- Email: [email protected]
Super-resolution noninvasive confocal quantum microscope
Optik - International Journal for Light and Electron Optics 198 (2019) 163209
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Super-resolution noninvasive confocal quantum microscope Sanjit Karmakar
T
Center for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore
A R T IC LE I N F O
ABS TRA CT
Keywords: Super-resolution Noninvasive Quantum microscope
Excellent resolving power, noninvasive and excellent image contrast are needed features of a high-quality microscope. To achieve the first feature, a shorter wavelength of light is necessary. On the other hand, a shorter wavelength of light destroys the second feature. The occurrence of the two features together is very critical. Recently, Karmakar et al. reported non-invasive high resolving power quantum microscope by using two-color ghost imaging technology [1,2]. However, this article reports a confocal microscope having super-resolution and noninvasive features by using another quantum mechanical phenomenon. In addition, this confocal microscope may provide very high image contrast.
1. Introduction For the progress of research and development in different areas of biomedical and biosciences, a very high resolving power and the noninvasive microscope is needed. The shorter wavelength of light provides high resolving power. On the other hand, to achieve noninvasive feature, i.e. feature of not damaging any cell or tissue, the longer wavelength is necessary. Thus, the development of such a microscope is critical. The discovery of the confocal microscope provides better resolving power as well as higher contrast over conventional wide-field microscopy [3]. By using a pin-hole in front of the detector, the light other than from the point of interest is blocked. As a consequence, the image becomes sharp. The light rays from the specimen pass the objective lens twice. As a result, the resolving power is improved. In 2010, Simon et al. demonstrated two approaches of confocal microscopy to improve the resolution in comparison of standard confocal microscopy. In one approach, they used photon correlation which can improve the resolution up to 50% [4]. In other approach, they used degenerate entangled photon pair generated from spontaneous parametric down-conversion (SPDC) to improve the resolution up to 77% [5]. They used non-linear crystal after the object, i.e. specimen. That means the specimen is illuminated by very high pump power to generate enough signal and idler photons. To have visible-light images with UV-level resolution, they suggested to use UV-pump to generate signal and idler in the visible range. Since the specimen is illuminated by UV-light with very high power, there is very high chances of damaging the specimen, i.e. the cells and tissues in the medical area. There are several other works on microscope to improve the resolving power and non-invasive features [6–8,1,2,9]. Their resolution is even smaller than the twin-photon confocal microscopes (reported by Simon et al.). But none of these confocal microscopes can achieve excellent resolution, noninvasive feature and high image contrast together. In this report, we wish to propose a confocal microscope capable of achieving very high resolving power (i.e. super-resolution) without any destruction of cells and tissues by using twin-photon two color confocal microscopy phenomenon. Additionally, this microscopy system may produce very high image contrast.
E-mail address: [email protected]. https://doi.org/10.1016/j.ijleo.2019.163209 Received 12 May 2019; Accepted 12 August 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 198 (2019) 163209
S. Karmakar
Fig. 1. Schematic of proposed experimental configuration. PBS represents polarized beam splitter.
