Double-slit interference of biphotons generated in spontaneous parametric downconversion from a thick crystal

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS J. Opt. B: Quantum Semiclass. Opt. 3 (2001 S50 S54 PII: S ( Double-slit intererence o biphotons generated in spontaneous parametric downconversion rom a thick crystal A F Abouraddy,BEASaleh, A V Sergienko andmcteich Quantum Imaging Laboratory, Departments o Electrical & Computer Engineering and Physics, Boston University, 8 Saint Mary s Street, Boston, MA 02215, USA raddy@bu.edu Received 4 July 2000, in inal orm 10 January 2001 Abstract In this paper we investigate the ourth-order coherence o biphoton beams rom spontaneous parametric downconversion resulting rom a nonlinear crystal as exempliied in a double-slit intererence coniguration where the signal and idler beams both pass through the same double slits. We ind that the angle o the crystal optical axis and the crystal length are important actors, along with the double-slit separation and system bandwidth, in determining the nature o the ourth-order intererence pattern obtained behind the slits. Only careul planning o system parameters and understanding o the tuning curves o the downconversion will illuminate the obtained coincidence patterns. Keywords: Quantum intererence, spontaneous parametric downconversion, ourth-order coherence, quantum entanglement, double-slit intererence 1. Introduction Young s double-slit intererence experiment is synonymous with the determination o the coherence properties o light. It has been used, classically, to determine the secondorder coherence unction by measuring the visibility o the ringes ormed behind the double slit as the slit separation changes [1]. The development o nonclassical sources o light has helped envisage new double-slit intererence experiments where both second- and ourth-order coherence properties are measured [2 5]. One such source is the light generated through the process o spontaneous parametric downconversion (SPDC. In this process a pump beam generates two highly correlated beams, called signal and idler beams, through a nonlinear interaction. Such twin beams are usually called entangled beams, and they have unusual and sometimes unintuitive properties compared to classical light sources. For example, the ourth-order coherence unction (two-photon coincidence pattern has an analogous orm to that o the second-order coherence unction o a partially coherent classical source [6]. Such behaviour can be investigated in double-slit conigurations. In one coniguration the signal beam is sent through the double slit and then detected by a single-photon detector, whilst the idler beam is detected by another single-photon detector, and the coincidence o detection at both detectors are registered while one o the detectors is scanned in the direction perpendicular to the expected classical secondorder ringes [2]. Such an experiment yields coincidence ringes analogous to the amiliar intensity ringes. In another coniguration both signal and idler beams are sent through the double slit [3 5]. This is the case to be studied in this paper and is the case most resembling that o the classical case in experimental set-up. We will show that such a coniguration yields a coincidence pattern that is highly sensitive to the source (crystal characteristics, giving more degrees o reedom compared to the classical case. 2. Theory In a recent paper [6] we studied in detail the duality between partial coherence and partial entanglement in the context o biphoton beams produced by SPDC. In that paper we developed a set o equations that can be used to calculate the ourth-order coherence unction, which is a measure o coincidence o the photon pairs, comprising a biphoton, ater they traverse an optical system. In this paper we briely review the equations developed there and apply them to the case o double-slit intererence /01/ $ IOP Publishing Ltd Printed in the UK S50

2 Double-slit intererence o biphotons generated in SPDC Assume an optical pump beam (usually a collimated laser beam, hereinater reerred to as the pump and denoted by p impinges on a nonlinear crystal (NLC, normal to its surace. I the crystal has an appreciable second-order nonlinearity, the nonlinear interaction will lead to the pump photons spontaneously disintegrating into two photons known (or historical reasons as the signal (s and idler (i. The signal and idler photons are highly correlated in requency and direction since their generation ollows the rules o conservation o energy and momentum, ω p = ω s + ω i and kp = ks + ki, respectively, also known as the phase matching conditions, where ω and k represent angular requency and momentum. We next assume that the signal and idler beams traverse