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JETP Letters, Vol. 73, No. 5, 2001, pp. 214218. Translated from Pis'ma v Zhurnal иksperimental'nooe i Teoreticheskooe Fiziki, Vol. 73, No. 5, 2001, pp. 248252. Original Russian Text Copyright © 2001 by Korystov, Kulik, Penin.

Rozhdestvenski Hooks in Two-Photon Parametric Light Scattering
D. Yu. Korystov, S. P. Kulik, and A. N. Penin
Faculty of Physics, Moscow State University, Vorob'evy gory, Moscow, 119899 Russia
Received February 8, 2001

The interference of spontaneous parametric radiation was observed from two nonlinear crystals separated by a macroscopic layer of a linear substance and excited by a common pump beam. In the presence of strong dielectric dispersion in the layer, the frequencyangular spectra display features analogous to the Rozhdestvenski hooks. The hook wavelength corresponds to the compensation of group velocity dispersions in the nonlinear crystals and the layer. © 2001 MAIK "Nauka / Interperiodica". PACS numbers: 42.50.Dv; 42.70.Mp

By now, a number of experimental procedures are known for the observation of the interference of twophoton (biphoton) radiation in either the second or fourth order in field [1]. Such an interference underlies many effects in quantum optics [2]. Because of the lack of adequate classical analogues of the biphoton states of the electromagnetic field, two-photon interference (TI) has found a purely quantum interpretation. Indeed, some experimentally detected quantities have values that are basically unattainable in the classical experiments. At the same time, a qualitative comparison of the classical and quantum optical interference effects reveals many common regularities. In some cases, the known analogues serve as the only "bridges" connecting the quantum and classical worlds at a level of interpretation of the experimental results [3]. In this work, the distortion effect observed in the interference pattern at frequencies close to the absorption band of a layer placed between two crystals spontaneously radiating photon pairs is discussed. Such distortions are caused by a phenomenon that was first employed by Rozhdestvenski in 1912 in his hook method. This elegant method was later used for the accurate quantitative investigation of anomalous dispersion in metal vapors [4]. Spontaneous parametric light scattering (SPS) in crystals with quadratic susceptibility is a source of twophoton radiation [5]. The SPS can be interpreted as resulting from the decay of pump photons with frequency p into pairs of correlated photons with frequencies s and i, according to the scheme p s + i, in a crystal with quadratic susceptibility (2). In the case of a homogeneous crystal and a plane monochromatic pump wave, the spontaneous radiation of photon pairs occurs predominantly in the directions determined by the stationarity s + i p = 0 and spatial homogeneity k ks + ki kp = 0 conditions

[kj (j = s, i, and p) are the wave vectors of the signal (s), idler (i), and pump (p) modes inside the crystal]. These expressions, called matching conditions, together with the dispersion law (k) of the crystal, define the relation s = s(s) between the frequencies and angles of the scattered radiation. The presence of spatial inhomogeneity in the scattering volume materially alters the frequencyangular SPS spectrum. Let the biphotons be emitted from two identical plane nonlinear crystals separated by a layer of a transparent substance (Fig. 1). Then the dependence of the scattering line shape (i.e., of the normalized intensity) on the scattering angle and frequency is determined from the condition sin ( /2 ) + ' 2 g 0 ( s, s ) = ------------------- cos ------------ , 2 /2 (1)

where and ' are the wave detunings in the nonlinear crystal and the layer, respectively, ( s, s) = L = ( k p k s k i ) L , '( s, s) = ' L' = ( k 'p k 's k 'i ) L', (2)

s is the angle inside the crystal, and kj and k 'j are the projections of the wave vectors onto the direction perpendicular to the layers. Equation (1) is derived without regard for absorption in the intermediate layer. The function g(s, s) is proportional to the square of the modulus of the biphoton amplitude F
2 ks k
i

2 (2) = g ( s, s ) ----- E p mL s i , c

2

which determines the contribution of the two-photon Fock's states to the wave function of the SPS field for a given pump field Ep [6] (m = 2 is the number of nonlinear crystals). Equation (1) describes the TI effect in the

0021-3640/01/7305-0214$21.00 © 2001 MAIK "Nauka / Interperiodica"