2. Proposed method The idea of twin-photon confocal microscopy is used here. To avoid the chances of damaging the specimen, the specimen is used after the non-linear crystal. In this proposed experiment, the specimen is illuminated by signal and idler photon pair. These signal and idler beams are on the visible and infrared ranges with low power. So this configuration can easily protect the specimen from destruction. In this case, these two different color signal and photons are used to improve the resolution. The schematic of this proposed experimental configuration is shown in Fig. 1. Here, the pump of 351 nm is used to generate signal (1178 nm) and idler (500 nm) pair by spontaneous parametric down-conversion (SPDC) process using a very thin BBO type II crystal. This 351 nm pump is generated from an Ar-ion laser. The output signal and idler beams are co-linear with the pump beam. The crystal is placed at the focal point of the pump beam. The specimen is placed immediately after the thin BBO crystal. The signal and idler photon pairs from the specimen are passed through the objective lens to a pinhole. This pinhole reduces the noise by blocking the noise from unnecessary rays from the specimen. These filtered pairs are then passed through a polarized beam splitter to separate the orthogonally polarized signal and idler photons. After the beam splitter, these pairs are collected by two photodetectors and the image is observed in pointby-point of the specimen in terms of coincidence counts between these two photodetectors. Since the image is formed in terms of coincidences, this confocal microscope provides a very high degree of image contrast. 3. Discussion In this configuration, a pump beam of beam waist wo and angular frequency ωp is focused on the face of a thin nonlinear crystal by ρ ) is placed where → ρ is the transverse using a lens of focal length f. On the other face of the crystal, a specimen of transmittance of t (→ position. The pump field inside the crystal can be written as − Ep (→ r , t ) = E0 e−i (ωp t + kp z ) e
where E0 is a constant and r02 =
|→ r⊥ |2 2r02 λp2 f 2 2π 2w02
under this experimental condition.
The outgoing two-photon state generated from the nonlinear crystal is defined as [5,10]
|ψ〉 = A
→
→
∫ d3 ks ∫ d3 ki δ (ωs + ωi − ωp) ∫0 →
→
→ −
× ei (kpz− ksz− kiz) ze−i ( ks ⊥+ ki ⊥)· r⊥ e † † × a→ a→ |0〉 ks
L
dz
|→ r⊥ |2 2r02
(1)
ki
where all the constants are absorbed in the constant A. The outgoing signal and idler field propagate s0 distances and then pass the objective lens of focal length f and again propagate sI 1 1 1 distance to reach the pinhole with satisfying the thin-lens condition of s + s = f . If the positive frequency field operator of the o I (+) → → † , then the field at the detector placed at a distance of so + sI is given by [5] outgoing beam at the crystal is Eˆ j ( r⊥ , t ) = E j(+) ( r⊥ , t ) aˆ → kj
(+) Eˆ j (→ r⊥ , t j ) =
∫
→′ k′j → 2 →′ → d3 k j e−i so | r⊥ | e−i k j ⊥· r⊥ e−iωj τ j
s →′ † r⊥ + o k j ⊥ ⎟⎞ E0′ aˆ → × somb ⎜⎛→ ′ kj ′j k ⎝ ⎠ where the propagation factor a confocal system is
(2) k′j →′ → →2 e−i so | r⊥ | e−i k j ⊥· r⊥ e−iωj τ j somb ⎛→ r
⎝
2
⊥
+
so →′ k ⎞, k ′j j ⊥
⎠
τj = t1 −
so + sI , c
j = s, i and E0′ is a constant.
Optik - International Journal for Light and Electron Optics 198 (2019) 163209
S. Karmakar
Here symbol subscript s and i represent signal and idler photons respectively. The image spot is at the pinhole and the detector is supposed to be at position of the pinhole. But for the ease of alignment and design, the signal and idler pair are separated immediately after the pinhole by a polarized beam splitter and then collected by two detectors 1 and 2. Here the signal and idler go to detector 1 and 2 respectively. Then the ‘s’ and ‘i’ can be replaced by ‘1’ and ‘2’ respectively. Now, the image forming correlation can be written as
C (τ1, τ2) = |A (τ1, τ2)|2
(3)
where A(τ1, τ2) is a coincidence amplitude. Without any specimen, the coincidence amplitude is as follows
A (τ1, τ2) =
∫ d2→r⊥ 〈0|E2(+) (→r⊥, τ2) E1(+) (→r⊥, τ1)|ψ〉