optical systems having classical optical impulse response unctions h s (x 1,x; ω s and h i (x 2,x; ω i, where the subscripts s and i again reer to signal and idler respectively, x 1 and x 2 are two points on the output observation plane, and x is a point on the input ace o the crystal used to describe the impinging pump electric-ield distribution, E p (x. We have shown that or a monochromatic pump, NLC o thickness l, and spectral ilters o bandwidth, the coincidence rate at the output plane is given by [6, equation (4.14] C(x 1 = ψ(x 1 ; ω s 2 dω s, (1 where ψ(x 1 ; ω s is a spectral probability amplitude given by ψ(x 1 ; ω s = 1 (q 4π 2 s,q i ; ω s H s (x 1,q s ; ω s H i (x 2,q i ; ω p ω s dq s dq i. (2 The quantity H s (x 1,q s ; ω s is the Fourier transorm o h s (x 1,x; ω s with respect to the parameter x, and H i (x 2,q i ; ω is deined similarly; q s and q i are the transverse components o the signal and idler wavevectors, respectively, and the kernel (q s,q i ; ω s is related to the pump ield distribution and the degree o phase-matching through the relation [6, equation (5.9] (q s,q i ; ω s = Ẽ p (q s + q i ζ(q s,q i ; ω s. (3 Here Ẽ p (q is the Fourier transorm o the pump spatial distribution E p (x and ζ(q s,q i ; ω s is a unction that describes the degree o phase-matching between the signal and idler beams [6, equation (5.10]: ζ(q s,q i ; ω s = lsinc l ωp 2n2 (ω p (q 2π c 2 s + q i 2 ω 2 s n 2 (ω s q c 2 2 s (ω p ω s 2 n 2 (ω p ω s q 2 c 2 i. (4 Through ζ(q s,q i ; ω s the eect o the length o the crystal l and its physical characteristics: its indices o reraction at the pump, signal, and idler requencies and polarizations, along with the angle o the optical axis o the anisotropic crystal with respect to the normal to the crystal surace (i.e. the optical axis o the external optical system is maniested. The sinc unction gives us the probability amplitude o each pair o signal/idler requencies having a certain pair o signal/idler directions. The unction ζ(q s,q i ; ω s assumes two limiting orms that are important in the sequel. The thin-crystal limit is obtained when l is very small whereas the thick-crystal limit results when l is very large. In the thin-crystal limit the sinc unction o ζ(q s,q i ; ω s becomes almost constant over a wide range o requencies and directions. The kernel (q s,q i ; ω s in (2 then becomes simply Ẽ p (q s + q i and the physical characteristics o the crystal have little eect. In the thick-crystal limit the width o the sinc unction becomes very small and tends towards a delta unction. This means that each requency is spread over a very narrow range o directions. 3. Tuning curves As mentioned in the previous section, the unction ζ(q s,q i ; ω s determines the nature o the relation between the requencies o a signal/idler biphoton pair and the directions at which the signal and idler modes will be emitted [7]. For each signal mode requency, there is a set o directions at which we expect it to be emitted with high probability (where the sinc unction peaks. There will be low probability or this signal mode requency to be emitted in all other directions (the tails o sinc unction. We calculated the tuning curves or three thicknesses (0.1, 1 and 10 mm o a Beta-Barium Borate (BBO crystal pumped at a wavelength o 325 nm (which corresponds to an ultraviolet line o a He Cd laser or type-i downconversion (signal and idler both have ordinary polarization with respect to the crystal optical axis, whilst the pump has extraordinary polarization. The calculations were perormed or three orientations o the crystal with respect to the normal direction to the crystal surace (denoted by ϕ: 36.30,36.50 and Figure 1 shows the resulting tuning curves o this BBO crystal with the above mentioned parameters. In all igures, the horizontal axes reer to the emitted requencies normalized to the pump requency, and the vertical axes reer to the angle at which the photons are emitted in radians, with respect to the normal to the surace o the crystal. The irst column shows the results or ϕ = as we proceed rom a 0.1 to a 10 mm thick crystal. We clearly observe the decreasing level o uncertainty o the angles at which each requency is being emitted rom the crystal. The same observation is clear in the second and third columns. Another eature is the drastic change in the shape o the tuning curves with change in the crystal angle ϕ. For example, the irst column shows that there is no emission at the degenerate requency (ω 0 = ω p /2. The collinear downconversion (i.e. the emission angle is 0 occurs with a non-degenerate pair. In the second column we have a noncollinear degenerate case. In the third column we have a collinear degenerate case. We shall show how this change in the shape o the tuning curves leads to dierent results in a double-slit intererence coniguration. 4. Double-slit coniguration We now turn to the intererometric coniguration that is the principal purpose o this paper. Let us consider the signal S51

