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Geometrical optics in Fresnel quantum holography
Optik 124 (2013) 4005–4011
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Optik journal homepage: www.elsevier.de/ijleo
Geometrical optics in Fresnel quantum holography Lu Gao ∗ , Shi-jian Wang, Zhang-xiang Cheng School of Science, China University of Geosciences, Beijing 100083, China
a r t i c l e
i n f o
Article history: Received 26 August 2012 Accepted 1 December 2012
PACS: 42.40.Ht 42.30.Wb 42.50.Ar
a b s t r a c t The geometrical optics in Fresnel quantum holography for quantum entangled two-photon source and classical thermal light source based on the second-order correlation measurement has been discussed. We theoretically prove that the Fresnel quantum hologram of the detected object can be obtained through second-order correlation measurement and the optical reconstruction process can be accomplished by making use of a point light source. © 2013 Elsevier GmbH. All rights reserved.
Keywords: Entangled two-photon source Thermal light source Quantum holography
1. Introduction The surprising quantum phenomenon of two-photon entangled state generated by spontaneous parametric down-conversion (SPDC), such as “ghost” imaging, “ghost” interference, sub-wavelength interference and so on have drawn much attention [1–5]. Recent studies have shownthat a classical thermal light source can play a role similar to that of a two-photon entangled source in “ghost” imaging, “ghost” interference and sub-wavelength interference [6–20]. The classical spatially incoherent thermal light source can be produced by a laser beam disorderly scattered by a material, such as a rotating ground glass [19]. Thus the optical field in the transverse plane across the beam is a random variable of transverse position. According to the statistical optics, the source possesses a spatial intensity correlation between transverse positions [18]. Interestingly, the spatial correlation of thermal light has the analogous form to that of entangled photon pair generated in the SPDC process. This is the physical interpretation of that the classical correlation can mimic quantum entanglement in “ghost” imaging, “ghost” interference and so on. In Ref. [15] quantum holography with entangled-photon pairs was proposed methodologically based on the second-order correlation of the quantum entangled light source. Then the question is raised whether the quantum holography only requires the quantum entanglement. In Ref. [21] a phase-sensitive imaging experiment using a spatially incoherent light source was reported, which is based on the first-order field correlation of the classical thermal light. In this paper, we focus on the geometrical optics in the Fresnel quantum holography based on the second-order correlation measurement. And the quantum holography schemes with entangled two-photon light source and classical thermal light source are proposed, respectively. The correlation measurement method of the “ghost interference” has been used, which provides a reconstruction process by making use of the point light source. Theoretical analysis prove that the Fresnel quantum holography can be carried out with both the quantum entangled light source and spatially incoherent thermal light source. Furthermore, special reconstruction processes based on the hologram recording optical path of the quantum holography with a point light source has been discussed. And the position of the reconstructed original image has been determined through theoretical derivation.
∗ Corresponding author. Tel.: +86 10 82321062; fax: +86 10 82321062. E-mail address: [email protected] (L. Gao). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.02.018
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L. Gao et al. / Optik 124 (2013) 4005–4011
T
d3 BS
d1
x
D1 x1
d0
x0 d2
x2
D2 Fig. 1. Sketch of the hologram recording process of the quantum holography with entangled two-photon light source; LS, the entangled two-photon light source; BS, the polarized beam splitter; T, the detected object; D1 and D2 , the detectors.
2. Fresnel quantum holography with entangled two-photon light source 2.1. The hologram recording process The sketch of the hologram recording process of the Fresnel quantum holography with entangled two-photon light source is shown in Fig. 1. ELS is the entangled two-photon light source. The entangled two-photon pairs emitted from the light source are split by the beam-splitter (BS). One photon is sent into a system including detector D1 which is called the test system, and the other photon is sent into another system including detector D2 which is called the reference system. In the test system the photon beam is split again into two paths. One path spreads through the detected object T(x) to the detector D1 , and the other path spreads freely to D1 . The two paths of the test system perform an interferometer [22]. The other photon beam spreads freely to the detector D2 in the reference system. The quantum hologram of the detected object T(x) can be obtained by second-order coincident measurement, and the theoretical calculation process is shown as below. The ideal two-photon entangled state is written as
† †
i d s ı(ωi + ωs − ωp )ı( i + s − p )ai as |0, d
| = C0
j (j = i, s, p) are the frequency and transverse wave vector of the signal, idle and pumping light. The average second-order where ωj and coincidence counts is proportional to I1 (x1 ) I2 (x2 ) = (+)
(+) E1
(+) (x1 ) E2
(−) (x2 ) E2
x2
(−) E1
2 (+) (+) x1 = 0|E1 (x1 )E2 (x2 )| = | h1 (x1 , x0 ) h2 (x2 , x0 ) dx0 |2 .
(1)
(−)
Ej (xj ) and Ej (xj ) (j = 1, 2) are the positive and negative frequency parts of the optical field in the detection plane of detector D1 and D2 . hj (xj , x0 ) (j = 1, 2) is the impulse response functions of the test system and the reference system. xj (j = 0, 1, 2) is the transverse position across the beam of the light source plane and the detection planes of detector D1 and D2 . The impulse response functions of the test system and the reference system are given by
h1 (x1 , x0 ) =
+
h2 (x2 , x0 ) =
ik(x0 − x1 )2 k −i + ikd0 + exp 4 2d0 2d0 k
2
exp
d1 d3
−i 2
+ ik(d1 + d3 ) ×
ik(x2 − x0 )2 k −i exp + ikd2 + 4 2d2 2d2
ik(x − x0 )2 ik(x1 − x)2 T (x) exp + 2d3 2d1
dx,
(2)
.