where E1(+) and E2(+) are the positive-frequency fields at the above detectors, 1 and 2. ψ is the outgoing state vector from the non-linear crystal. r⊥ + → ρ ) where → ρ is the scanning point The specimen is considered at output face of the crystal with its transmission function t (→ unit of the microscope. The image is formed in terms of coincidence counts between two detectors. After simplification [5,10], the coincidence amplitude is calculated as follows
∫ d2→r⊥ t (→r⊥ + →ρ ) 〈0|E2(+) (→r⊥, τ2) E1(+) (→r⊥, τ1)|ψ〉 = A1 Π(τ1 − τ2) e
i 2π | → r |2 so λp ⊥
4πa|→ ρ |⎞ somb ⎜⎛ ⎟ λ s c ⎠ s o ⎝
|→ ρ |2 4πa|→ ρ | ⎞ − 2r 2 0 × somb ⎜⎛ ⎟e ⎝ λi so c ⎠
(4)
where Π(τ1 − τ2) is unit step function in which nonzero value has in the range of 0 < τ1 − τ2 < DL, D is the inverse group velocity difference of orthogonally polarized signal and idler photons, λp is the wavelength of the pump beam and A1 is a constant. Here λs and λi are the wavelength of the signal and idler beam respectively and a is the radius of both lenses. Here the object is a point with r⊥ + → ρ ) = δ (2) (→ r⊥ + → ρ ). perfectly transmissive and hence the transmittance function can be treated as a delta function, i.e. t (→ After substituting the coincidence amplitude in Eq. (4) into Eq. (3), the image forming correlation of a point of the specimen is given by ρ |2 → − |→ 4πa|→ ρ |⎞ 2 ⎛ 4πa| ρ | ⎞ e r02 somb C = A2 somb2 ⎜⎛ ⎟ ⎜ ⎟ ⎝ λs so c ⎠ ⎝ λi so c ⎠
(5)
where A2 is constant and coincidence time window is DL. This is the coincidence time window in an ideal detection system. In a realistic situation, this coincidence time would increase due to the finite response time of the detection system. In the one-dimensional case, the calculated point-spread function for the twin-photon microscope can be written as follows
PSFtwin (y ) y2
4π ay ⎞ 4π ay ⎞ − r 2 somb2 ⎛ e 0 = somb2 ⎛ λ s c ⎝ s o ⎠ ⎝ λi so c ⎠ ⎜
⎟
⎜
⎟
(6)
where y = y1 − y2 is the relative position of the scanning detector. From Eq. (6), one can say that resolving power of the microscope could be increased by three ways: (1) by decreasing object distance, so, (2) by decreasing f-number, f/2a and (3) by increasing the wavelength ratio of signal and idler beam, λs/λi. For comparison purpose, the lenses with having a = f are considered. To achieve maximum resolution for this case, the pump size is considered as w0 = a . For example for twin-photon cases, so = sI = 2f is considered. Now the calculated point spread function for both degenerate and non-degenerate cases of twin-photon confocal and standard confocal are shown in Fig. 2. This figure shows that it is possible to improve the resolution by 2% by using two different colors of signal and idler photons with respect to the same color signal and photons. 1 1 1 Now I assume another twin-photon case with the condition of so = 1.5f and sI = 3f to satisfy the thin lens equation s + s = f . In o I this case acceptance angle of objective lens increases and as a consequence resolution is improved. This condition of the twin-photon case is shown in Fig. 3. For this case of two different colors, the resolution is improved by 4% than the same color signal and photons. Next another twin-photon confocal case is assessed where I imply the condition of so = 1.5f and sI = 3f as above. One more condition of λs = 1595 nm and λi = 450 nm with the pump of 351 nm is considered where the ratio of signal and idler wavelengths is increased than the previous case. The comparison of standard confocal, degenerate twin-photon and non-degenerate twin-photon cases are shown in Fig. 4. In this case, the resolution is improved by 6% than the degenerate twin-photon case with the same condition. For all the above cases, the f-number of the objective lens of 0.5, i.e. a = f, is assumed. If we can use a smaller f-number objective lens, then the acceptance angle of the objective lens is also increased. Now I use the f-number of the objective lens of 0.25. As a consequence, the resolution, in this case, is improved by 10% than the standard degenerate twin-photon confocal case with the same 3
Optik - International Journal for Light and Electron Optics 198 (2019) 163209
S. Karmakar
Fig. 2. Comparison of resolution for three cases: (1) standard confocal microscope, (2) twin-photon confocal microscope with degenerate case and (3) twin-photon confocal microscope with non-degenerate case.
