectrum of the biphoton field is determined by the frequency dependence of the phase mismatch k() and its width, by the condition ­2 k() 2 . (15) L L Thus, in order to obtain the biphoton field with a nar row spectrum, a sufficiently long crystal can be taken and be placed into the cavity for additional frequency selection. In this case, the spectrum narrower than 3.0 MHz can be obtained [36]. The problem of obtaining the biphoton field with a broad spectrum is more difficult. As follows from Eq. (15), the trivial solution would be the use of a short crystal. In particular, a field with a spectral width of 174 nm (106 THz was obtained in [37] with the use of the BBO crystal 0.1 mm in thickness in the orthogonal polarization modes in the degenerate regime at a wavelength of 702 nm. However, the intensity of radi ation proportional to the square of the length of the crystal, Eq. (14), decreases in the case of the thin crys
4

4

F ( ) dz exp [ i k ( ) z ] ,
0



(13)
As a characteristic of the spectral width of the biphoton field, the range width is taken, because it is certainly related to the
(2) correlation times (1) and min and the short correlation time is responsible for the value of the broad spectrum in most appli cations.

where z is the coordinate along the axis directed along the pump propagation, L is the length of the crystal in which generation occurs, and k = kp ­ ks ­ ki is the

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Vol. 112

No. 1

2011


CONTROL OF THE SPECTRUM OF THE BIPHOTON FIELD

25

tal. The integral intensity decreases linearly with a decrease in L [38]. Another method for obtaining the broad spectrum of the biphoton field is the choice of the phase match ings such that the mismatch function k() depends weakly on in a certain interval near the exact phase matching (k = 0). By definition k() = kp ­ ks ­ ki or, in representation (2), k ( ) = k p ­ k s ( s0 + ) ­ k i ( i0 ­ ) (16) = k p ­ k s0 ( ) ­ k i0 ( ­ ) , where k
s, i 0

( ) = k s, i (

s, i 0

+ ).

(17)

Expanding k() in a Taylor series, we obtain k ( ) = [ k p ­ k s0 ­ k i0 ] (18) 2 1 ­ [ k 's0 ­ k 'i0 ] ­ [ k '' + k ''0 ] ­ ... , s0 i 2 where all derivatives of the functions ks, i0() are taken at zero. Consequently, the broadband phase matching requires the conditions k p ­ k s0 ­ k i0 = 0 , (19) k 's0 ­ k 'i0 = 0 , k '' + k '' = 0 . s0 i0 (20) (21)

where kg = 2/ and is the period of the polarization of the square susceptibility of the crystal. Then, condi tion (19) is modified to the form (23) k p ­ k s0 ­ k i0 ­ k g = 0 , which makes it possible to satisfy conditions (20) and (21) by choosing the dispersion of the crystal and the wavelength of the pump and to satisfy condition (23) by choosing the period of induced polarization. It was shown in [40] that the generation of the biphoton field with a spectral width of 1080 nm (91 THz) can be ensured in a periodically polarized lithium niobate (LiNbO3) crystal 1 cm in thickness with a period of = 27.4 m under the condition of collinear type I phase matching degenerate at a wavelength of 1885 nm. Another method for locally smoothing the k() dependence was demonstrated in [5]. The feature of the proposed scheme is the use of elements introduc ing angular dispersion (see Fig. 5). It was shown in [41­43] that a light pulse passing through a system of two diffraction gratings (or prisms) surrounding a medium with the walk off effect is transformed as if it propagated through a medium with changed deriva tives of the dispersion function, k'() and k''() ~ k ' = k ' + , (24) tan , = tan , = c where is the angle between the wave vector and Poynting vector (walk off angle), is the pulse front tilt angle [44], which appears after the first dispersing element and is compensated by the second dispersing element, and c is the speed of light. The angle depends on the parameters of the dispersing element and on the central wavelength of the pulse. Using Eqs. (24), one can choose the dispersing elements such as to ensure conditions (20) and (21) [5]. In particular, the spectral width in a BBO crystal 2 mm in thickness was experimentally increases from 5.2 nm (2.4 THz) to 41 nm (19 THz) for the case of collinear type II phase matching degenerate at a wavelength of 810 nm. The broadening of the spectrum from 96 nm (44 THz) to 465 nm (213 THz) for the case of degenerate type I phase matching was theoretically predicted. However, the correctness of approximations (24) used for such a broad spectral range is doubtful in view of the initial limit of the spectral width. Additional possibilities of varying the function k() appear in the case of noncollinear phase matching. Let us consider the generation of spontane ous parametric downconversion in a periodically polarized crystal by a monochromatic pump having a certain angular distribution f (q) with a width of q, where q is the transverse component of the wave vector (see Fig. 6) [45]. It is convenient to represent the phase mismatch k in the form of the sum of two compo
Vol. 112 No. 1 2011

~ k '' = k '' ­ , k

2

The first condition is the exact phase matching for the central frequencies of the signal and idle photons, the second condition means the equality of their group velocities, and the third condition ensures the absence of the dispersion of group velocities. Note that conditions (19) and (20) are satisfied automatically for the degenerate phase matching of type I, because the functions ks0() and ki0() are identical in this case and kI 2. In the case of non degenerate phase matching or type II phase match ing, where the polarizations of the signal and idle pho tons are orthogonal, kII . However, kI 4 for degenerate type I phase matching under condition (21) and kII 2 for type II phase matching under condition (20). It is very difficult, though possible to choose the medium simultaneously satisfying conditions (19)­ (21), i.e., to ensure the local smoothing of the k() dependence near the exact satisfaction of the phase matching. In particular, it was shown in [39] that the width of collinear degenerate type I phase matching for a 14 mm thick BBO crystal and a 728 nm pump is about 750 nm (106 THz). To simplify the problem of simultaneous satisfac tion of conditions (19)­(21), it was proposed to use periodically polarized crystals in which the phase matching is satisfied taking into account the reciprocal superlattice vector kg: k = kp ­ ks ­ ki ­ kg , (22)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