(3)
k is the wave number of the light source, and k = 2/ ( is the wavelength of the light source). T(x) designates the transmission function of the object, which is generally complex. x is the transverse position across the object T(x) plane, and dm (m = 0, 1, 2, 3) is the longitudinal optical distance. d0 is the optical distance from the light source plane to the detector D1 , d1 is the optical distance from the light source plane to the object T(x) and d3 is the optical distance from the object T(x) to the detector D1 ; d2 is the optical distance from the light source to the detector D2 . Substituting Eqs. (2) and (3) into Eq. (1) the second-order coincidence counts is obtained as
L. Gao et al. / Optik 124 (2013) 4005–4011
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Fig. 2. Sketch of the reconstruction process of the quantum holography with entangled two-photon light source; H, the hologram; P, the reconstructed point light source; IM, the reconstructed image.
I1 (x1 )I2 (x2 )
=|
h1 (x1 , x0 )h2 (x2 , x0 )dx0 |2
2 k ik(x1 − x2 ) k −i −i + ik(d0 + d2 ) + + ik(d1 + d2 + d3 ) × = exp + exp 4 2 2(d0 + d2 ) 2(d0 + d2 ) 2 d3 (d1 + d2 )
T (x) exp
2
2
ik(x − x2 ) ik(x1 − x) + 2d3 2(d1 + d2 )
2 dx . (4)
In Ref. [15] the coincident measurement is proposed by “bucket measurement”. However, here it is proposed that the quantum hologram can be achieved by the measurement method of the “ghost interference” experiment instead of the “bucket measurement” [2]. The premise is that the scale of the detected object T and the range of the detection planes of D1 and D2 satisfy the Fresnel diffraction approximation. In this way of coincident measurement, detector D1 records the intensity at a fixed position and detector D2 detects the transverse intensity distribution. So Eq. (4) can be regarded as the transmission function of the hologram, where x1 is a constant representing the fixed position coordinate of detector D1 . Eq. (4) has four items, two of which contain both the amplitude and phase information of the detected object T(x). They are expressed as t(x2 ) and its conjugate t∗ (x2 ), and t(x2 ) =
k 3/2 2
1 (d0 + d2 )d3 (d1 + d2 )
× exp
−i 4
+ ik(d1 +d3 − d0 ) ×
−ik(x1 − x2 )2 ik(x1 − x)2 ik(x − x2 )2 + T (x) exp + 2(d0 + d2 ) 2d3 2(d1 + d2 )
dx. (5)
2.2. The reconstruction process The process of quantum holography reconstruction can be implemented by computer program to obtain the digital reconstructed image from the recorded digital hologram, or by a coherent light source to illuminate the hologram recorded on a photographic film. The coincident measurement method described above provides a kind of optical reconstruction way by making use of a point light source. So here we focus on discussing the optical reconstruction with a point light source based on the hologram recording optical path. And the reconstruction geometrical optical path is shown in Fig. 2. H is the hologram film; P is the reconstruction point light source. The hologram
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H is placed at the symmetrical position of detector D2 relative to the BS (beam-splitter). So the transverse coordinate of the hologram plane is also x2 . P is the reconstruction point light source with the same wavelength as the light source. The transverse coordinate of the reconstruction light source P is xp and the distance from P to the ELS plane is zp . So the complex amplitude on the hologram plane from the light source P can be expressed as
−ik(xp − x2 )2 ap P(x2 ) = exp zp 2zp
.
(6)
Then the transmitted complex amplitude through the hologram is proportional to P(x2 )I1 (x1 )I2 (x2 ), which also contains four items. Two items represent the holographic reconstructed images, and they are given by R(x2 ) and R (x2 ).
R(x2 ) = P(x2 )t(x2 ) = C
T (x) exp
ik
2zi
R (x2 ) = P(x2 )t ∗ (x2 ) = C
T ∗ (x) exp
(x2 − xi )2 dx,
(7)
ik (x2 − xi )2 dx, 2zi
(8)
where C and C are the formulae having no business with x2 . And xi =
(x/(d1 + d2 ) − x1 /(d0 + d2 ) − xp /zp ) , (1/(d1 + d2 ) − 1/(d0 + d2 ) − 1/zp )
xi =
(−x/(d1 + d2 ) + x1 /(d0 + d2 ) − xp /zp ) , (−1/(d1 + d2 ) + 1/(d0 + d2 ) − 1/zp )
zi =
1 1 1 − − zp d1 + d2 d0 + d2
zi =
−
−1
1 1 1 + − zp d1 + d2 d0 + d2
,
−1 .