3 A F Abouraddy et al Emission angle (in radians Normalized Frequency η Figure 1. Tuning curves or type-i downconversion rom a BBO crystal. Pump wavelength is 325 nm. Proceeding rom top to bottom we change the crystal length rom 0.1 to 1 mm and then 10 mm. In all igures, the horizontal axes reer to the emitted requencies normalized to the pump requency, and the vertical axes reer to the angle at which the photons are emitted in radians, with respect to the normal to the surace o the crystal. The irst column corresponds to ϕ = 36.30, the second to ϕ = 36.44, and the third to ϕ = and idler photons traversing a 4 system (which serves as a generic linear shit-invariant system as shown in igure 2. The aperture t(xin the Fourier plane is, in our case, a double slit, given by ( ( x a/2 x + a/2 t(x = rect b + rect b, (5 where the unction rect(x is a uniorm unction o width unity, a is the distance between the centres o the slits, and b is the width o each slit. We shall deine p = b/a as the relative width o the slit with respect to the separation o the centres. In this case, the transer unctions o both the signal and idler are ( c H s (x, q; ω = H i (x, q; ω e ixq t s ω q, (6 where t s (x = t i (x = t(x. Assuming a pump width large in comparison to all the apertures in the system, Ẽ p (q becomes approximately a delta unction, which leads to a spectral probability amplitude ψ(x 1 ; ω s dq s e qs(x1 x2 ζ ( ( c c (q s, q s ; ω s t s q s t i q s. (7 ω s ω p ω s t(x Figure 2. Schematic o the 4 system used in the calculations. The coincidence rate is obtained by substituting this spectral probability amplitude into (1. The coincidence rate at the output plane is a unction o only the separation o the two points under consideration, a direct result o assuming a large pump width. We now introduce some normalized parameters to cast the previous equations into a simpler orm. We deine a normalized requency η = ω s /ω p and a normalized bandwidth ρ = /ω p so that η will thus assume values o (1 ρ/2 to(1+ρ/2, with a maximum value o ρ = 1. We urther deine θ = a/, the angle subtended by the double slits at the crystal, and a characteristic distance = 2λ p /θ, where λ p is the pump x 1 x 2 S52

4 Double-slit intererence o biphotons generated in SPDC 0.25 x x10-3 (a 0.20 C(x Normalized Separation (x 1 Figure 3. Coincidence rate as a unction o normalized separation or ϕ = 36.44, l = 0.1 mm, θ = 3, p = 0.1, ρ = 5. wavelength. Equation (7 may now be written as ( ψ(x 1 ; η = dq ξ(q, q; η exp i2πq x 1 x 2, A (8 where q = qs, with limits o integration A that depend on η 2π and p as ollows { (1 η(1 p η(1+p& η(1+p (1 η(1 p, or η< 1 2 q = η(1 p (1 η(1+p& (1 η(1+p η(1 p, or η> 1 2. (9 The range o η is (1 p/2 <η<(1+p/2 ip<ρand (1 ρ/2 <η<(1+ρ/2 ip > ρ. The kernel o the integration in (8 is given by ( l ξ(q, q; η = lsinc λ p [ n(ω p η 2 n 2 (ω s (qθ/2 2 ] (1 η 2 n 2 (ω p ω s (qθ/2 2. (10 It is interesting to note the mixture o normalized bandwidth and slit width in determining the period o integration. This is a direct result o the high correlation between directions and requencies in SPDC. Both the size o the physical apertures and the spectral bandwidth o the optical system determine the eective aperture o the system. Using equations (8 (10 we can calculate the coincidence rate at the output plane as a unction o the normalized separation (x 1 x 2 / or the crystal orientations and thicknesses illustrated in the tuning curves o the previous section. The coincidence rate C(x 1 may now be calculated by substitution o (8 into (1. Let us start with the case o ϕ = and l = 0.1 mm. Taking ρ = 5, θ = 3 rad, and p = 0.1, we obtain the result in igure 3. This is a coincidence pattern (ourth-order coherence unction similar to the intensity intererence pattern (second-order coherence unction that would be obtained by a coherent classical source o light. Reerring to the corresponding tuning curve, we notice that these parameters correspond to downconverted light rom the centre o the tuning curve, i.e. where the peak o the sinc unction lies, and is essentially constant. The main actor determining the coincidence pattern would be the optical system, leading to the results shown. We obtain C(x 1 C(x C(x x10-4 x Normalized Separation (x 1 Figure 4. Coincidence rate as a unction o normalized separation or ϕ = 36.50, l = 10 mm, p = 0.1, (a θ = 3, ρ = 5, (b θ = 7, ρ = 5, and (c θ = 7, ρ = 0.1. similar results or the two other orientations o the crystal i all other parameters are held ixed, since or such a thin crystal the tuning curves exhibit enough