26 PBS Gr2 s

KATAMADZE, KULIK D1 Mono1 q k
g p

k

p

k

s


i

s

i Mono2 D2 Pump Gr1
Fig. 5. Local smoothing of the k() dependence due to angular dispersion. The generation of spontaneous para metric downconversion in type II phase matching occurs in a nonlinear crystal located between two diffraction grat ings. Grating Gr1 creates an inclination of the pump pulse front, which holds for the pulse front of spontaneous para metric downconversion and is compensated by lattice Gr2. As a result, the phase matchings are modified to Eqs. (25). Polarization beam splitter PBS separates the signal and idle photons into two spatial modes, which are then regis tered by detectors D1 and D2. To determine the spectrum of the biphoton field, monochromators Mono1 and Mono2 are used.

z

k
(2)

i

CC

Fig. 6. Orientation of the wave vectors under the noncol linear phase matching of spontaneous parametric down conversion in a periodically polarized sample.

According to Eq. (28), the k||() is weakened with an increase in the angles s and i. At the same time, the condition k = 0 together with Eq. (27) implies the constraint on the spectral width, which is attributed to the width of the angular distribution of the pump q and is enhanced with an increase in the angles s and i. Let us consider the case of degenerate type I phase matching, where the signal and idle modes have the ordinary polarization. In this case, ks0 = ki0 = k0, s = i = 0, and under the exact phase matching k p ­ k g ­ k s0 cos s ­ k i0 cos i = 0 (29) Eqs. (27) and (28) have the form k = q + 2 k '0 sin 0 + ... , (30)
2 k || = ­ 2 k '' cos 0 + ... (31) 0 The condition k = 0 implies the constraint on the spectral width: 1 (32) q ' 2 k 0 sin 0 and the condition k|| 2/L provides

nents k and k|| orthogonal and parallel to the z axis, respectively, coinciding with the pump propagation direction. In this case, the phase matchings have the form 2 k = 0 , ­ 2 k || . (25) L L In the approximation of the narrow angular spectrum of the pump, q k p = k p sin p k p p , k p || = k p cos p k p . In this case, k and k|| can be represented in the form k = q + k s sin s ­ k i sin i , (26) k || = k p ­ k s cos s ­ k i cos i ­ k g . Taking into account representation (16), we expand Eqs. (26) into the Taylor series in in the form similar to Eq. (18): k ( ) = q + [ k s0 sin s ­ k i0 sin i ] + [ k 's0 sin s + k 'i0 sin i ]
2 1 + [ k '' sin s ­ k '' sin i ] + ... , s0 i0 2 k || ( ) = [ k p ­ k g ­ k s0 cos s ­ k i0 cos i ]

(27)

2 (33) . '' Lk 0 cos 0 Thus, the spectrum can be strongly broadened for the strongly focused pump radiation. In particular, a pump focusing induced increase in the spectral width from 6.2 nm (2.8 THz) to 148 nm (67 THz) was exper imentally demonstrated in [46] for the case of degen erate type I phase matching at a wavelength of 812 nm. Note that this method leads to angular broad ening, which is a direct consequence of the phase matchings. Note that the phase matchings for the case of the transverse phase matching (s = i = 90°) are degener ate and have the form = 1 q; (34) 2 k '0 i.e., the form of the frequency spectrum of the bipho ton field under closed phase matching completely cor responds to the angular spectrum of the pump. This provides the possibility of controlling the shape of the frequency spectrum.
Vol. 112 No. 1 2011

­ [ k 's0 cos s ­ k 'i0 cos i ]
2 1 ­ [ k '' cos s + k '' cos i ] + ... s0 i0 2

(28)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


CONTROL OF THE SPECTRUM OF THE BIPHOTON FIELD

27

The spectral width of the biphoton field can be increased due to the broadening of not only the angu lar, but also frequency spectrum of the pump. In this case, the frequency anticorrelation condition is weak ened (s + i const), but the small broadening of the frequency spectrum of the pump under certain condi tions can lead to the strong broadening of the spec trum of biphotons. Taking into account the width of the pump spectrum, Eqs. (2) are represented in the form p = p0 + p , s = s0 + s , i = i0 ­ i , (35)

where the frequency phase matching is satisfied both for the central frequencies, p0 = s0 + i0, and for mismatchs, p = s ­ i. Let us expand k in a power series in p, s, and i. Hereinafter, we retain only the first power of p, assuming that the spectral width of the pump is much smaller than the spectral width of biphotons: ' ' ' k = [ k p0 ­ k s0 ­ k i0 ] 0 + [ k p0 p ­ k s0 s + k i0 i ] 1 '' 2 '' 2 ­ [ k s0 s + k i0 i ] 2 ­ ... 2
1

(36)

Let us consider the case of degenerate type I phase matching (ks0 = ki0 = k0). Taking into account that kp0 ­ ks0 ­ ki0 = 0 under the exact satisfaction of the phase matching for central frequencies, we obtain
2 k = ( k 'p0 ­ k '0 ) p ­ k '' [ s ­ p s ] 2 . s0