Eqs. (7) and (8) can be seen as the complex amplitude on the hologram plane of a spherical wave with paraxial approximation, respectively. The spherical wave center positions are (xi , zi ) and (xi , zi ). If the position of P is specially chosen at xp = x1 , zp = − (d0 + d2 ), and thus Eqs. (7) and (8) are obtained as
R(x2 ) = M
T (x) exp
ik(x2 − x)2 2(d1 + d2 )
R (x2 ) = M
T ∗ (x) exp
−ik 2
dx,
2 1 − d1 + d2 d0 + d2
(9)
(x2 − x)2 dx,
(10)
where M and M are the formulae having no business with x2 . Eq. (9) is the spherical wave centered at the space coordinate of (x, − d1 − d2 ), and represents the original image of T at a distance of (d1 + d2 ) behind the hologram. And the original image of T is shown by IM in Fig. 2. Eq. (10) is the spherical wave centered at the space coordinate of x, (1/(d1 + d2 ) − 2/(d0 + d2 ))−1 , which represents the conjugated image of T. And the position of the conjugated image depends on the value of (1/(d1 + d2 ) − 2/(d0 + d2 ))−1 . According to the calculation result shown by Eq. (9), if the reconstructed point light source P is placed at the symmetrical position of D1 relative to the ELS, and thus the reconstructed original image IM is at the symmetrical position of the object T relative to the ELS. The physical interpretation of this result is that the quantum entangled two-photon light source emits a pair of correlated rays with opposite transverse wave vectors, one to the test system and the other to the reference system, and the quantum source acts as a mirror [9]. So that a part of reconstruction optical path can be performed through reflecting the test system by the ELS. The BS also plays the role of a mirror. The reference system is reflected by the BS to perform the other part of the reconstruction optical path. Thus, H is at the symmetrical position of D2 relative to the BS; P is at the symmetrical position of D1 relative to the ELS; the reconstructed original image IM is at the mirror imaging position of the detected object T relative to the ELS. Above all, if P is placed at the symmetrical position of D1 , the position of the reconstructed original image can be determined by geometrical optical path based on the correlation character of the quantum entangled two-photon light source. 3. Fresnel quantum holography with classical thermal light source 3.1. The hologram recording process The sketch of the hologram recording process of the quantum holography with classical thermal light source is shown in Fig. 3. Here we focus on the quantum holography with spatially incoherent classical thermal light, which can be produced by illuminating a laser beam onto a rotating ground glass. CLS is the classical thermal light source. And the thermal light beams are split into two parts by a beam-splitter (BS). The transmitted light beams pass through the test system and the reflected beams pass through the reference system. Two detectors D1 and D2 are set at the end of the test system and reference system, respectively. The beams in the test system are separated into two parts again, and one part illuminates the detected object T then to the detector D1 and the other part spreads freely to D1 . The two optical paths in the test system perform an unbalanced interferometer [21,22]. However, the hologram can also be achieved by the intensity correlation measurement between the light beams arriving at detectors D1 and D2 . The theoretical analysis are shown as below. The transmission function of the quantum hologram with classical thermal light source is proportional to the spatial intensity correlation I1 (x1 )I2 (x2 ) = E1∗ (x1 )E2∗ (x2 )E2 (x2 )E1 (x1 ) =I1 (x1 )I2 (x2 ) + |E1∗ (x1 )E2 (x2 )|2 ,
(11)
L. Gao et al. / Optik 124 (2013) 4005–4011
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Fig. 3. Sketch of the hologram recording process of the quantum holography with classical thermal light source; LS, the classical thermal light source; BS, the non-polarized beam splitter; T, the detected object; D1 and D2 , the detectors.
Ij (xj ) =
h∗j (xj , x0 )hj (xj , x0 )S(x0 − x0 )dx0 dx0 ,
j = 1, 2,
|E1∗ (x1 )E2 (x2 )|2 = |
(12)
h∗1 (x1 , x0 )h2 (x2 , x0 )S(x0 − x0 )dx0 dx0 . 2
(13)
Ej (xj )(j = 1, 2) is the optical field with the transverse position xj in the test system and reference system. hj (xj , x0 )(j = 1, 2) is the corresponding impulse response function describing the field propagation in each beam, which can also be expressed by Eqs. (2) and (3). S(x0 − x0 ) is the first-order spatial correlation for the thermal light source. For simplicity we consider the broadband limit for the source, S(x0 − x0 ) = S0 ı(x0 − x0 ). S0 denotes the intensity of the source. I1 (x1 ) and I2 (x2 ) are the intensity distribution functions of the detection planes of detector D1 and D2 , respectively. |E1∗ (x1 )E2 (x2 )|2 is the cross correlation function containing the holography information of the detected object. Substituting Eqs. (2) and (3) into Eq. (13), the cross correlation function is obtained as
|E1∗ (x1 )E2 (x2 )|2
h∗1 (x1 , x0 )h2 (x2 , x0 )dx0
= |S0
+ S0
2
k d3 (d1 − d2 )
exp
= S0
i 2
k 2 (d0 − d2 )
× exp
+ ik(d2 − d1 − d3 ) ×
i ik (x1 − x2 )2 + ik(d2 − d0 ) − 4 2 d0 − d2
ik(x − x1 )2 ik(x − x2 )2 T (x) exp − − 2d3 2(d1 − d2 ) ∗
dx|2 .