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second order in field [7]; the phase incursions of all three modes in the intermediate layer L' modulate the line shape following the law cos2[( + ')/2]. According to Eq. (1), spontaneous radiation is fully suppressed for some frequencies and angles. Since the dispersion properties of the intermediate layer influence only the modulation multiplier, the TI can be used for determining the dispersion law for the substance placed between the crystals [8, 9]. The distinguishing feature of such spectroscopy is that the scattering intensity is determined by the nonlinear SPS process, whereas the interference modulation is driven by the phase incursion in the layer, where, in the general case, (2) = 0; i.e., the substance may be linear. By expressing the wave detunings and ' explicitly in terms of frequencies and angles [8], one can determine the condition under which the slopes of the interference maxima ( + ' = 2m) are zero, i.e., s/s = 0, where is the external (observed) scattering angle: s ------- s ( s ) + ( ' 'i ) = -----------------------------------i---------------s------------------------------- = 0. s 2 1 1 1 1 s ----- L --- + --- + L' --- + --- c k 's k 'i k s k i (3)

Fig. 1. Diagram of a nonlinear interferometer. Two optically nonlinear crystals of thickness L are separated by a layer of the optically linear substance L' and are excited by a common pump beam.

Condition (3) is derived in the approximation of small scattering angles (s 1) and without regard for the anisotropy of the group and phase velocities in the nonlinear crystals. In the noncolinear scattering regime (s 0), the derivative s/s is zero if ( s i ) + ( 's 'i ) = 0. (4) Here, s = L/us, i = L/ui, 's = L'/ u 's , and 'i = L'/ u 'i have the sense of times of the signal- and idler-photon transition across the nonlinear crystal (q) and the intermediate layer ( 'q ) and uq = dq/dkq (u 'q = dq/d k 'q ) are the corresponding group velocities (q = s and i). When detecting the SPS by the crossed-dispersion method [10], the output of the spectral instrument provides a two-dimensional intensity distribution of the scattered radiation in the {s, s} coordinates [see Eq. (1)]. In these coordinates, the interference pattern has a form of alternating fringes with slopes determined by the dispersion of the nonlinear crystals and intermediate layer. The interference maximum forms a hook if the fringe slopes become zero (s/s = 0). The dispersions of the group velocities in the nonlinear crystals and intermediate layer are compensated at the corresponding frequencies of the signal (s) and idler (i = p s) waves; i.e., the differences in the times of signal and idler photon transitions through the crystal and the layer coincide in magnitude and are opposite in sign: = '. The slope increases in the range of strong dispersion in the substance, i.e., near the absorpJETP LETTERS Vol. 73 No. 5 2001

tion bands. If the nonlinear crystals are transparent over a broad spectral range for all three frequencies while the layer is transparent in the range of signal and pump waves but has resonances of dielectric constant at idler frequencies (in the IR range), then the interference pattern will reflect a change in the dispersion of idler waves in the intermediate layer. The specific feature of the TI is that the absorption in the layer between the crystals, which increases as the frequency approaches resonance, does not diminish the integrated intensity of the scattered radiation. In this case, only the interference visibility is reduced and in the limit where the absorption coefficient becomes of the order of the layer reciprocal thickness 'i 1/L' the visibility tends to zero [11]. The expression for the scattering line shape, with allowance made for the absorption at the idler frequencies, takes the form 1 + e i cos ( + ' ) 2 g 1( s, s) = sin c -- ----------------------------------------------- . 2 2
' L' ' L'

(5)

As follows from Eq. (5), by the visibility is meant the e i function. The unity in the second multiplier in Eq. (5) appears because of the spontaneous nature of the TI effect. The frequencyangular TI spectra and their features are physically similar to the interference patterns obtained by Rozhdestvenski. In his hook method, the two-dimensional interference spectra are also recorded by using the crossed-dispersion method, when the pattern at the interferometer output is that which is projected onto the entrance slit of the spectrograph. The substance under study is placed in one of the interferometer arms and glass plates of different thicknesses

216 (a) 2000
0

KORYSTOV et al. i, cm1 4000 (b) 2000 3000 i, cm1 4000

3000

5 4 3 2

0

0

0

10 5400 5400 5600 s, е 5800 6000 5600 s, е 5800 6000

Fig. 2. The frequencyangular SPS spectra: (a) two LiNBO 3 crystals (L = 440 µm) are separated by a layer (L' = 50 µm) of paraffin oil; hooks are indicated by the lines parallel to the angular axis; (b) one LiNBO 3 crystal.