(37)

We emphasize that all considered methods for broadening the spectrum of the biphoton field are reduced to the local smoothing of the k() depen dence, which ensures the satisfaction of the phase matchings in a wider wavelength range. This leads, on one hand, to the broadening of the absolute value of the spectral amplitude and, on the other hand, to a weak frequency dependence of the spectral amplitude. This makes it possible to narrow not only the first order correlation function, but also the second order correlation function. Nevertheless, such a method for broadening of the spectrum of spontaneous paramet ric downconversion is strongly restricted by the disper sion relations in the medium and is applicable only for the case of comparatively small frequency mismatchs , when the several first terms of the Taylor expansion are sufficient. Moreover, this method is inapplicable for the nondegenerate regime. Recall that Eq. (14) for the spectral amplitude was obtained under the assumption that the crystal in which spontaneous parametric downconversion occurs is spatially homogeneous and the phase mis match k is independent of z. The use of spatially inhomogeneous structures makes it possible to simul taneously satisfy the phase matchings for different fre quency pairs in different regions of the crystal. As a result, spontaneous parametric downconversion com ponents appearing in different parts of the crystal are added taking into account phases and output radiation has a broad spectrum with a complex shape (due to interference) and a nontrivial frequency dependence of the spectral amplitude:
L

The condition k = 0 provides the square equation with respect to s whose solution has the form [4, 6] = p ± p p , 2 k ' ­ k '0 . where = p0 k '' 0

F ( ) dz exp [ i k ( , z ) z ] .
0



(39)

(38)

Note that the coefficient is nonnegative for the case of normal dispersion. Thus, the width of the spectrum of the biphoton field is = p for the fixed spectral width of the pump p. The broadening of the spectrum to 197 nm (84 THz) at a central wavelength of 840 nm was exper imentally demonstrated in [47], where the spectral width of the pump was 7.7 THz and lithium iodate (LiIO3) was used as a nonlinear crystal. A similar broadening of the spectrum in the nondegenerate regime at the wavelengths of 741 and 909 nm to about 30 THz, which is much larger than the width of the typical spectrum of spontaneous parametric down conversion in the nondegenerate regime, was demon strated in [48], where a BBO crystal was used as the nonlinear crystal 3 mm in length and the spectral width of the pump was 6.5 THz.

Thus, the resulting radiation can be Fourier limited and, to reduce (2), compression methods should be additionally used [49­51]. To obtain the spatial dependence k(z), it is possi ble to use periodically polarized crystals, where polar ization period increases [52] so as to ensure a linear increase (chirp) in the reciprocal superlattice vector (see Fig. 7a): k g ( z ) = k g0 + z . In this case, the mismatch has the form k ( , z ) = k p ­ k s0 ( ) ­ k i0 ( ­ ) ­ k g ( z ) . (41) (40)

The spectral width of the resulting radiation can be estimated as the difference ~ ~ (42) (z = 0) ­ (z = L), ~ where (z) is the solution of the equation k(, z) = 0. In particular, spontaneous parametric downconver sion radiation at a degenerate wavelength of 812 nm with a spectral width from 17 nm (7.7 THz) to 300 nm
Vol. 112 No. 1 2011

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


28

KATAMADZE, KULIK (a)

|F(s, i)|2 1.0 0.8 0.6 0.4 1 0.2 0 1000 s, i, nm 2 (b)

600

800

Fig. 7. Broadening of the spectrum in the periodically polarized structure with linear chirp. (a) Periodically polarized crystal with a linear variation of the polarization period in the radiation propagation direction. The arrows indicate the directions of the polar ization in different parts of the crystal. (b) The wavelength dependence of the spectral intensity of the biphoton field at = (1) 0.2 â 10­7 and (2) 9.7 â 10­6 m­2.

(136 THz) was obtained in [53] when the growth parameter in the stoichiometric lithium tantalate (SLT) 18 mm in length increases from 0.2 â 10­7 to 9.7 â 10­6 m. In that work, the spectrum had a com plex shape and included several peaks (see Fig. 7b). Radiation obtained in hyperparametric scattering [35] can be used as an alternative source of the bipho ton field with a broad spectrum. Since hyperparamet ric scattering is due to the third order nonlinear sus ceptibility (3), it can be observed in media with an inversion center, e.g., in an optic fiber. The smallness of (3) can be compensated by the length of the fiber. Such a method for obtaining the biphoton field has certain advantages as compared to the methods dis cussed above, because radiation obtained in the fiber can be matched with other fiber optical devices. The radiation of spontaneous parametric downconversion obtained in the crystal is not diffraction limited and it is difficult to introduce it in the fiber with small losses. Moreover, phase matchings for hyperparametric scat tering are much weaker and the phase matching width is much larger. The problem is that it is difficult to detect hyperparametric scattering in the degenerate regime, because emission occurs at the pump fre quency. In addition, there are a number of incidental effects induced by (3) nonlinearity that are compara ble in intensity with hyperparametric scattering; Raman scattering is the strongest of these effects. Sep arating hyperparametric scattering radiation from "supercontinuum," one can obtain a spectrum with a

width of up to 10 THz at a mismatch of 25 THz from the degenerate regime at a wavelength of 741 nm [54]. 5. INHOMOGENEOUS BROADENING OF THE SPECTRUM DUE TO THE SPATIAL MODULATION OF THE REFRACTIVE INDICES ALONG THE NONLINEAR CRYSTAL Another method for creating the dependence of k on z was proposed in [52]. It consists of varying the refractive indices np, ns, and ni using their dependence on the external parameters such as temperature [55], electrostatic field, and pressure. This method is con sidered experimentally and theoretically in this work. 1. Let us consider the control of the spectrum due to the temperature modulation of the refractive index. The temperature distribution T(z) induced along the crystal leads to the spatial dependence k(T(z)); con sequently, the integral amplitude is given by the spec tral function:
L