uncertainty so as to render the three orientations almost the same over a wide range o requencies and directions (centred around the direction o the pump wavevector. As the thickness o the crystal increases, care must be exercised in choosing the parameters o the set-up. We shall take the case o ϕ = For our given choice o bandwidth we must be careul in our choice o double-slit separation. I we choose an angle θ (angle subtended by the slits at the crystal, in this case 3 rad that does intersect with the peak ridge o the sinc unction, we shall obtain a coincidence pattern, shown in igure 4(a, that is similar to that previously obtained in igure 3. I, however, we increase the double-slit separation to 7 rad we shall obtain a coincidence pattern that is centred on a value other than zero (igure 4(b. This is attributed to the occurrence o the intererence between the sidelobes o the sinc unction in the tuning curves. These sidelobes are modulated with a rate which depends on the crystal thickness. The new intererence pattern is centred on a point proportional to the modulation requency and thus the crystal length. The zero- (b (c S53

5 A F Abouraddy et al C(x,x 1 2 C(x,x 1 2 x x (a (b particular SPDC process. I this is not the case, and the double slit intersects the sidelobes o the tuning curve, a coincidence pattern is obtained but is shited rom the centre o the previous pattern and exhibits a dierent ringe width. Changing the crystal thickness, system bandwidth, or the direction o the crystal optical axis permits us to go rom one case to the other with a ixed double-slit separation. Such a system thereore exhibits more degrees o reedom than its classical counterpart. These results can be explained in another ashion. The double slit can be viewed as sampling the ourth-order coherence unction o the SPDC process in the NLC. This coherence unction has a striking property: directions are strongly correlated to requencies in accordance with the tuning curves o the parametric process. There is no classical counterpart to this behaviour. It is clear that the double-slit intererence experiment continues to yield valuable insights into the properties o light. Acknowledgment Norm alized Separation (x 1 Figure 5. Coincidence rate as a unction o normalized separation or ϕ = 36.30, l = 10 mm, θ = 5, (a ρ = 0.2, and (b ρ = 2. centred intererence pattern can be restored by increasing the normalized bandwidth o the system to 0.1, i.e. by accepting a wider range o requencies at the detectors (igure 4(c. In this case the double-slits will intersect the sinc unction at its peak and the eect o the sidelobes will diminish. We next examine the case o ϕ = In this case or a wide normalized bandwidth o 0.2 and an angle θ o 5 we obtain the coincidence pattern in igure 5(a. Decreasing the bandwidth while keeping the same double-slit separation will increase the contribution o the sidelobes to the intererence as seen in igure 5(b when the normalized bandwidth is decreased to Conclusions We have studied the coincidence pattern ormed behind a double slit when both signal and idler beams resulting rom SPDC are transmitted through it. The coincidence pattern is sensitive to several parameters. The obtained coincidence pattern is analogous to the second-order intererence pattern o a classical partially coherent source as long as the slits intersect the directions o the peak o the tuning curve in the This work was supported by the U.S. National Science Foundation. Reerences [1] Saleh BEAandTeich M C 1991 Fundamentals o Photonics (New York: Wiley chapter 10 Mandel L and Wol E 1995 Optical Coherence and Quantum Optics (Cambridge: New York Born M and Wol E 1999 Principles o Optics (Cambridge: Cambridge University Press [2] RibeiroPHS,Pádua S, Machado da Silva J C and Barbosa G A 1994 Controlling the degree o visibility o Young s ringes with photon coincidence measurements Phys. Rev. A Řeháček J and Peřina J 1996 Two-slit experiment with downconverted beams Opt. Commun Strekalov D V, Sergienko A V, Klyshko D N and Shih Y H 1995 Observation o two-photon ghost intererence and diraction Phys. Rev. Lett [3] Fonseca EJS,Monken C H and Pádua S 1999 Measurement o the de Broglie wavelength o a multiphoton wave packet Phys. Rev. Lett [4] Hong C K and Noh T G 1998 Two-photon double-slit intererence experiment J. Opt. Soc. Am. B [5] Abouraddy A F, Nasr M B, Saleh BEA,Sergienko AVand Teich M C Demonstration o the complementarity o entanglement and coherence in the spatial domain at press [6] Saleh BEA,Abouraddy A F, Sergienko A V and Teich M C 2000 Duality between partial coherence and partial entanglement Phys. Rev. A [7] Malygin A A, Penin A N and Sergienko A V 1981 Eicient generator o a two-photon ield o visible radiation Sov. J. Quantum Electron S54