are placed in the other. Because the dispersion strongly varies near the absorption band of the substance under study, there will always be a wavelength for which the action of the substance is exactly compensated by the action of a glass plate, so that the slope of the interference curve will pass through zero at this point; to the left of this wavelength, the curves go down and, to the right, they go up (or vice versa), thus forming a hook, whose position can be accurately measured on the wavelength scale [12]. Of course, the Rozhdestvenski method deals with the linear effect, i.e., with the compensation of light velocities in different substances at

the same frequency under the condition of broadband illumination of the interferometer. The requirement on the frequency corresponding to the hook top can be expressed in terms of the group and phase delay times in the substance and plate: ( 'g r g r ) + ( 'vac vac ) = 0. (6)

Here, gr = L(d/dk)1, vac = L/c, 'vac = L'/c, and 'g r = L'(d/dk')1 are the group delay times in the substance (of length L) and the plate (of length L'). As in Eq. (3), the small-angle approximation m 1 is used, where m is the order of interference. Therefore, for the hook to appear in the Rozhdestvenski method, it is necessary that the difference in group delays 'g r gr in the substance and plate coincide in magnitude with the difference in group (phase) delays 'vac vac in the vacuum and be of the opposite sign to it. A comparison of Eqs. (4) and (6) shows that in both cases the hooks can be observed only if the dispersion of the substance under study is compensated. With TI, the differences in the transition times of the signal and idler photons in the nonlinear crystal and the intermediate layer are compensated. In the Rozhdestvenski method, the dispersion of the substance appears as a difference in the group delay times in the studied and reference substances relative to the delays in the vacuum. The following important feature of two-photon interferometry is noteworthy: although the recording is in the visible region, the contribution to the interference
JETP LETTERS Vol. 73 No. 5 2001

Fig. 3. Frequency-dependent refraction index of paraffin oil in the vicinity of the 0 = 2950 cm1 band as derived from the TI spectra.

ROZHDESTVENSKI HOOKS (c)

217

5890

5896 (d)

(е)

5 4 3 2

2500

3000 3500 i, cm1

4000

Fig. 4. Qualitative comparison of the (a, c) hook method and (b, d) two-photon interferometry. (a, b) Schemes of the respective interferometers. (c, d) Observed interference patterns: (c) taken from [4] and (d) calculated TI modulation function for L = 440 µm and L' = 50 µm. The envelope sin c2(/2) restricting the observed frequencyangular TI spectrum is shown by the dotted line.

pattern is determined by the group delay at a conjugated frequency lying in the IR range. In the experiment, the SPS was excited in two thin (L = 440 µm) lithium niobate crystals arranged in tandem and exposed to a common argon laser beam. These crystals are transparent over a wide range (0.4-5.0 µm) and their parametric scattering spectra are well understood. The gap between the crystals was filled with a layer of paraffin oil. The layer thickness was varied within 1 < L' < 50 µm. The oil has an isolated absorption band at a frequency of 2950 cm1 with a width of 95 cm1. The photograph of a fragment of the frequencyangular TI spectrum of the crystaloilcrystal system is given in Fig. 2a. For comparison, the SPS spectrum of one of the lithium niobate crystals is shown in Fig. 2b (L = 440 µm), i.e., in the absence of TI. The resonance-induced distortion of a monotonic progression of the TI orders is clearly seen near 2950 cm1 (signal wavelength s = 5700 е). The visibility of the interference pattern within the absorption line width is close to zero. In this frequency range, the spontaneous radiation from two crystals is independent, because the idler
JETP LETTERS Vol. 73 No. 5 2001

photons are absorbed in the gap. A sharp change in the slope of the interference maxima occurs in close vicinity of the resonance. It is in this range that the oil refraction index n'(i) strongly depends on frequency. For some frequencies (i 3105 and 2805 cm1), the slope of the orders is zero and hooks (indicated by lines) appear in the spectra: the differences in the times of transition through the lithium niobate and paraffin oil coincide for the signal and idler photons. By varying the thickness L' of the gap between the crystals, one can, according to Eq. (4), shift the hooks on the frequency axis, as it also happens in the Rozhdestvenski method. It is notable that the zero slope of the interference orders can be observed away from the resonance as well, because the compensation is also possible in the case of a weak dispersion variation in the substance, but the hooks are then widely extended on the frequency scale. The TI spectra were used to calculate the n'(i) dependence for mineral oil. Using the data on the dispersion n() in lithium niobate [13] at frequencies i1 3105 cm1 (s1 = 5752 е) and i2 2805 cm1

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KORYSTOV et al.