F ( ) dz exp [ i k ( , T ( z ) ) z ] .
0



(43)

The temperature dependence of the phase mismatch k is described as follows. Let the refractive indices be linear functions of the temperature, nj = nj 0 + j T , j = p, s, i .
Vol. 112 No. 1

(44)
2011

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


CONTROL OF THE SPECTRUM OF THE BIPHOTON FIELD Copper cooler Thin heat conducting layer Textolite baffles Cold junctions of differential thermocouples Crystal Thermocouples Low ohmic resistors

29

Water Water

To the feed To the voltage circuit

Water with ice

Fig. 8. Five section heater ensuring the inhomogeneous heating of the crystal. Each section contains 15 resistors as heating elements and copper­constantan thermocouples. The cold junctions of thermocouples are located outside and are immersed in the mixture of water and ice in order to fix their temperature at 0°C. The feed circuit makes it possible to independently vary the voltage/power at each resistor.

For simplicity, we assume that s i , which is valid near the degenerate regime. In this case, the mis match is given by the expression k = k0 + p ( p ­ ) T . c (45)

If the temperature of the sample is a linear function of the longitudinal coordinate, T ( z ) = T0 + z , the dependence k(z) has the form k ( z ) = k0 + z , = p . c (46)

ments. To control the temperature in each section of the heater and in the radiator, thermocouples were located near the surface on which the crystal was placed. Voltages on all resistors could be controlled independently, thus ensuring the control of the tem perature distribution T(z) along the crystal. To ensure the thermal contact, each section was filled with a heat conducting compound and the sections were separated by textolite plates 1.2 mm in thickness. Note that one of the demerits of this method is the impossibility of controlling the temperature inside the crystal. Even if the pump beam maximally approaches the surface of the heater, the temperatures inside the crystal and on its surface can be strongly different at temperatures above 100°C. Figure 9 shows the layout of the experimental setup. An Ar laser operating in the continuous regime produced a 351.1 nm pump beam with an angular divergence of about 0.2 mrad. The beam was guided to the optical system using prism P separating the neces sary spectral mode and mirror M. A portion of light passed through the mirror was detected by photoresis tor D1 to control the pump power. After the passage through the vertically oriented polarization prism V, the laser beam arrives at a potassium dihydrogen phos phate (KDP) crystal cut at an angle 50° to the optical axis in order to ensure the collinear degenerate phase matching (crystal dimensions, 20 â 7 â 7 mm3). The crystal is placed on the heater and the temperature dis tribution T(z) can be created along the crystal by applying a voltage on different sections. Filter F and
Vol. 112 No. 1 2011

Thus, the mismatch depends not only on the parame ters of the crystal and pump, but also on the external control parameter . To estimate the spectral width, modified expres sion (42) can be used: ~ ~ (47) (T(z = 0)) ­ (T(z = L)), ~ where (T) is the solution of the equation k(, T) = 0. The Sellmeier formulas [56] can be used to deter mine the dependence k(, T). To create a given temperature gradient along the crystal, a five section heater was created (see Fig. 8). In order to ensure the maximum temperature gradi ent, a copper radiator with a one through cooling water system was placed on one of the sides of the heater. Low ohmic resistors were used as heating ele

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


30 D1 V KDP

KATAMADZE, KULIK

H

ISP 51

M

F

O

D2



PC Ar 351 nm P

Fig. 9. Layout of the experimental setup.

horizontally oriented polarization prism H intercept parasitic pump radiation and luminescence and trans mit spontaneous parametric downconversion radia tion. Objective O located downstream focuses radia tion onto the input slit of the ISP 51 spectrograph. Detector D2, which is a silicon avalanche photodiode (PerkinElmer C3090E) operating in the photon count regime, is placed behind the spectrograph. Since the slit of the spectrograph is in the focal plane of the objective, the objective ensures the transition from the angle to the coordinate; therefore, angular scanning can be performed by moving the detector through the vertical. Wavelength scanning is performed by rotating the prism of the spectrograph. The frequency and
, deg 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 30 50

Temperature, °C 75 45 Power, W 2 0 12345 Section no.

70

90 , THz

Fig. 10. Relation between the widths of the frequency, , and angular, , spectra of biphotons under a change in the orientation of the crystal and unchanged temperature distribution T(z). The squares are the experimental points corresponding to the inhomogeneously heated crystal. The triangle is the point corresponding to the crystal at room temperature. The histograms show the powers and temper atures of the sections of the heater (the zeroth section cor responds to the heater).

