Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://qi.phys.msu.ru/kulik/Papers/JETP1_04KulikLO.pdf Äàòà èçìåíåíèÿ: Thu Dec 25 15:40:24 2003 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:05:03 2012 Êîäèðîâêà:

Journal of Experimental and Theoretical Physics, Vol. 98, No. 1, 2004, pp. 31­38. Translated from Zhurnal èksperimental'nooe i Teoreticheskooe Fiziki, Vol. 125, No. 1, 2004, pp. 38­47. Original Russian Text Copyright © 2004 by Kulik, Maslennikov, Merkulova, Penin, Radchenko, Krasheninnikov.

NUCLEI, PARTICLES, AND THEIR INTERACTION

Two-Photon Interference in the Presence of Absorption
S. P. Kulika,*, G. A. Maslennikova, S. P. Merkulovab, A. N. Penina, L. K. Radchenkoa, and V. N. Krasheninnikovb
Moscow State University, Leninskie gory, Moscow, 119992 Russia e-mail: postmast@qopt.phys.msu.ru b Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow oblast, 141290 Russia
Received July 17, 2003
a

Abstract--Experiments on two-photon interference are discussed in the case when there is absorption of all the modes participating in the process of spontaneous parametric down-conversion (SPDC) of light. The object of investigation are 10­80-å-thick ultrathin gold films deposited on fused-silica substrates. Conditions are determined under which the effect of absorption of the signal and pump waves on the interference pattern is small. It is shown that, under these conditions, the visibility of the interference pattern and the shape of the frequency­angular spectrum at signal frequency are determined by the optical parameters of the medium at idler frequency, which belongs to the near-infrared region. © 2003 MAIK "Nauka / Interperiodica".

1. INTRODUCTION Over two decades, two-photon interference [1­3] has been attracting the attention of physicists. This interest is primarily associated with the interpretation of a number of experiments based on two-photon interference, as well as with the fact that two-photon (nonclassical) states of light are relatively easy to obtain. The most efficient source of such states is the spontaneous parametric down-conversion (SPDC) of light. Recently, various aspects of two-photon interference have been investigated in the context of the physics of quantum information: various fields of quantum information, such as generation of entangled states, quantum cryptography, quantum teleportation, etc. [4], intensively use the experience in the preparation, transformation, and measurement of two-photon light. This experience seems to be useful for developing quantum communication devices that employ nonclassical states of light as information carriers. At the same time, another property of two-photon interference, which may be useful in spectroscopy, has not been paid due attention. The point is that the interference pattern itself bears information about the properties of the medium (or several media) in which the generation and transformation of two-photon light occurs. Hence, one can solve the inverse problem; namely, one can recover the properties of the scattering and/or transforming medium from the interferograms of two-photon interference, as it is done in Raman or polariton spectroscopy. The latter method is the limiting case of the SPDC when the frequency of one of the waves falls within the range of lattice oscillations of a nonlinear crystal [5]. In this sense, the interferometry of spontaneous parametric down-conversion is a generalization of the method of polariton spectroscopy to nonlinear media, where (2) = 0.

The line shape of two-photon SPDC in a separate layer was considered in [6, 7] when the interference phenomena associated with the reflection and absorption of all the waves participating in the process were taken into account. The spectroscopic aspects of twophoton interference were discussed in the literature in the context of nonlinear diffraction [8, 9]. In [10], a method of diagnosis of quasiregular domain structures by the frequency­angular spectra of SPDC was considered. In [11­14], the authors analyzed the capabilities of two-photon interferometry as a method that enables one to evaluate the optical parameters of substances placed in a nonlinear interferometer. In the present paper, we discuss the application of the method of two-photon interference to the study of thin metal films deposited on fused-silica substrates. 2. TWO-PHOTON INTERFERENCE IN THE MACH­ZEHNDER SCHEME 2.1. Mach­Zehnder Nonlinear Interferometer with Several Layers Consider a system of plane layers (Fig. 1). The first and the last layers (m = 1, 2) have a nonzero value of quadratic susceptibility (2) , while, in the intermediate layers, this parameter is equal to zero, and the layers differ only in their permittivity n , where n is the layer number. In the literature, such a system was called the Mach­Zehnder nonlinear interferometer (MZNI) [15]. This term reflects the fact that a laser beam propagating across the layers induces nonlinear polarization in the first and the last layers due to ; these layers are analogous to beam-splitters that divide/mix spatial­frequency modes. The optical fields with new frequencies generated by the nonlinear process have different phase delays while propagating across intermediate layers