(14)
k is the wave number of the thermal light source. dj (j = 0, 1, 2, 3) is the longitudinal optical distance. d0 is the optical distance from the CLS to the detector D1 , and d1 is the optical distance from the CLS to the object Twhile d3 is the distance from T to the detector D1 . d2 is the optical distance from the CLS to the detector D2 . Eq. (14) has four items, two of which include both the amplitude and phase informations of the detected object T. The two items are conjugated of each other, and we set them as t(x2 ) and t∗ (x2 ),
t(x2 ) =
S02
k 3/2 2
1 d3 (d1 − d2 )
ik(x1 − x2 )2 × exp − 2(d0 − d2 )
dx.
d0 − d2
i
× exp −
4
+ ik(d1 + d3 − d0 ) ×
ik(x − x2 )2 ik(x − x1 )2 T (x) exp + 2d3 2(d1 − d2 )
(15)
Here the correlation measurement method of “ghost interference” is also carried out, and thus detector D1 records the intensity at a fixed position and detector D2 detects the transverse intensity distribution. So x1 is a constant representing the coordinate of detector D1 in Eq. (15).
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Fig. 4. Sketch of the reconstruction process of the quantum holography with classical thermal light source; H, the hologram; P, the reconstructed point light source; IM, the reconstructed image.
3.2. The reconstruction process Here we also focus on discussing the optical reconstruction process with a point light source. The reconstruction optical paths is based on the hologram recording scheme,which is shownin Fig.4. H is the hologram; P is the reconstruction point light source. H is placed at the detection plane of detector D2 . The position of P is xp , zp , which is at a distance of zp from H. The spherical wave function emitted from P on the hologram plane is expressed by
R(x2 ) =
ap ik(xp − x2 )2 exp − zp 2zp
.
(16)
The transmission function of the hologram is proportional to R(x2 )I1 (x1 )I2 (x2 ). And the reconstructed images are included in R(x2 )t(x2 ) and R(x2 )t∗ (x2 ), and they are given by
R(x2 )t(x2 ) = G
T (x) exp
−ik 2zi
R(x2 )t ∗ (x2 ) = G
T ∗ (x) exp
(x2 − xi )2 dx,
(17)
−ik (x2 − xi )2 dx, 2zi
(18)
where G and G are the formulae having no business with x2 . And
(xp /zp ) − x1 /(d2 − d0 ) + x/(d2 − d1 ) , xi = (1/zp ) − 1/(d2 − d0 ) + 1/(d2 − d1 )
xi
zi =
(xp /zp ) + x1 /(d2 − d0 ) − x/(d2 − d1 )) , = (1/zp ) + 1/(d2 − d0 ) − 1/(d2 − d1 )
1 1 1 + − zp d2 − d1 d2 − d0
zi
=
−1
1 1 1 − + zp d2 − d1 d2 − d0
,
−1 .
Both Eqs. (17) and (18) can be regarded as the spherical wave functions with paraxial approximation. Eq. (17) is the spherical wave with the sphere center at (xi , zi ), which represents the original image of T. Eq. (18) is the spherical wave with sphere center at (xi , zi ), which represents the conjugated image of T. If the reconstructed point light source P is placed at the position of (x1 , d2 − d0 ) with the same wavelength as the thermal light source, Eqs. (17) and (18) become
R(xp )t(x2 ) = N
ik(x2 − x)2 T (x) exp − 2(d2 − d1 )
∗
R(xp )t (x2 ) = N exp
−ik 2
dx,
2 1 − d2 − d0 d2 − d1
(19)
×
2x1 /(d2 − d0 ) − x/(d2 − d1 ) x2 − 2/(d2 − d0 ) − 1/(d2 − d1 )
2 dx,
(20)
where N and N are the formulae having no business with x2 . Eq. (19) represents the original image of T, which is at a distance of (d2 − d1 ) from the hologram; Eq. (20) represents the conjugated image of T, which is at a distance of (2/(d2 − d0 ) − 1/(d2 − d1 ))−1 from the hologram.
L. Gao et al. / Optik 124 (2013) 4005–4011
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According to the derivation results above, if the hologram H is located at the detection plane of detector D2 and P is placed at the symmetrical position of detector D1 relative to the BS, thus the reconstructed original image is at the symmetrical position of T relative to the BS. And the geometrical optical paths are shown in Fig. 4. The results can also be explained by the nature of the self-correlation of the wave vectors of the classical thermal light source. The classical spatially incoherent thermal light source acts as a phase-conjugate mirror [9]. The reconstruction scheme shown in Fig. 4 can be seen as reflecting the test system by the BS: P is at the symmetrical position of D1 and thus the original image IM is at the symmetrical position of T relative to the BS, respectively. 4. Conclusion To summarize, we have reported on the geometrical optics in the Fresnel quantum holography with quantum entangled two-photon source and classical spatially incoherent thermal light source, respectively. The correlation measurement method of “ghost interference” has been carried out, which provides a kind of reconstruction process by making use of point light source. According to the theoretical derivation above, both of the quantum entangled two-photon light source and the classical spatially incoherent thermal light source are competent for the quantum holography. Although each of the beams in the test system or reference system is incoherent, whatsoever, the quantum entanglement of the entangled two-photon light source and the intensity correlation of the classical thermal light source permit interference and hence offer the possibility of holography. For the quantum holography with entangled two-photon light source, the correlation happens between the entangled two photons of an emitted photon pair and the entangled light source acts as a mirror. However, the correlation of the classical thermal light happens between the separate light beams of the same spatial mode, and the thermal light source acts as a phase-conjugate mirror. The Fresnel quantum holography based on the second-order correlation measurement provides the possibility of nonlocal holography technology with incoherent light source, such as X-ray or neutron source. Acknowledgment This work was supported by Beijing Natural Science Foundation (4133086), and Research Fund for the Doctoral Program of Higher Education of China (20110022120002). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Geometrical optics in Fresnel quantum holography Lu Gao ∗ , Shi-jian Wang, Zhang-xiang Cheng School of Science, China University of Geosciences, Beijing 100083, China
a r t i c l e
i n f o
Article history: Received 26 August 2012 Accepted 1 December 2012
PACS: 42.40.Ht 42.30.Wb 42.50.Ar
a b s t r a c t The geometrical optics in Fresnel quantum holography for quantum entangled two-photon source and classical thermal light source based on the second-order correlation measurement has been discussed. We theoretically prove that the Fresnel quantum hologram of the detected object can be obtained through second-order correlation measurement and the optical reconstruction process can be accomplished by making use of a point light source. © 2013 Elsevier GmbH. All rights reserved.