angle scanning is performed using a PC controlled step motor. Three series of the experiments were carried out. In the first series, we determined the variation of the widths of the angular and frequency spectra of the biphoton field generated in the inhomogeneously heated crystal under a small change in the angle between its optical axis and the pump propagation direction. The angular spectrum was recorded in the frequency degenerate regime, the frequency spectrum was detected in the collinear regime, and the temper ature distribution along the crystal remained unchanged. The results are shown in Fig. 10, where it is seen that the frequency broadening can be trans formed to the angular broadening by varying the ori entation of the crystal. At the maximum frequency broadening, the width of the angular spectrum is close to the width of the angular spectrum of the cold crystal and the frequency broadening disappears at the maxi mum broadening of the angular spectrum. Thus, these experiments show that the frequency and angular broadenings of the spontaneous parametric downcon version spectra can be separated by appropriately choosing the orientation of the crystal. In the second series, the dependence of the width of the frequency spectrum in the collinear regime on the difference between the minimum and maximum temperatures on the crystal was determined. Since the heater was longer than the crystal, its thermal contact with the radiator was not ensured; for this reason, the temperature of the radiator was disregarded when cal culating this difference. The frequency spectrum was detected for two cases. (a) The degenerate regime of the generation of spontaneous parametric downconversion. The crystal was oriented so as to ensure the maximum spectral width.
Vol. 112 No. 1 2011

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


CONTROL OF THE SPECTRUM OF THE BIPHOTON FIELD , THz 240 (a) 200 60 160 120 80 20 40 40 , THz 80 (b)

31

0

50

100

150

200 T, K

0

50

100

150

200 T, K

Fig. 11. Spectral width versus the difference T between the maximum and minimum temperatures on the crystal. The points are the experimental data and the dash­dotted and dashed lines are theoretical estimates by Eq. (47) for the minimum tempera tures of 24 and 100°C, respectively.

Integral intensity, arb. units 1.0 (a)

Integral intensity, arb. units 1.0 (b)

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

50

100

150 , THz

0

10

20

30

40 50 , THz

Fig. 12. Integral intensity versus the spectral width in the (a) degenerate and (b) nondegenerate cases.

(b) The frequency nondegenerate regime of spon taneous parametric downconversion, when the differ ence between the frequencies nearest to the degener ate case was 84 THz. The results are shown in Figs. 11a and 11b, respec tively. The lines are the lower theoretical estimates

obtained by Eq. (47). Since the minimum temperature of the crystal increased slowly in spite of water cooling, the estimates were performed for two cases, where the minimum temperatures were 24 and 100°C. The experimental data are in good agreement with the the oretical results when the temperature gradient is no
Vol. 112 No. 1 2011

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


32 Intensity, arb. units 1.0 0.8 0.6 0.4 0.2 0 600 650 700 Wavelength, nm (a)
Temperature, °C 65 45 25 Power, W 2 1 0 12345 Section no.

KATAMADZE, KULIK Intensity, arb. units 1.0 0.8 0.6 0.4 0.2 0 600 650 700 Wavelength, nm (b)
Temperature, °C 65 45 25 Power, W 2 1 0 12345 Section no.

Intensity, arb. units 1.0 0.8 0.6 0.4 0.2 0 600 650 700 Wavelength, nm (c)
Temperature, °C 65 45 25 Power, W 2 1 0 12345 Section no.

Fig. 13. Control of the shape of the spectrum. The histograms show the powers and temperatures of the section of the heater (the zeroth section is the radiator).

more than 75°C. Above this value, the experimental width of the spectrum is smaller than the theoretical value. The reason is that the temperature distribution inside the crystal at high temperatures (200°C and above) can be strongly different from that detected by thermocouples located in the heater near it surface. As a result of multiple heatings of the crystal, the spectral width was increased from 21 to 154 THz in the degen erate regime and from 3.5 to 48 THz in the nondegen erate regime. Note that the maximum currently achieved width of the spectrum of the experimentally obtained biphoton field is 136 THz (see Fig. 7) [40]. In addition, the dependence of the integral inten sity (i.e., the area under the envelope of the frequency spectrum) on the spectral width was determined for both degenerate and nondegenerate cases. In this case, the photon count rate was normalized to the pump power. The results are shown in Fig. 12. A large error in the determination of the intensity is caused by the instability of the operation of the laser and by the dif ficulty of the inclusion of background light from both external sources and the luminescence of the optical elements of the setup. It is seen in the plots that the integral intensity decreases. In particular, the integral intensity decreases by 84 and 72% for the cases of the maximum broadening of the spectrum up to 154 THz in the degenerate regime and up to 48 THz in the non degenerate regime, respectively. This is apparently due to the simultaneous broadening of the angular spec trum in the transition to the significantly nondegener ate regime. In the third series of the experiments, we studied the dependence of the shape of the spectrum on the temperature distribution along the crystal. Since the shape of the spectrum near the degenerate regime depends strongly on the orientation of the crystal, the spectrum was recorded in the nondegenerate regime for the central wavelength of one of the maxima near 644 nm (at a degenerate wavelength of 702.2 nm). Fig

ure 13 shows the three spectra obtained with various temperature distributions. In case (a), only the extreme section of the heater was switched on; for this reason, the most part of the crystal was not heated and the maximum of the spectrum is closer to the degener ate regime. In case (b), additional three sections of the heater were switched on, the crystal was heated better, and the shape of the spectrum was close to rectangular. In case (c), two sections maintain a high temperature on one half of the crystal, where the second half is not heated and the spectrum exhibits two maxima at the edges of the spectrum near and far from the degenerate regime. Figures 14 and 15 exemplify the experimental fre quency and angular spectra. It is worth noting that the resulting spectral widths are not maximum possible. First, a broader spectrum can be obtained with a better inhomogeneous heating (or cooling) of the crystal. Second, a better result can be obtained with other crystals. The main characteristic of the crystal neces sary for such a method for broadening the spectrum is the parameter (p) ­ (s) introduced in Eqs. (44) and (45). It is physically determined by the temperature dependence of the dispersion of the refractive index at the frequencies p and s i p/2. In the KDP crystal for the degenerate regime and p = 351 nm, this parameter is = ­5.5 â 10 6 K­1 [56]. However, a much larger broadening of the spectrum can be obtained using a medium with a larger parameter . For example, it is an order of magnitude larger, = 2.4 â 10­5 K­1, for a lithium niobate crystal (p = 750 nm, s i = 1500 nm) [57]. 2. Let us consider the possibility of controlling the spectrum of the biphoton field owing to the modula tion of the refractive indices induced by the electro
Vol. 112 No. 1 2011