1063-7761/04/9801-0031$26.00 © 2004 MAIK "Nauka / Interperiodica"


32 x
1 s i (2) (2) i 1
s

KULIK et al.
Recording system Detector

2 3

p

s

i

l

l1 l2 l ''

z

of the signal and idler photons: k s = ­ k i or ks sin s = - ki sin i, where s and i are the angles between the z axis and the scattering directions of the signal and idler photons in the crystals, respectively. In the low-absorption approximation, which is valid in the transparency regions of the crystals, the dispersion relation (2) contains the real parts of the wave vector ki 2nj /j (j = s, p, i) and the dielectric permittivity. Formally, the intensity of the SPDC as a function of frequency and the scattering angle (the line shape) is proportional to the squared modulus of the sum of amplitudes of biphoton fields emitted from different macroscopic regions [18]. In our case, there are two such regions; these are nonlinear crystals in the MZNI:
x x

Fig. 1. Scheme of the Mach­Zehnder nonlinear interferometer with two nonlinear crystals.

I s ( s, s ) and, hence, may interfere at the output of the system if they are coherent.1 The phase delays of the field components are determined by the dispersion of dielectric permittivity l(). The oscillating behavior of the intensity of the generated fields as a function of a certain parameter of the system, for example, the optical thickness of intermediate (linear) layers, has been a subject of study by the method of nonlinear interferometry [12, 16]. In our case, a biphoton field is generated in the first and the second nonlinear crystals during the SPDC [17]. Recall that, during the SPDC, a photon (p) of the laser pump spontaneously decays into a pair of photons, the so-called signal (s) and idler (i) photons. In the stationary case, the photon frequencies are related by the energy conservation law: p = s + i , (1)


m=1

2

2

f

m

=f

2 1

+f

2 2

+ 2Re ( f 1 f * ) . (4) 2

while the propagation direction of the generated wave is determined by the dispersion law of the medium, k = --- ( ) c through the phase-matching condition kp + ks + ki + D . (3) (2)

The amplitudes fm determine the wave function of the biphoton filed. They depend on the intensity of the laser pump field, the quadratic susceptibility of a crystal, the frequencies of interacting fields, etc. The third term in (4) describes the periodic modulation of the scattering intensity as a function of the relative phase of the two amplitudes. In stationary experimental conditions, two-photon interference, or interference of biphotons in the second order in the field, manifests itself as alternating maxima and minima in the intensity of the frequency­angular spectra of the SPDC; the relative phase in (4) depends on the variation of the direction and / or frequency of observation [11]. If the crystals are transparent at all three frequencies p, s, and i , then the intensity of the observable (signal) wave as a function of frequency and scattering angle is given by I s ( s, s ) 1 sin ( l /2 ) ----------------------- cos -- L + l /2 2


q=1

p

'q l 'q ,

2

(5)

Here, kj (j = p, s, i) are the wave vectors and D is the wave mismatch associated with the dimensions of nonlinear crystals. Since the layers are assumed to be infinite in the transverse direction, we have 0, where = k p ­ k s ­ k i . When the pump wave propax x x

gates along the z axis, we have kp = k p . As a result, we obtain a strict relation between the scattering directions
1

z

The coherence of the components of interfering fields is guaranteed by common laser pumping, which is assumed to be classical and fixed.

where ' = k 'p ­ k 's ­ k 'i ­ z is the z component of the wave mismatch in the intermediate media. The first coefficient in (5) describes the frequency­angular line shape of the spontaneous parametric down-conversion in a plane nonlinear layer [17]. The second coefficient is due to the interference between the signal fields generated in the extreme layers; this coefficient is responsible for the modulation of the line shape due to the contributions of the phase mismatches ' = 'l ' in the intermediate materials. The explicit form of the wave mismatches and ' as functions of the parameters observed in the experiment (the wavelength s and the angle s) is given in [12].
Vol. 98 No. 1 2004