Keywords: Entangled two-photon source Thermal light source Quantum holography
1. Introduction The surprising quantum phenomenon of two-photon entangled state generated by spontaneous parametric down-conversion (SPDC), such as “ghost” imaging, “ghost” interference, sub-wavelength interference and so on have drawn much attention [1–5]. Recent studies have shownthat a classical thermal light source can play a role similar to that of a two-photon entangled source in “ghost” imaging, “ghost” interference and sub-wavelength interference [6–20]. The classical spatially incoherent thermal light source can be produced by a laser beam disorderly scattered by a material, such as a rotating ground glass [19]. Thus the optical field in the transverse plane across the beam is a random variable of transverse position. According to the statistical optics, the source possesses a spatial intensity correlation between transverse positions [18]. Interestingly, the spatial correlation of thermal light has the analogous form to that of entangled photon pair generated in the SPDC process. This is the physical interpretation of that the classical correlation can mimic quantum entanglement in “ghost” imaging, “ghost” interference and so on. In Ref. [15] quantum holography with entangled-photon pairs was proposed methodologically based on the second-order correlation of the quantum entangled light source. Then the question is raised whether the quantum holography only requires the quantum entanglement. In Ref. [21] a phase-sensitive imaging experiment using a spatially incoherent light source was reported, which is based on the first-order field correlation of the classical thermal light. In this paper, we focus on the geometrical optics in the Fresnel quantum holography based on the second-order correlation measurement. And the quantum holography schemes with entangled two-photon light source and classical thermal light source are proposed, respectively. The correlation measurement method of the “ghost interference” has been used, which provides a reconstruction process by making use of the point light source. Theoretical analysis prove that the Fresnel quantum holography can be carried out with both the quantum entangled light source and spatially incoherent thermal light source. Furthermore, special reconstruction processes based on the hologram recording optical path of the quantum holography with a point light source has been discussed. And the position of the reconstructed original image has been determined through theoretical derivation.
∗ Corresponding author. Tel.: +86 10 82321062; fax: +86 10 82321062. E-mail address: [email protected] (L. Gao). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.02.018
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L. Gao et al. / Optik 124 (2013) 4005–4011
T
d3 BS
d1
x
D1 x1
d0
x0 d2
x2
D2 Fig. 1. Sketch of the hologram recording process of the quantum holography with entangled two-photon light source; LS, the entangled two-photon light source; BS, the polarized beam splitter; T, the detected object; D1 and D2 , the detectors.
2. Fresnel quantum holography with entangled two-photon light source 2.1. The hologram recording process The sketch of the hologram recording process of the Fresnel quantum holography with entangled two-photon light source is shown in Fig. 1. ELS is the entangled two-photon light source. The entangled two-photon pairs emitted from the light source are split by the beam-splitter (BS). One photon is sent into a system including detector D1 which is called the test system, and the other photon is sent into another system including detector D2 which is called the reference system. In the test system the photon beam is split again into two paths. One path spreads through the detected object T(x) to the detector D1 , and the other path spreads freely to D1 . The two paths of the test system perform an interferometer [22]. The other photon beam spreads freely to the detector D2 in the reference system. The quantum hologram of the detected object T(x) can be obtained by second-order coincident measurement, and the theoretical calculation process is shown as below. The ideal two-photon entangled state is written as
† †
i d s ı(ωi + ωs − ωp )ı( i + s − p )ai as |0, d
| = C0
j (j = i, s, p) are the frequency and transverse wave vector of the signal, idle and pumping light. The average second-order where ωj and coincidence counts is proportional to I1 (x1 ) I2 (x2 ) = (+)
(+) E1
(+) (x1 ) E2
(−) (x2 ) E2
x2
(−) E1
2 (+) (+) x1 = 0|E1 (x1 )E2 (x2 )| = | h1 (x1 , x0 ) h2 (x2 , x0 ) dx0 |2 .
(1)
(−)
Ej (xj ) and Ej (xj ) (j = 1, 2) are the positive and negative frequency parts of the optical field in the detection plane of detector D1 and D2 . hj (xj , x0 ) (j = 1, 2) is the impulse response functions of the test system and the reference system. xj (j = 0, 1, 2) is the transverse position across the beam of the light source plane and the detection planes of detector D1 and D2 . The impulse response functions of the test system and the reference system are given by
h1 (x1 , x0 ) =
+
h2 (x2 , x0 ) =
ik(x0 − x1 )2 k −i + ikd0 + exp 4 2d0 2d0 k
2
exp
d1 d3
−i 2
+ ik(d1 + d3 ) ×
ik(x2 − x0 )2 k −i exp + ikd2 + 4 2d2 2d2
ik(x − x0 )2 ik(x1 − x)2 T (x) exp + 2d3 2d1
dx,
(2)
.