Fig. 3. Comparison of resolution for three cases with higher acceptance angle of objective lens by decreasing the object distance in twin-photon case.
4
Optik - International Journal for Light and Electron Optics 198 (2019) 163209
S. Karmakar
Fig. 4. Comparison of resolution for three cases with higher acceptance angle of objective lens by decreasing the object distance and higher ratio of signal and idler wavelength in twin-photon case.
Fig. 5. Comparison of resolution for three cases with higher acceptance angle of objective lens by decreasing both the object distance and the fnumber of the lens and also with higher ratio of signal and idler wavelength in twin-photon case. 5
Optik - International Journal for Light and Electron Optics 198 (2019) 163209
S. Karmakar
condition. This is shown in Fig. 5. From the above results of the non-degenerate case, one can say that the resolution can be improved further by increasing the signal and idler wavelength ratio and also by increasing the acceptance angle of the objective lens. The resolution shown in Fig. 2 in the degenerate case, 350 nm, is improved to the resolution of 315 nm described in the non-degenerate case of Fig. 5. The resolution is determined by measuring the full-width-half-maximum (FWHM) of the peaks presented in Figs. 2 and 5. Here the specimen is illuminated by signal idler photons which have very low intensity and also the signal and idler beams are either visible or infrared regime. Here, CW laser is used to generate signal and idler pair through SPDC process by using a nonlinear crystal. As a result, the specimen is illuminated by very low power. Hence the specimen is very safe during the process. In coincidence detection system, the simultaneously generated signal and idler pair are collected from the point of interest. That means the unwanted photons other than signal-idler photons are not collected. By discarding these noise producing unwanted photons in a coincidence system, the image contrast is improved. In addition, the image contrast is also improved by blocking unwanted noise producing photons using a pinhole in a confocal system. In this configuration, the confocal system and coincidence measurement together may provide very high image contrast. 4. Conclusion From this study, one may say that this proposed configuration promises to have a confocal microscope with three important features:(1) excellent resolution of approximately 315 nm, (2) noninvasive and (3) very high image contrast. By increasing the signal and idler wavelength ratio and also by increasing the acceptance angle of the objective lens, the resolution can be improved further. This proposed microscope might be very important to advance the progress of research and development as well as technology. Conflict of interest None. Acknowledgements The authors thank Y. Shih, R. Meyers, K. Deacon and A. Tunick for helpful discussions with them. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
S. Karmakar, R. Meyers, Y. Shih, Non-invasive high resolving power quantum microscope, Proc. SPIE 8875 (2013) 88750A. S. Karmakar, R. Meyers, Y. Shih, Noninvasive high resolving power entangled photon quantum microscope, J. Biomed. Opt. 20 (2015) 016008. J.B. Pawley, Handbook of Biological Confocal Microscopy, Springer Business Science & Business Media Inc., New York, 1995. D.S. Simon, A.V. Sergienko, The correlation confocal microscope, Opt. Express 18 (2010) 9765–9779. D.S. Simon, A.V. Sergienko, Twin-photon confocal microscopy, Opt. Express 18 (2010) 22147–22157. C. Olsovsky, R. Shelton, O. Carrasco-Zevallos, B.E. Applegate, K.C. Maitland, Chromatic confocal microscopy for multi-depth imaging of epithelial tissue, Biomed. Opt. Express 4 (2013) 732–740. J. Chen, Y. Xu, X. Lv, X. Lai, S. Zeng, Super-resolution differential interference contrast microscopy by structured illumination, Opt. Express 21 (2013) 112–121. G. Zeng, R. Horstmeyer, C. Yang, Wide-field, high-resolution Fourier ptychographic microscopy, Nat. Photonics 7 (2013) 739–745. C. Boudoux, S.H. Yun, W.Y. Oh, W.M. White, N.V. Iftimia, M. Shishkov, B.E. Bouma, G.J. Tearney, Rapid wavelength-swept spectrally encoded confocal microscopy, Opt. Express 13 (2005) 8214. T.B. Pittman, D.V. Strekalov, D.N. Klyshko, M.H. Rubin, A.V. Sergienko, Y.H. shih, Two-photon geometric optics, Phys. Rev. A 53 (1996) 2804–2815.