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


CONTROL OF THE SPECTRUM OF THE BIPHOTON FIELD Intensity, arb. units 1.0 0.8 0.6 0.4 0.2 0 340 360 (a) Temperature, °C 330 280 230 180 130 80 Power, W 4 3 2 1 380 400 420 440 460 480 500 520 Frequency, THz 12345 Section no. Temperature, °C 330 280 230 180 130 80 Power, W 4 3 0.2 0 2 1 340 360 380 400 Frequency, THz 0 12345 Section no. 0

33

Intensity, arb. units 1.0 0.8 0.6 0.4 (b)

Fig. 14. Examples of the broadening of the frequency spectrum of the biphoton field. The dashed lines are the spectra of the inho mogeneously heated crystal (T = 239 K). The histograms show the powers and temperatures of the section of the heater (the zeroth section is the radiator). The dash­dotted lines are the spectra of the crystal at room temperature.

optical effect. Similar to the preceding case, the spec tral amplitude is expressed in terms of the integral
L

F ( ) dz exp [ i k ( , E ( z ) ) z ] ,
0



(48)

voltages between the electrodes. The layout of the experimental setup shown in Fig. 9, but the crystal subjected to the uniform electrostatic field was used instead of the inhomogeneously heated crystal. Figure 17 shows the spectra for various applied fields. The asymmetry of the spectra with respect to a degenerate wavelength of 702 nm is attributed to the frequency dependence of losses in the optical channel. It is seen that the applying the field in the degenerate regime eliminates degeneracy and leads to an increase in the distance between the peaks in the nondegener ate regime. The estimates show that the application of the spatially nonuniform static field varying along the pump propagation direction from 0 to 15 kV/cm is accompanied by an increase in the spectral width from 41 nm (25 THz) to 71 nm (43 THz) in the degenerate regime and from 6.9 to 10 THz in the nondegenerate
Vol. 112 No. 1 2011

and the spectral width is estimated as ~ ~ (E(z = 0)) ­ (E(z = L)).

(49)

According to Eq. (48), the method is based on the dependence k(E). For this reason, to prove the pos sibility of controlling the spectral width, we only dem onstrate a noticeable change in the spectrum when a uniform electrostatic field is applied to the crystal. To this end, electrodes were placed on the faces of the crystal (see Fig. 16) and the spectrum of spontaneous parametric downconversion was detected at various

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


34 Intensity, arb. units

KATAMADZE, KULIK

Temperature, °C 1.0 0.8 0.6 0.4 0.2 0 -1.2 0 70 60 50 12345 Power, W 2

-0.8

-0.4

0

0.4

0.8 1.2 Angle, deg

12345 Section no.

Fig. 15. Examples of the broadening of the angular spectrum of the biphoton field. The dashed lines are the spectra of the inho mogeneously heated crystal (T = 25 K). The histograms show the powers and temperatures of the section of the heater (the zeroth section is the radiator). The dash­dotted lines are the spectra of the crystal at room temperature.

regime. Such an inhomogeneous field distribution can be obtained by, e.g., the sectioning of electrodes. 6. DISCUSSION OF THE RESULTS The method for controlling the spectrum owing to the spatial modulation of the refractive index has all advantages and disadvantages of the methods based on the use of the spatially inhomogeneous structures as compared to the methods for locally smoothing the frequency dependence of the phase mismatch. The spectrum of the biphoton field is not Fourier limited, complicating the use of this method to obtain fields with small second order correlation time (compres sion methods are necessary). At the same time, it allows a much larger broadening of spectra and is applicable even in the nondegenerate regime. Since the spectral width in this case is no longer determined by the dispersion characteristics of the crystal, there is
+ Nonlinear crystal

a problem of the physical constraints for the maximum experimentally available spectral width of the bipho ton field. On one hand, it is seemingly limited either by technical capabilities of varying the modulation period or by the range of the temperature drop in the field along the sample. On the other hand, as was men tioned above, the broadening of the spectrum is accompanied by a decrease in the integral intensity, which complicates its detection. This problem requires additional investigation, which is beyond the scope of this work. We also note that all known meth ods for controlling the frequency spectrum of the biphoton field are also applicable to control its angular spectrum; for this reason, it is reasonable to consider the problems of controlling the frequency and angular spectra together. To conclude, we point to a number of technical advantages and disadvantages of the methods for con trolling the spectrum based on the use of spatially inhomogeneous structures. The application of chirped structures is associated with a complex technological process of their manufacture. After the completion of the repolarization cycle, the spatial structure of the sample specifying the spectrum remains then unchanged. In this context, the application of the temperature gradient to the crystal seems to be a more efficient solution of the problem. In addition, not only the width, but also the shape of the spectrum can be controlled by varying the powers of the sections of the heater. At the same time, there are obvious disadvan tages of this method associated with the difficulty of the determination of the real temperature inside the crystal and the creation of an arbitrary (preset) tem perature distribution.
Vol. 112 No. 1 2011