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


TWO-PHOTON INTERFERENCE IN THE PRESENCE OF ABSORPTION

33

Formula (5) is a particular case of the expression obtained in [19] for the MZNI containing several transparent linear media. This formula is obtained when one takes into account the contribution of the dispersion of all intermediate layers (without taking into account reflections) to the propagators of the signal and idler modes, as well as of the pump mode. Note that, in the real interferometer schemes used in the experiments in [12­14], the observed line shape of down-conversion is given precisely by (5) because there always exist air gaps between nonlinear crystals and the dispersive substance, so that the whole scheme actually consists of five, rather than three, layers. 2.2. Taking into Account Losses at Idler Frequency From the viewpoint of interpretation of experimental results, the following case is of interest. In the scheme shown in Fig. 1, one detects a signal wave, while the idler (nonobservable) modes experience losses due to, for example, absorption or reflection. This case was first considered by Mandel and colleagues in [20]. In the scheme suggested, they succeeded in spatially separating the signal and idler modes generated in different nonlinear crystals. In this configuration, it is convenient to control the transmission in the idler mode by inserting filters with different optical densities. The effect observed in this case was called induced coherence because the visibility of the interference pattern observed in a signal mode depends on the transmission coefficient of the filter inserted into the idler mode. In [11­14], it was pointed out that the induced coherence can find application in spectroscopy when the idler modes fall within the infrared region of the spectrum, while the signal mode is detected in the visible region. The physical scheme considered in the present paper (Fig. 1) does not essentially differ from the scheme proposed in [20]. However, from the experimental point of view, the MZNI scheme considered here is more convenient because the optical-path difference between the signal and idler modes is maintained constant automatically [11]. In this scheme, the line shape of the signal wave for a finite amplitude of the transmission coefficient i of a certain intermediate layer at idler frequencies is determined by the following expression, which takes losses into account: 1 l /2 I s ( s, s ) -- sin ---------- l /2 2 â 1 + i cos l +
2

field generated in the first crystal has a characteristic spectral luminosity of the order of 10­8 1 photons per mode and does not influence the process of SPDC in the second crystal (a spontaneous regime); this yields the relation ^ ^ ^ a i2 ( i ) = i a i1 ( i ) + r i a vac ( i ) .
eff

(7)

The second term in (7) is attributed to the unitarity of ^ the transformation: the operator a vac describes the vacuum field that is admixed to the idler mode with the eff weight r i for |i |2 < 1; in this case,
eff 2 i

+r

eff 2 i

= 1.

The coefficient r i describes losses due to reflection and absorption. According to the scheme shown in Fig. 1, for identical nonlinear crystals (1 4 and 1 4), the idler modes ki1 and ki2 are degenerate; therefore, formally, formula (7) allows one to take losses into account. A detailed analysis of (6) in the multimode case was carried out in [21]. Formula (6) implies that the visibility of interference pattern, which is defined in a standard way [22] by I max ­ I min V = ----------------------- , I max + I min falls to zero as For instance, if (8)

0; this fact was pointed out in [20]. exp { ­ i l' } ,
2

where i is the Bouguer absorption coefficient at idler frequency, attenuation of the idler mode due to absorption in the intermediate medium deteriorates the visibility of the interference pattern. Formula (6) displays an essential property of two-photon interference: losses in the idler mode do not change the integral intensity of the SPDC but only affect its shape. This property underlies the two-photon interferometric method for estimating the absorption coefficient of nonlinear crystals in the near-infrared region [14, 23]. 2.3. Taking into Account Losses at Pump and Signal Frequencies If the linear layers between two nonlinear crystals introduce losses in the p and s modes, then, on the one hand, the pump amplitude in the second crystal decreases, E
Vol. 98 No. 1
(2) p


q=1

p

' q l 'q .