6
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Super-resolution noninvasive confocal quantum microscope Sanjit Karmakar
T
Center for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore
A R T IC LE I N F O
ABS TRA CT
Keywords: Super-resolution Noninvasive Quantum microscope
Excellent resolving power, noninvasive and excellent image contrast are needed features of a high-quality microscope. To achieve the first feature, a shorter wavelength of light is necessary. On the other hand, a shorter wavelength of light destroys the second feature. The occurrence of the two features together is very critical. Recently, Karmakar et al. reported non-invasive high resolving power quantum microscope by using two-color ghost imaging technology [1,2]. However, this article reports a confocal microscope having super-resolution and noninvasive features by using another quantum mechanical phenomenon. In addition, this confocal microscope may provide very high image contrast.
1. Introduction For the progress of research and development in different areas of biomedical and biosciences, a very high resolving power and the noninvasive microscope is needed. The shorter wavelength of light provides high resolving power. On the other hand, to achieve noninvasive feature, i.e. feature of not damaging any cell or tissue, the longer wavelength is necessary. Thus, the development of such a microscope is critical. The discovery of the confocal microscope provides better resolving power as well as higher contrast over conventional wide-field microscopy [3]. By using a pin-hole in front of the detector, the light other than from the point of interest is blocked. As a consequence, the image becomes sharp. The light rays from the specimen pass the objective lens twice. As a result, the resolving power is improved. In 2010, Simon et al. demonstrated two approaches of confocal microscopy to improve the resolution in comparison of standard confocal microscopy. In one approach, they used photon correlation which can improve the resolution up to 50% [4]. In other approach, they used degenerate entangled photon pair generated from spontaneous parametric down-conversion (SPDC) to improve the resolution up to 77% [5]. They used non-linear crystal after the object, i.e. specimen. That means the specimen is illuminated by very high pump power to generate enough signal and idler photons. To have visible-light images with UV-level resolution, they suggested to use UV-pump to generate signal and idler in the visible range. Since the specimen is illuminated by UV-light with very high power, there is very high chances of damaging the specimen, i.e. the cells and tissues in the medical area. There are several other works on microscope to improve the resolving power and non-invasive features [6–8,1,2,9]. Their resolution is even smaller than the twin-photon confocal microscopes (reported by Simon et al.). But none of these confocal microscopes can achieve excellent resolution, noninvasive feature and high image contrast together. In this report, we wish to propose a confocal microscope capable of achieving very high resolving power (i.e. super-resolution) without any destruction of cells and tissues by using twin-photon two color confocal microscopy phenomenon. Additionally, this microscopy system may produce very high image contrast.
E-mail address: [email protected]. https://doi.org/10.1016/j.ijleo.2019.163209 Received 12 May 2019; Accepted 12 August 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 198 (2019) 163209
S. Karmakar
Fig. 1. Schematic of proposed experimental configuration. PBS represents polarized beam splitter.