(3)
k is the wave number of the light source, and k = 2/ ( is the wavelength of the light source). T(x) designates the transmission function of the object, which is generally complex. x is the transverse position across the object T(x) plane, and dm (m = 0, 1, 2, 3) is the longitudinal optical distance. d0 is the optical distance from the light source plane to the detector D1 , d1 is the optical distance from the light source plane to the object T(x) and d3 is the optical distance from the object T(x) to the detector D1 ; d2 is the optical distance from the light source to the detector D2 . Substituting Eqs. (2) and (3) into Eq. (1) the second-order coincidence counts is obtained as
L. Gao et al. / Optik 124 (2013) 4005–4011
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Fig. 2. Sketch of the reconstruction process of the quantum holography with entangled two-photon light source; H, the hologram; P, the reconstructed point light source; IM, the reconstructed image.
I1 (x1 )I2 (x2 )
=|
h1 (x1 , x0 )h2 (x2 , x0 )dx0 |2
2 k ik(x1 − x2 ) k −i −i + ik(d0 + d2 ) + + ik(d1 + d2 + d3 ) × = exp + exp 4 2 2(d0 + d2 ) 2(d0 + d2 ) 2 d3 (d1 + d2 )
T (x) exp
2
2
ik(x − x2 ) ik(x1 − x) + 2d3 2(d1 + d2 )
2 dx . (4)
In Ref. [15] the coincident measurement is proposed by “bucket measurement”. However, here it is proposed that the quantum hologram can be achieved by the measurement method of the “ghost interference” experiment instead of the “bucket measurement” [2]. The premise is that the scale of the detected object T and the range of the detection planes of D1 and D2 satisfy the Fresnel diffraction approximation. In this way of coincident measurement, detector D1 records the intensity at a fixed position and detector D2 detects the transverse intensity distribution. So Eq. (4) can be regarded as the transmission function of the hologram, where x1 is a constant representing the fixed position coordinate of detector D1 . Eq. (4) has four items, two of which contain both the amplitude and phase information of the detected object T(x). They are expressed as t(x2 ) and its conjugate t∗ (x2 ), and t(x2 ) =
k 3/2 2
1 (d0 + d2 )d3 (d1 + d2 )
× exp
−i 4
+ ik(d1 +d3 − d0 ) ×
−ik(x1 − x2 )2 ik(x1 − x)2 ik(x − x2 )2 + T (x) exp + 2(d0 + d2 ) 2d3 2(d1 + d2 )
dx. (5)
2.2. The reconstruction process The process of quantum holography reconstruction can be implemented by computer program to obtain the digital reconstructed image from the recorded digital hologram, or by a coherent light source to illuminate the hologram recorded on a photographic film. The coincident measurement method described above provides a kind of optical reconstruction way by making use of a point light source. So here we focus on discussing the optical reconstruction with a point light source based on the hologram recording optical path. And the reconstruction geometrical optical path is shown in Fig. 2. H is the hologram film; P is the reconstruction point light source. The hologram
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H is placed at the symmetrical position of detector D2 relative to the BS (beam-splitter). So the transverse coordinate of the hologram plane is also x2 . P is the reconstruction point light source with the same wavelength as the light source. The transverse coordinate of the reconstruction light source P is xp and the distance from P to the ELS plane is zp . So the complex amplitude on the hologram plane from the light source P can be expressed as
−ik(xp − x2 )2 ap P(x2 ) = exp zp 2zp
.
(6)
Then the transmitted complex amplitude through the hologram is proportional to P(x2 )I1 (x1 )I2 (x2 ), which also contains four items. Two items represent the holographic reconstructed images, and they are given by R(x2 ) and R (x2 ).
R(x2 ) = P(x2 )t(x2 ) = C
T (x) exp
ik
2zi
R (x2 ) = P(x2 )t ∗ (x2 ) = C
T ∗ (x) exp
(x2 − xi )2 dx,
(7)
ik (x2 − xi )2 dx, 2zi
(8)
where C and C are the formulae having no business with x2 . And xi =
(x/(d1 + d2 ) − x1 /(d0 + d2 ) − xp /zp ) , (1/(d1 + d2 ) − 1/(d0 + d2 ) − 1/zp )
xi =
(−x/(d1 + d2 ) + x1 /(d0 + d2 ) − xp /zp ) , (−1/(d1 + d2 ) + 1/(d0 + d2 ) − 1/zp )
zi =
1 1 1 − − zp d1 + d2 d0 + d2
zi =
−
−1
1 1 1 + − zp d1 + d2 d0 + d2
,
−1 .