(6)

The transmission coefficient i relates the annihilation operators of photons in idler modes after the first and second nonlinear crystals [21]. Note that the idler

= p Ep ,

(1)

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS

2004


34 0.8 0.7 Transmission 0.6 0.5 0.4 0.3 300

KULIK et al. 1 2 3 5

mainly attributed to the losses in the idler (nonobservable) mode. This fact serves as a basis for the analysis of experimental data obtained in the present work. Under condition (10), formula (6) gives a relation between the visibility of two-photon interference, which can be measured experimentally, and the losses at idler frequency: = i . (11) If we take into account that ||2 exp{­ i l '3 }, then the absorption coefficient at idler frequency proves to be logarithmically related to V: 2 ln V i = ­ ------------ . l '3 (12)

8 nm 400 500 600 700 Wavelength, nm 800 900

Fig. 2. Transmission coefficient of samples as a function of wavelength.

therefore, the amplitude of the signal wave generated in the second crystal is given by f 2 E
(2) p

= p Ep .

(1)

On the other hand, due to the losses of the signal wave generated in the first crystal while passing through intermediate layers, this amplitude decreases at the input of the second crystal: f where f
losses 1 right

f

1 left

= s f 1 s Ep , 3. EXPERIMENT 3.1. Description of Samples We used 0.2-mm-thick polished fused-silica plates with an area of 15 â 15 mm2 as the substrates. A gold film was deposited on these substrates by the cathode sputtering method. We investigated films with integral thickness of 10, 20, 30, 50, and 80 å in the working area. Due to the small thickness, the films did not continuously cover the substrate; they were characterized by a cluster structure when the thickness was less than 30 å and by a porous structure for greater thickness. The transmission coefficient of the samples was measured by an M400 spectrophotometer as a function of the wavelength. The results of these measurements are shown in Fig. 2. 3.2. Experimental Setup and Measurement Technique In the experiment, we measured two-dimensional frequency­angular spectra of spontaneous parametric down-conversion of light emitted from the MZNI. As the nonlinear media, we used lithium niobate crystals doped with magnesium oxide, LiNbO3 : MgO (5%). Between these nonlinear media, we placed gold films of various thicknesses deposited on fused-silica substrates. The thickness of crystals was 440 µm, and the
Vol. 98 No. 1 2004

(1)

Thus, measuring the visibility of two-photon interference, one can directly evaluate the absorption coefficient of a substance (in the infrared region) placed between nonlinear crystals. Note that the losses in the s and p modes do not lead to a decrease in the integral intensity of the signal of parametric down-conversion. Therefore, it would be interesting to verify experimentally which of the two factors proves to be dominant as losses increase in all the modes s, p, and i: the disappearance of interference under a still appreciable total intensity of the signal of parametric down-conversion or the total disappearance of the signal.

1 right

is the amplitude on the right boundary of

the first crystal and f 1 left is the amplitude on the left boundary of the second crystal. Thus, amplitudes that differ not only in phase but also in absolute value contribute to the interference of biphoton fields. This may deteriorate the visibility of two-photon interference. The difference in amplitude is the more conspicuous, the greater the frequency dispersion of the transmission coefficient s, p(), or the greater the difference of the parameter s/p from unity. Using (4), we can estimate the visibility of interference pattern in two-photon interference for different amplitudes: 2 s / p V -------------------------- . 2 ( s / p ) + 1 (9)

It follows from (9) that, even for s/p = 2, the visibility of interference is still sufficiently high: V 80%. Thus, we can assert that losses in the pump modes and in the signal (observable) mode do not affect the visibility of two-photon interference while the parameter s/p is not too different from unity. For instance, V 90% if 0.6 s / p 1.6 . (10) Hence, under condition (10), the observable deterioration of the visibility of two-photon interference is