2. Proposed method The idea of twin-photon confocal microscopy is used here. To avoid the chances of damaging the specimen, the specimen is used after the non-linear crystal. In this proposed experiment, the specimen is illuminated by signal and idler photon pair. These signal and idler beams are on the visible and infrared ranges with low power. So this configuration can easily protect the specimen from destruction. In this case, these two different color signal and photons are used to improve the resolution. The schematic of this proposed experimental configuration is shown in Fig. 1. Here, the pump of 351 nm is used to generate signal (1178 nm) and idler (500 nm) pair by spontaneous parametric down-conversion (SPDC) process using a very thin BBO type II crystal. This 351 nm pump is generated from an Ar-ion laser. The output signal and idler beams are co-linear with the pump beam. The crystal is placed at the focal point of the pump beam. The specimen is placed immediately after the thin BBO crystal. The signal and idler photon pairs from the specimen are passed through the objective lens to a pinhole. This pinhole reduces the noise by blocking the noise from unnecessary rays from the specimen. These filtered pairs are then passed through a polarized beam splitter to separate the orthogonally polarized signal and idler photons. After the beam splitter, these pairs are collected by two photodetectors and the image is observed in pointby-point of the specimen in terms of coincidence counts between these two photodetectors. Since the image is formed in terms of coincidences, this confocal microscope provides a very high degree of image contrast. 3. Discussion In this configuration, a pump beam of beam waist wo and angular frequency ωp is focused on the face of a thin nonlinear crystal by ρ ) is placed where → ρ is the transverse using a lens of focal length f. On the other face of the crystal, a specimen of transmittance of t (→ position. The pump field inside the crystal can be written as − Ep (→ r , t ) = E0 e−i (ωp t + kp z ) e
where E0 is a constant and r02 =
|→ r⊥ |2 2r02 λp2 f 2 2π 2w02
under this experimental condition.
The outgoing two-photon state generated from the nonlinear crystal is defined as [5,10]
|ψ〉 = A
→
→
∫ d3 ks ∫ d3 ki δ (ωs + ωi − ωp) ∫0 →
→
→ −
× ei (kpz− ksz− kiz) ze−i ( ks ⊥+ ki ⊥)· r⊥ e † † × a→ a→ |0〉 ks
L
dz
|→ r⊥ |2 2r02
(1)
ki
where all the constants are absorbed in the constant A. The outgoing signal and idler field propagate s0 distances and then pass the objective lens of focal length f and again propagate sI 1 1 1 distance to reach the pinhole with satisfying the thin-lens condition of s + s = f . If the positive frequency field operator of the o I (+) → → † , then the field at the detector placed at a distance of so + sI is given by [5] outgoing beam at the crystal is Eˆ j ( r⊥ , t ) = E j(+) ( r⊥ , t ) aˆ → kj
(+) Eˆ j (→ r⊥ , t j ) =
∫
→′ k′j → 2 →′ → d3 k j e−i so | r⊥ | e−i k j ⊥· r⊥ e−iωj τ j
s →′ † r⊥ + o k j ⊥ ⎟⎞ E0′ aˆ → × somb ⎜⎛→ ′ kj ′j k ⎝ ⎠ where the propagation factor a confocal system is
(2) k′j →′ → →2 e−i so | r⊥ | e−i k j ⊥· r⊥ e−iωj τ j somb ⎛→ r
⎝
2
⊥
+
so →′ k ⎞, k ′j j ⊥
⎠
τj = t1 −
so + sI , c
j = s, i and E0′ is a constant.