(s2 = 5654 е), where the hooks are observed (see Fig. 2a), the differences in the group delay times in oil were determined from Eq. (4). They proved to be 's 1 'i 1 = 330 fs and, respectively, 's 2 'i 2 = 312 fs. After measuring the oil dispersion in the visible range by the prism method, Eq. (5) was used to find the group velocities on different sides of the resonance frequency 0 = 2950 cm1: u '1 (i1) u '2 (i2) = (8.3 ± 0.5) в 109 cm/s. These data agree, within the experimental error, with the results obtained by estimating u' from the measured n'(i) dependence (Fig. 3). In conclusion, let us turn to Fig. 4, where the two methods are pictorially compared with each other. The schemes of linear and nonlinear interferometers [14] are shown in Figs. 4a and 4b. The intermediate layer (L') is an analogue of the substance under study (of length L') and the nonlinear crystals (L) are analogues of the compensation plates (of length L). The respective delays are shown by the shaded rectangles. The frequencyangular TI spectra and the Rozhdestvenski interference patterns borrowed from [4] are shown in Figs. 4c and 4d. For clarity, only the modulation TI component cos2[( + ')/2] is shown, without taking into account the envelope sin c2(L/2) (shown by the dotted line). The qualitative similarity of both pictures confirms that the analogues are adequate. It should also be emphasized that the mechanism of formation of the frequencyangular TI spectra near the absorption bands of the intermediate substance has much in common with the formation of the polariton-scattering spectra observed in crystals without center of inversion [15]. We are grateful to A.N. Nozdryakov and S.V. Ivanchenko for assistance in obtaining the IR spectra. This work was supported by the Russian Foundation for Basic Research, project nos. 99-02-16418, 99-02-16419, and 00-15-96541.

REFERENCES
1. A. V. Burlakov, M. V. Chekhova, D. N. Klyshko, et al., Phys. Rev. A 56, 3214 (1997). 2. L. Mandel, Rev. Mod. Phys. 71, S274 (1999). 3. D. N. Klyshko, Usp. Fiz. Nauk 164, 1187 (1994) [Phys. Usp. 37, 1097 (1994)]. 4. D. S. Rozhdestvenskioe, Works on Anomalous Dispersion of Metal Vapors (Akad. Nauk SSSR, Moscow, 1951). 5. D. N. Klyshko, Photons and Nonlinear Optics (Nauka, Moscow, 1980). 6. A. V. Belinsky and D. N. Klyshko, Laser Phys. 2, 112 (1992). 7. D. N. Klyshko, Zh. иksp. Teor. Fiz. 104, 2676 (1993) [JETP 77, 222 (1993)]. 8. D. Yu. Korystov, S. P. Kulik, and A. N. Penin, Kvantovaya иlektron. (Moscow) 30, 922 (2000). 9. A. V. Burlakov, S. P. Kulik, A. N. Penin, and M. V. Chekhova, Zh. иksp. Teor. Fiz. 113, 1991 (1998) [JETP 86, 1090 (1998)]. 10. D. N. Klyshko, A. N. Penin, and B. F. Polkovnikov, Pis'ma Zh. иksp. Teor. Fiz. 11, 11 (1970) [JETP Lett. 11, 5 (1970)]. 11. A. V. Burlakov, Yu. B. Mamaeva, A. N. Penin, and M. V. Chekhova, Zh. иksp. Teor. Fiz. (in press) [JETP (in press)]. 12. G. S. Landsberg, Optics (Nauka, Moscow, 1976). 13. G. Kh. Kitaeva, K. A. Kuznetsov, I. I. Naumova, and A. N. Penin, Kvantovaya иlektron. (Moscow) 30, 726 (2000). 14. It was demonstrated in [1] that the schemes shown in Figs. 1 and 4b are physically equivalent. 15. Yu. N. Polivanov, Usp. Fiz. Nauk 126, 185 (1978) [Sov. Phys. Usp. 21, 805 (1978)].

Translated by V. Sakun

JETP LETTERS

Vol. 73

No. 5

2001