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


TWO-PHOTON INTERFERENCE IN THE PRESENCE OF ABSORPTION

35 i, cm­1

thickness of the fused-silica substrates was 200 µm. An argon laser operating at the 488-nm line with an output power of 1 W and a beam diameter of 2 mm2 served as the pump source; the scattered field was collimated by an objective lens and focused onto the input slit of an ISP-51 spectrograph. To obtain panoramic spectra, we used a photographic technique that is a standard one for the SPDC spectroscopy [24]. Quantitative information about the line shape of down-conversion was obtained by two methods. In the first method, the angular distribution of intensity at several signal wavelengths was recovered after taking into account the nonlinear dependence of the blackening of photographic film as a function of light intensity. In the second method, the line shape was recorded directly by the angular scanning of twodimensional spectra at a fixed wavelength. A Hamamatsu R5600U photomultiplier tube was placed in the focal plane of the spectrographic camera and could move along two coordinates (the angle s and the wavelength s). The output pulses of the photomultiplier tube were amplified, subjected to amplitude discrimination, and fed to a counter. The entire electronic part of the receiving system was assembled in the CAMAC standard. Typical photographs of the spectra and the corresponding angular intensity distributions that were obtained during scanning are shown in Figs. 3 and 4. 4. DISCUSSION OF THE RESULTS The diagrams shown in Fig. 2 allow one to estimate the parameter s/p introduced in the preceding section. For the signal wavelengths 5685, 5707, and 5731 å (the corresponding idler frequencies, defined by (1), are equal to 2902, 2970, and 3043 cm­1, respectively) at which the spectra were processed, the values of the parameter s/p are given in the table. One can see that this parameter satisfies condition (10). Thus, at these wavelengths, the visibility of two-photon interference is mainly determined by losses at idler frequencies in the infrared region. The visibility of two-photon interference as a function of losses was experimentally investigated when gold films of various thicknesses were placed into the MZNI. Figures 3 and 4 show that interference phenomena in two-photon light virtually disappear for a film thickness of 50 å. A test photograph of a 80-å-thick sample shows that there is no angular­frequency modulation. At the same time, we certainly observed an SPDC signal even in films of thickness 100 å. Hence, we can conclude that the method of two-photon interference is more sensitive to the losses i at idler frequencies than to the losses p and s at pump and signal frequencies, which result in a decrease in the integral intensity of the spectra. The transmission coefficient at a frequency of 2970 cm­1 determined by formulas (6) and (11a) from

(a) s, deg 5 4 3 2 1 565.3 s, deg 5 4 3 2 1 565.3 s, deg 5 4 3 2 1 565.3 581.7 595.7 s, nm 581.7 595.7 s, nm 581.7 595.7 s, nm 3309 3713

(b) 3309 3713 i, cm­1

(d) 3309 3713 i, cm­1

Fig. 3. Photographs of angular-frequency spectra of the SPDC obtained in the system nonlinear crystal­fused-silica substrate­gold film­nonlinear crystal for films of various thicknesses; (a) in the absence of a film, (b) for a film thickness of 10 å, and (c) for a film thickness of 50 å.

the angular scans of interferograms is shown in Fig. 5 as a function of the film thickness. We did not observe an appreciable difference in the behavior of these functions in the range of idler frequencies from 2900 to 3040 cm­1. This fact suggests that the absorption coefficient of the films shows weak dispersion in this spectral domain. A similar conclusion can be drawn from a visual analysis of the frequency­angular spectra (Fig. 3) that were obtained in a wider range (2400­ 3700 cm­1): in these spectrograms, one cannot distinguish regions where the visibility is appreciably varied. According to (11), the visibility of the interference pattern depends on the transmission coefficient at idler frequency. In this range, the optical properties of metals are primarily determined by free electrons. A relation
Vol. 98 No. 1 2004

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


36 0.0735 0.0730 Intencity, arb. units. 0.0725 0.0720 0.0715 0.0710 0.0705 0.0700 4.1 280 Number of detector counts in 100 ms 4.2 4.3 4.4 s, deg 4.5

KULIK et al. 320 (a) Number of detector counts in 100 ms (b) 300

280

260 3.6 240 3.8 4.0 4.2 s, deg 4.4 4.6

4.6

4.7

(c) 260

Number of detector counts in 100 ms

(d)

220

240

200

220 3.6 3.8 4.0 4.2 s, deg 4.4 4.6

180

3.4

3.6

3.8

4.0 4.2 s, deg

4.4

4.6

Fig. 4. Angular distribution of the SPDC intensity for films of various thicknesses; (a) in the absence of a film, (b) for a film thickness of 10 å, (c) for a film thickness of 30 å, and (d) for a film thickness of 50 å. Distribution (a) is obtained by recovering the intensity from the blackening level of a film by Photoshop 6.0 software for s = 5684 å. Distributions (b)­(d) are obtained by photoelectric recording of signals for s = 5707 å.