Eqs. (7) and (8) can be seen as the complex amplitude on the hologram plane of a spherical wave with paraxial approximation, respectively. The spherical wave center positions are (xi , zi ) and (xi , zi ). If the position of P is specially chosen at xp = x1 , zp = − (d0 + d2 ), and thus Eqs. (7) and (8) are obtained as
R(x2 ) = M
T (x) exp
ik(x2 − x)2 2(d1 + d2 )
R (x2 ) = M
T ∗ (x) exp
−ik 2
dx,
2 1 − d1 + d2 d0 + d2
(9)
(x2 − x)2 dx,
(10)
where M and M are the formulae having no business with x2 . Eq. (9) is the spherical wave centered at the space coordinate of (x, − d1 − d2 ), and represents the original image of T at a distance of (d1 + d2 ) behind the hologram. And the original image of T is shown by IM in Fig. 2. Eq. (10) is the spherical wave centered at the space coordinate of x, (1/(d1 + d2 ) − 2/(d0 + d2 ))−1 , which represents the conjugated image of T. And the position of the conjugated image depends on the value of (1/(d1 + d2 ) − 2/(d0 + d2 ))−1 . According to the calculation result shown by Eq. (9), if the reconstructed point light source P is placed at the symmetrical position of D1 relative to the ELS, and thus the reconstructed original image IM is at the symmetrical position of the object T relative to the ELS. The physical interpretation of this result is that the quantum entangled two-photon light source emits a pair of correlated rays with opposite transverse wave vectors, one to the test system and the other to the reference system, and the quantum source acts as a mirror [9]. So that a part of reconstruction optical path can be performed through reflecting the test system by the ELS. The BS also plays the role of a mirror. The reference system is reflected by the BS to perform the other part of the reconstruction optical path. Thus, H is at the symmetrical position of D2 relative to the BS; P is at the symmetrical position of D1 relative to the ELS; the reconstructed original image IM is at the mirror imaging position of the detected object T relative to the ELS. Above all, if P is placed at the symmetrical position of D1 , the position of the reconstructed original image can be determined by geometrical optical path based on the correlation character of the quantum entangled two-photon light source. 3. Fresnel quantum holography with classical thermal light source 3.1. The hologram recording process The sketch of the hologram recording process of the quantum holography with classical thermal light source is shown in Fig. 3. Here we focus on the quantum holography with spatially incoherent classical thermal light, which can be produced by illuminating a laser beam onto a rotating ground glass. CLS is the classical thermal light source. And the thermal light beams are split into two parts by a beam-splitter (BS). The transmitted light beams pass through the test system and the reflected beams pass through the reference system. Two detectors D1 and D2 are set at the end of the test system and reference system, respectively. The beams in the test system are separated into two parts again, and one part illuminates the detected object T then to the detector D1 and the other part spreads freely to D1 . The two optical paths in the test system perform an unbalanced interferometer [21,22]. However, the hologram can also be achieved by the intensity correlation measurement between the light beams arriving at detectors D1 and D2 . The theoretical analysis are shown as below. The transmission function of the quantum hologram with classical thermal light source is proportional to the spatial intensity correlation I1 (x1 )I2 (x2 ) = E1∗ (x1 )E2∗ (x2 )E2 (x2 )E1 (x1 ) =I1 (x1 )I2 (x2 ) + |E1∗ (x1 )E2 (x2 )|2 ,
(11)
L. Gao et al. / Optik 124 (2013) 4005–4011
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Fig. 3. Sketch of the hologram recording process of the quantum holography with classical thermal light source; LS, the classical thermal light source; BS, the non-polarized beam splitter; T, the detected object; D1 and D2 , the detectors.
Ij (xj ) =
h∗j (xj , x0 )hj (xj , x0 )S(x0 − x0 )dx0 dx0 ,
j = 1, 2,
|E1∗ (x1 )E2 (x2 )|2 = |
(12)
h∗1 (x1 , x0 )h2 (x2 , x0 )S(x0 − x0 )dx0 dx0 . 2
(13)
Ej (xj )(j = 1, 2) is the optical field with the transverse position xj in the test system and reference system. hj (xj , x0 )(j = 1, 2) is the corresponding impulse response function describing the field propagation in each beam, which can also be expressed by Eqs. (2) and (3). S(x0 − x0 ) is the first-order spatial correlation for the thermal light source. For simplicity we consider the broadband limit for the source, S(x0 − x0 ) = S0 ı(x0 − x0 ). S0 denotes the intensity of the source. I1 (x1 ) and I2 (x2 ) are the intensity distribution functions of the detection planes of detector D1 and D2 , respectively. |E1∗ (x1 )E2 (x2 )|2 is the cross correlation function containing the holography information of the detected object. Substituting Eqs. (2) and (3) into Eq. (13), the cross correlation function is obtained as
|E1∗ (x1 )E2 (x2 )|2
h∗1 (x1 , x0 )h2 (x2 , x0 )dx0
= |S0
+ S0
2
k d3 (d1 − d2 )
exp
= S0
i 2
k 2 (d0 − d2 )
× exp
+ ik(d2 − d1 − d3 ) ×
i ik (x1 − x2 )2 + ik(d2 − d0 ) − 4 2 d0 − d2
ik(x − x1 )2 ik(x − x2 )2 T (x) exp − − 2d3 2(d1 − d2 ) ∗
dx|2 .