Optical axis -
Fig. 16. Application of the uniform electrostatic field to the KDP crystal 20 mm in length. Electrodes are deposited on the faces of the crystal and the field inside the crystal varies depending on the voltage between the electrons.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


CONTROL OF THE SPECTRUM OF THE BIPHOTON FIELD Intensity, arb. units (a) 1.0 E E E E = = = = 0 7.3 kV/cm 11 kV/cm 15 kV/cm

35

0.8

0.6

0.4

0.2

0

650

660

670

680

690

700

710

720

730 740 750 Wavelength, nm

Intensity, arb. units 1.0 (b)

0.8

0.6 E=0 E = 7.3 kV/cm E = 14.7 kV/cm

0.4

0.2

0

640

660

680

700

720

740

760 780 Wavelength, nm

Fig. 17. Spectra of spontaneous parametric downconversion for various intensities of the uniform electrostatic field near the (a) degenerate and (b) nondegenerate regimes.

The method for broadening the spectrum owing to the nonuniform electrostatic field seems to be preferable in view of the simplicity of its implementation, although it was experimentally implemented incompletely. 7. CONCLUSIONS The known methods for controlling the spectrum of the biphoton field in the presence of the spontane

ous parametric downconversion of light have been analyzed. We have considered and implemented a rel atively simple method for controlling the spectrum of the biphoton field by creating the longitudinal temper ature gradient in the nonlinear crystal, as well as have analyzed its disadvantages and advantages as com pared to other methods. The discussed method makes it possible to control the shape of the spectral distribu tions, as well as to reach the record broadening of the
Vol. 112 No. 1 2011

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


36

KATAMADZE, KULIK 19. Z. Y. Ou and Y. J. Lu, Phys. Rev. Lett. 83, 2556 (1999). 20. H. Bechmann Pasquinucci and A. Peres, Phys. Rev. Lett. 85, 3313 (2000). 21. M. Bourennane, A. Karlsson, and G. BjÆrk, Phys. Rev. A: At., Mol., Opt. Phys. 64, 012 306 (2001). 22. D. Bruss and C. Machiavello, Phys. Rev. Lett. 88, 127 901 (2002). 23. F. Caruso, H. Bechmann Pasquinucci, and C. Macchi avello, Phys. Rev. A: At., Mol., Opt. Phys. 72, 032 340 (2005). 24. D. Kaszlikowski, D. K. L. Oi, M. Christandl, K. Chang, A. Ekert, L. C. Kwek, and C. H. Oh, Phys. Rev. A: At., Mol., Opt. Phys. 67, 012 310 (2003). 25. P. G. D. Kaszlikowski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, Phys. Rev. Lett. 85, 4418 (2000). 26. T. Durt, N. J. Cerf, N. Gisin, and M. Zukowski, Phys. Rev. A: At., Mol., Opt. Phys. 67, 012 311 (2003). 27. D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, Phys. Rev. Lett. 88, 040 404 (2002). 28. M. V. Fedorov, M. A. Efremov, P. A. Volkov, and J. H. Eberly, J. Phys.: Condens. Matter 39, S467 (2006). 29. G. Brida, V. Caricato, M. V. Fedorov, M. Genovese, M. Gramegna, and S. P. Kulik, Europhys. Lett. 87, 64 003 (2009). 30. M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, Phys. Rev. Lett. 91, 083 601 (2003). 31. V. Giovannetti, S. Lloyd, and L. Maccone, Science (Washington) 306, 1330 (2004). 32. J. Squier and M. MÝller, Rev. Sci. Instrum. 72, 2855 (2001). 33. A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, Phys. Rev. Lett. 85, 2733 (2000). 34. B. E. A. Saleh, B. M. Jost, H. Fei, and M. C. Teich, Phys. Rev. Lett. 80, 3483 (1998). 35. A. Valencia, G. Scarcelli, and Y. Shih, Appl. Phys. Lett. 85, 2655 (2004). 36. M. Scholz, L. Koch, and O. Benson, Phys. Rev. Lett. 102, 063 603 (2009). 37. E. Dauler, G. Jaeger, A. Muller, A. Sergienko, and A. Migdall, J. Res. Natl. Inst. Stand. Technol. 104, 1 (1999). 38. D. N. Klyshko, Photons and Nonlinear Optics (Nauka, Moscow, 1980; Gordon and Breach, London, 1988). 39. A. Pe'er, Y. Silberberg, B. Dayan, and A. A. Friesem, Phys. Rev. A: At., Mol., Opt. Phys. 74, 053 805 (2006). 40. K. A. O'Donnell and A. B. U'Ren, Opt. Lett. 32, 817 (2007). 41. C. R. Menyuk, J. Opt. Soc. Am. B 11, 2434 (1994). 42. J. P. Torres, S. Carrasco, L. Torner, and E. W. VanStry land, Opt. Lett. 25, 1735 (2000). 43. M. Hendrych, M. Mi c uda, and J. P. Torres, Opt. Lett. 32, 2339 (2007). 44. J. Hebling, Opt. Quantum Electron. 28, 1759 (1996). 45. S. Carrasco, J. P. Torres, L. Torner, A. Sergienko, B. E. A. Saleh, and M. C. Teich, Phys. Rev. A: At., Mol., Opt. Phys. 70, 043 817 (2004). ^
Vol. 112 No. 1 2011