between the dielectric permittivity and the basic optical constants is given by the following formulas [22]: Re = n ( 1 ­ ) ,
2 2

i) and is the extinction coefficient. From (12) and (13), we can derive the absorption coefficient 2 4 i = -------i n = ----- , c 0i (15)

(13) (14)

4 2 Im = --------- = 2 n ,

^ where n is the real part of the refractive index n = n(1 +
Table No. 1 2 3 4 5 Sample thickness, å 10 20 30 50 80 |s| 5685 å 0.79 0.75 0.71 0.69 0.58
2

where 0i is the idler wavelength in vacuum. Approximating the experimental curve (Fig. 5) by an exponential function (solid curve), we obtain i 2.8 â 106 cm­1 at a

|p|2 5731 å 0.79 0.74 0.70 0.69 0.58 4880 å 0.84 0.80 0.74 0.64 0.5 5685 å 0.974 0.968 0.98 1.038 1.077

s/

p

5707 å 0.79 0.74 0.71 0.69 0.58

5707 å 0.974 0.962 0.98 1.038 1.077
Vol. 98 No. 1

5731 å 0.974 0.962 0.973 1.038 1.077
2004

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


TWO-PHOTON INTERFERENCE IN THE PRESENCE OF ABSORPTION

37

frequency of i 3000 cm­1 (or 0i 3.3 µm). Unfortunately, we had no information about the optical constants of the films in the infrared region. Therefore, we used the data given in [25]. Substituting the values of the refractive index n 0.8 and the extinction coefficient 20 into (14), we obtain i 6.1 â 105 cm­1, which is about five times less than the value obtained in our experiments. The difference between the experimental and calculated values of the absorption coefficient, which is substantially greater than the measurement error, can be attributed to several factors. First, the optical constants of gold presented in [25] refer to thick films. It is well known that the optical constants in ultrathin films, of thickness less than 10 nm, are significantly affected by the film structure: an ultrathin film is considered as a set of separate islands. For instance, it was pointed out in [26] that the cluster structure leads to a significant increase in the absorption. A giant growth in the absorption of infrared radiation in metal particles was pointed out, for example, in [27]. The optical properties of fractal clusters, in particular, an anomalous behavior of susceptibility, was considered in [28]. Recently, the optical properties of metal island films near the percolation threshold have been intensively discussed in the literature [29]. Second, the measurement technique for the absorption coefficient, based on two-photon interference, may give results that essentially differ from those obtained earlier. The point is that, in conventional methods for investigating metal films, one measures the reflection and transmission coefficients of "free" waves, i.e., waves with nonzero mean occupation of modes incident to a sample from free space. The dispersion relations for the transmission coefficient of films in the visible region (Fig. 2) have been obtained precisely in this way. However, in the method of two-photon interference, the optical parameters at a nonobservable (idler) frequency depend on fluctuating vacuum fields with zero mean occupation number of modes. There are examples in the literature where a comparison of these two methods shows a significant discrepancy precisely when measuring the absorption coefficient: the absorption measured by the SPDC spectroscopy (field fluctuations) [30] proves to be about an order of magnitude greater than that obtained by four-wave coherent scattering by polaritons (excitation of polaritons by biharmonic pumping) [31] or by direct measurement of infrared transmission [32]. The physical nature of this discrepancy has not yet been revealed, and one may suggest that experiments on two-photon interference in the Mach­Zehnder scheme will give an answer to this question.2
2

1.0 0.8 Transmission 0.6 0.4 0.2

0

2

4 6 Film thickness, nm

8

Fig. 5. Amplitude transmission coefficient at a frequency of 2970 cm­1 as a function of film thickness, obtained from angular scans of interferograms. The solid line represents an exponential approximation of the experimental data.