(14)
k is the wave number of the thermal light source. dj (j = 0, 1, 2, 3) is the longitudinal optical distance. d0 is the optical distance from the CLS to the detector D1 , and d1 is the optical distance from the CLS to the object Twhile d3 is the distance from T to the detector D1 . d2 is the optical distance from the CLS to the detector D2 . Eq. (14) has four items, two of which include both the amplitude and phase informations of the detected object T. The two items are conjugated of each other, and we set them as t(x2 ) and t∗ (x2 ),
t(x2 ) =
S02
k 3/2 2
1 d3 (d1 − d2 )
ik(x1 − x2 )2 × exp − 2(d0 − d2 )
dx.
d0 − d2
i
× exp −
4
+ ik(d1 + d3 − d0 ) ×
ik(x − x2 )2 ik(x − x1 )2 T (x) exp + 2d3 2(d1 − d2 )
(15)
Here the correlation measurement method of “ghost interference” is also carried out, and thus detector D1 records the intensity at a fixed position and detector D2 detects the transverse intensity distribution. So x1 is a constant representing the coordinate of detector D1 in Eq. (15).
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Fig. 4. Sketch of the reconstruction process of the quantum holography with classical thermal light source; H, the hologram; P, the reconstructed point light source; IM, the reconstructed image.
3.2. The reconstruction process Here we also focus on discussing the optical reconstruction process with a point light source. The reconstruction optical paths is based on the hologram recording scheme,which is shownin Fig.4. H is the hologram; P is the reconstruction point light source. H is placed at the detection plane of detector D2 . The position of P is xp , zp , which is at a distance of zp from H. The spherical wave function emitted from P on the hologram plane is expressed by
R(x2 ) =
ap ik(xp − x2 )2 exp − zp 2zp
.
(16)
The transmission function of the hologram is proportional to R(x2 )I1 (x1 )I2 (x2 ). And the reconstructed images are included in R(x2 )t(x2 ) and R(x2 )t∗ (x2 ), and they are given by
R(x2 )t(x2 ) = G
T (x) exp
−ik 2zi
R(x2 )t ∗ (x2 ) = G
T ∗ (x) exp
(x2 − xi )2 dx,
(17)
−ik (x2 − xi )2 dx, 2zi
(18)
where G and G are the formulae having no business with x2 . And
(xp /zp ) − x1 /(d2 − d0 ) + x/(d2 − d1 ) , xi = (1/zp ) − 1/(d2 − d0 ) + 1/(d2 − d1 )
xi
zi =
(xp /zp ) + x1 /(d2 − d0 ) − x/(d2 − d1 )) , = (1/zp ) + 1/(d2 − d0 ) − 1/(d2 − d1 )
1 1 1 + − zp d2 − d1 d2 − d0
zi
=
−1
1 1 1 − + zp d2 − d1 d2 − d0
,
−1 .
Both Eqs. (17) and (18) can be regarded as the spherical wave functions with paraxial approximation. Eq. (17) is the spherical wave with the sphere center at (xi , zi ), which represents the original image of T. Eq. (18) is the spherical wave with sphere center at (xi , zi ), which represents the conjugated image of T. If the reconstructed point light source P is placed at the position of (x1 , d2 − d0 ) with the same wavelength as the thermal light source, Eqs. (17) and (18) become
R(xp )t(x2 ) = N
ik(x2 − x)2 T (x) exp − 2(d2 − d1 )
∗
R(xp )t (x2 ) = N exp
−ik 2
dx,
2 1 − d2 − d0 d2 − d1
(19)
×
2x1 /(d2 − d0 ) − x/(d2 − d1 ) x2 − 2/(d2 − d0 ) − 1/(d2 − d1 )
2 dx,
(20)
where N and N are the formulae having no business with x2 . Eq. (19) represents the original image of T, which is at a distance of (d2 − d1 ) from the hologram; Eq. (20) represents the conjugated image of T, which is at a distance of (2/(d2 − d0 ) − 1/(d2 − d1 ))−1 from the hologram.
L. Gao et al. / Optik 124 (2013) 4005–4011
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According to the derivation results above, if the hologram H is located at the detection plane of detector D2 and P is placed at the symmetrical position of detector D1 relative to the BS, thus the reconstructed original image is at the symmetrical position of T relative to the BS. And the geometrical optical paths are shown in Fig. 4. The results can also be explained by the nature of the self-correlation of the wave vectors of the classical thermal light source. The classical spatially incoherent thermal light source acts as a phase-conjugate mirror [9]. The reconstruction scheme shown in Fig. 4 can be seen as reflecting the test system by the BS: P is at the symmetrical position of D1 and thus the original image IM is at the symmetrical position of T relative to the BS, respectively. 4. Conclusion To summarize, we have reported on the geometrical optics in the Fresnel quantum holography with quantum entangled two-photon source and classical spatially incoherent thermal light source, respectively. The correlation measurement method of “ghost interference” has been carried out, which provides a kind of reconstruction process by making use of point light source. According to the theoretical derivation above, both of the quantum entangled two-photon light source and the classical spatially incoherent thermal light source are competent for the quantum holography. Although each of the beams in the test system or reference system is incoherent, whatsoever, the quantum entanglement of the entangled two-photon light source and the intensity correlation of the classical thermal light source permit interference and hence offer the possibility of holography. For the quantum holography with entangled two-photon light source, the correlation happens between the entangled two photons of an emitted photon pair and the entangled light source acts as a mirror. However, the correlation of the classical thermal light happens between the separate light beams of the same spatial mode, and the thermal light source acts as a phase-conjugate mirror. The Fresnel quantum holography based on the second-order correlation measurement provides the possibility of nonlocal holography technology with incoherent light source, such as X-ray or neutron source. Acknowledgment This work was supported by Beijing Natural Science Foundation (4133086), and Research Fund for the Doctoral Program of Higher Education of China (20110022120002). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
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