spectrum, which was 253 nm for a central wavelength of 702 nm or 154 THz, at the temperature difference between the extreme sections of the heater T = 156 K. The broadening of the frequency spectrum in the nondegenerate regime, as well as the broadening of the frequency and angular spectra, has been demon strated. In addition, the fundamental possibility of control ling the spectrum owing to the electro optical effect has been shown. ACKNOWLEDGMENTS We are deeply grateful to E.G. Yakimova and A.V. Korolev for assistance in the experiments with the electrostatic field. This work was supported by the Russian Foundation for Basic Research, project nos. 10 02 00204 a and 08 02 00741 a. REFERENCES
1. M. V. Chekhova, Pis'ma Zh. Eksp. Teor. Fiz. 75 (5), 271 (2002) [JETP Lett. 75 (5), 225 (2002)]. 2. A. V. Belinsky and D. N. Klyshko, Laser Phys. 4, 663 (1994). 3. D. N. Klyshko, Physical Principles of Quantum Elec tronics (Nauka, Moscow, 1986) [in Russian]. 4. Y. M. Mikhailova, P. A. Volkov, and M. V. Fedorov, Phys. Rev. A: At., Mol., Opt. Phys. 78, 062 327 (2008). 5. M. Hendrych, A. V. X. Shi, and J. P. Torres, Phys. Rev. A: At., Mol., Opt. Phys. 79, 023 817 (2009). 6. M. V. Fedorov, Y. M. Mikhailova, and P. A. Volkov, J. Phys. B: At., Mol. Opt. Phys. 42, 175 503 (2009). 7. D. V. Strekalov, Y. H. Kim, and Y. Shih, Phys. Rev. A: At., Mol., Opt. Phys. 60, 2685 (1999). 8. T. B. Pittman, D. V. Strekalov, A. Migdall, M. H. Rubin, A. V. Sergienko, and Y. H. Shih, Phys. Rev. Lett. 77, 1917 (1996). 9. C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987). 10. A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, and S. P. Kulik, Phys. Rev. A: At., Mol., Opt. Phys. 64, 041 803(R) (2001). 11. I. Abram, R. K. Raj, J. L. Oudar, and G. Dolique, Phys. Rev. Lett. 57, 2516 (1986). 12. K. S. Choi, H. Deng, J. Laurat, and H. J. Kimble, Nature Lett. 452, 67 (2008). 13. L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Nature (London) 414, 413 (2001). 14. D. N. Klyshko, Kvantovaya Elektron. (Moscow) 4, 1056 (1982). 15. D. N. Klyshko and A. N. Penin, Usp. Fiz. Nauk 152 (4), 653 (1987) [Sov. Phys.--Usp. 30 (8), 716 (1987)]. 16. J. G. Rarity, P. R. Tapster, and E. Jakeman, Opt. Com mun. 62, 201 (1987). 17. T. Aichele, A. I. Lvovsky, and S. Schiller, Eur. Phys. J. D 18, 237 (2002). 18. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Rev. Mod. Phys. 74, 145 (2002).

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


CONTROL OF THE SPECTRUM OF THE BIPHOTON FIELD 46. S. Carrasco, M. B. Nasr, A. V. Sergienko, B. E. A. Saleh, M. C. Teich, J. P. Torres, and L. Torner, Opt. Lett. 31, 253 (2006). 47. M. B. Nasr, G. D. Giuseppe, B. E. A. Saleh, A. V. Ser gienko, and M. C. Teich, Opt. Commun. 246, 521 (2005). 48. S. Y. Baek and Y. H. Kim, Phys. Rev. A: At., Mol., Opt. Phys. 80, 033 814 (2009). 49. S. E. Harris, Phys. Rev. Lett. 98, 063 602 (2007). 50. G. Kh. Kitaeva, M. V. Chekhova, and O. A. Shu milkina, Pis'ma Zh. Eksp. Teor. Fiz. 90 (3), 190 (2009) [JETP Lett. 90 (3), 190 (2009)]. 51. G. M. Brida, M. V. Chekhova, I. P. Degiovanni, M. Genovese, G. K. Kitaeva, A. Meda, and O. A. Shu milkina, Phys. Rev. Lett. 103, 193 602 (2009).

37

52. S. Carrasco, J. P. Torres, L. Torner, A. Sergienko, B. E. A. Saleh, and M. C. Teich, Opt. Lett. 29, 2429 (2004). 53. M. B. Nasr, S. Carrasco, B. E. A. Saleh, A. V. Ser gienko, M. C. Teich, J. P. Torres, L. Torner, D. S. Hum, and M. M. Fejer, Phys. Rev. Lett. 100, 183 601 (2008). 54. J. Fan and A. Migdall, Opt. Express 15, 2915 (2007). 55. D. A. Kalashnikov, K. G. Katamadze, and S. P. Kulik, Pis'ma Zh. Eksp. Teor. Fiz. 89 (5), 264 (2009) [JETP Lett. 89 (5), 224 (2009)]. 56. G. C. Ghosh and G. C. Bhar, IEEE J. Quantum Elec tron. 18, 143 (1982). 57. G. G. Gudzoryan, V. G. Dmitriev, and D. N. Nikogo syan, Nonlinear Optical Materials (Radio i Svyaz', Mos cow, 1991) [in Russian].

Translated by R. Tyapaev

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

Vol. 112

No. 1

2011