Note that the approach considered in the present paper does not allow us to make any conclusions about the dispersion of the real part of the refractive index of films. This is associated with the small thickness of the films. As was shown in [11, 13], the distinct features of the dispersion of the refractive index that arise near resonance frequencies must be accompanied by the variation of the curvature of interference orders or with the appearance of crooks in the spectra of two-photon interference. This is associated with the fact that, by definition, the wave mismatch ' = ' l '3 of the film involves the real parts of the wave vectors ^ ^ Re ( k ) = --- Re ( n ) = --- n . c c The strong dispersion of n leads to an increase in the wave mismatch ', which leads to a variation in the phase of the interference pattern versus frequency / angle. The value of n of gold ranges from 0.2 to 1.2 as the wavelength varies from 0.3 to 1 µm. In the near infrared region, when i 3 µm, the value of n proves to be of the order of unity and weakly increases with the wavelength [24, 34, 35]. However, since the film thickness is about tens of angstroms and l '3 l, l '2 , the corresponding mismatch '3 is small, and its contribution to the line shape (6) is negligible compared with the contributions of the mismatches in the nonlinear crystals, = l, and in the quartz substrate, '2 = ' l '2 . 5. CONCLUSION We have discussed experiments on two-photon interference in the presence of absorption in all the
Vol. 98 No. 1 2004

Note that recent publications (see, for example, [30]) on the spectroscopic applications of the so-called frequency-entangled photon pairs do not answer the question posed since these works deal with the contribution of the direct transmission of a real idler wave to the distribution of photocount coincidences.

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS


38

KULIK et al. 12. D. Yu. Korystov, S. P. Kulik, and A. N. Penin, Kvantovaya èlektron. (Moscow) 30, 921 (2000). 13. D. Yu. Korystov, S. P. Kulik, and A. N. Penin, Pis'ma Zh. èksp. Teor. Fiz. 73, 248 (2001) [JETP Lett. 73, 214 (2001)]. 14. A. V. Burlakov, M. V. Chekhova, O. A. Karabutova, et al., Laser Phys. 12, 825 (2001). 15. A. V. Burlakov, M. V. Chekhova, D. N. Klyshko, et al., Phys. Rev. A 56, 3214 (1997). 16. R. Stolle, G. Marovsky, E. Schwartzberg, and G. Berkovie, Appl. Phys. B 63, 491 (1996). 17. D. N. Klyshko, Photons and Nonlinear Optics (Nauka, Moscow, 1980), p. 256. 18. A. V. Belinsky and D. N. Klyshko, Laser Phys. 4, 663 (1994). 19. D. N. Klyshko, Zh. èksp. Teor. Fiz. 104, 2676 (1993) [JETP 77, 222 (1993)]. 20. L. J. Wang, X. Y. Zou, and L. Mandel, Phys. Rev. Lett. 67, 318 (1991). 21. L. J. Wang, X. Y. Zou, and L. Mandel, Phys. Rev. A 44, 4614 (1991). 22. M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon Press, Oxford, 1969; Nauka, Moscow, 1970). 23. A. V. Burlakov, Yu. B. Mamaeva, A. N. Penin, and M. V. Chekhova, Zh. èksp. Teor. Fiz. 120, 67 (2001) [JETP 93, 55 (2001)]. 24. D. N. Klyshko, A. N. Penin, and B. F. Polkovnikov, Pis'ma Zh. èksp. Teor. Fiz. 11, 11 (1970) [JETP Lett. 11, 5 (1970)]. 25. Physikalisch­Chemische Tabellen (Springer, Berlin, 1929), Vol. 165B. 26. R. P. Devaty and A. J. Sievers, Phys. Rev. Lett. 52, 1344 (1984). 27. A. V. Plyukin, A. K. Sarychev, and A. M. Dykne, Phys. Rev. B 59, 1685 (1999). 28. V. M. Shalaev and M. I. Shtokman, Zh. èksp. Teor. Fiz. 92, 509 (1987) [Sov. Phys. JETP 65, 287 (1987)]. 29. A. K. Sarychev, V. A. Shubin, and V. M. Shalaev, Phys. Rev. E 59, 7239 (1999). 30. A. V. Burlakov, M. V. Chek