Optik - International Journal for Light and Electron Optics 198 (2019) 163209
S. Karmakar
Here symbol subscript s and i represent signal and idler photons respectively. The image spot is at the pinhole and the detector is supposed to be at position of the pinhole. But for the ease of alignment and design, the signal and idler pair are separated immediately after the pinhole by a polarized beam splitter and then collected by two detectors 1 and 2. Here the signal and idler go to detector 1 and 2 respectively. Then the ‘s’ and ‘i’ can be replaced by ‘1’ and ‘2’ respectively. Now, the image forming correlation can be written as
C (τ1, τ2) = |A (τ1, τ2)|2
(3)
where A(τ1, τ2) is a coincidence amplitude. Without any specimen, the coincidence amplitude is as follows
A (τ1, τ2) =
∫ d2→r⊥ 〈0|E2(+) (→r⊥, τ2) E1(+) (→r⊥, τ1)|ψ〉
where E1(+) and E2(+) are the positive-frequency fields at the above detectors, 1 and 2. ψ is the outgoing state vector from the non-linear crystal. r⊥ + → ρ ) where → ρ is the scanning point The specimen is considered at output face of the crystal with its transmission function t (→ unit of the microscope. The image is formed in terms of coincidence counts between two detectors. After simplification [5,10], the coincidence amplitude is calculated as follows
∫ d2→r⊥ t (→r⊥ + →ρ ) 〈0|E2(+) (→r⊥, τ2) E1(+) (→r⊥, τ1)|ψ〉 = A1 Π(τ1 − τ2) e
i 2π | → r |2 so λp ⊥
4πa|→ ρ |⎞ somb ⎜⎛ ⎟ λ s c ⎠ s o ⎝
|→ ρ |2 4πa|→ ρ | ⎞ − 2r 2 0 × somb ⎜⎛ ⎟e ⎝ λi so c ⎠
(4)
where Π(τ1 − τ2) is unit step function in which nonzero value has in the range of 0 < τ1 − τ2 < DL, D is the inverse group velocity difference of orthogonally polarized signal and idler photons, λp is the wavelength of the pump beam and A1 is a constant. Here λs and λi are the wavelength of the signal and idler beam respectively and a is the radius of both lenses. Here the object is a point with r⊥ + → ρ ) = δ (2) (→ r⊥ + → ρ ). perfectly transmissive and hence the transmittance function can be treated as a delta function, i.e. t (→ After substituting the coincidence amplitude in Eq. (4) into Eq. (3), the image forming correlation of a point of the specimen is given by ρ |2 → − |→ 4πa|→ ρ |⎞ 2 ⎛ 4πa| ρ | ⎞ e r02 somb C = A2 somb2 ⎜⎛ ⎟ ⎜ ⎟ ⎝ λs so c ⎠ ⎝ λi so c ⎠
(5)
where A2 is constant and coincidence time window is DL. This is the coincidence time window in an ideal detection system. In a realistic situation, this coincidence time would increase due to the finite response time of the detection system. In the one-dimensional case, the calculated point-spread function for the twin-photon microscope can be written as follows
PSFtwin (y ) y2
4π ay ⎞ 4π ay ⎞ − r 2 somb2 ⎛ e 0 = somb2 ⎛ λ s c ⎝ s o ⎠ ⎝ λi so c ⎠ ⎜
⎟
⎜
⎟
(6)
where y = y1 − y2 is the relative position of the scanning detector. From Eq. (6), one can say that resolving power of the microscope could be increased by three ways: (1) by decreasing object distance, so, (2) by decreasing f-number, f/2a and (3) by increasing the wavelength ratio of signal and idler beam, λs/λi. For comparison purpose, the lenses with having a = f are considered. To achieve maximum resolution for this case, the pump size is considered as w0 = a . For example for twin-photon cases, so = sI = 2f is considered. Now the calculated point spread function for both degenerate and non-degenerate cases of twin-photon confocal and standard confocal are shown in Fig. 2. This figure shows that it is possible to improve the resolution by 2% by using two different colors of signal and idler photons with respect to the same color signal and photons. 1 1 1 Now I assume another twin-photon case with the condition of so = 1.5f and sI = 3f to satisfy the thin lens equation s + s = f . In o I this case acceptance angle of objective lens increases and as a consequence resolution is improved. This condition of the twin-photon case is shown in Fig. 3. For this case of two different colors, the resolution is improved by 4% than the same color signal and photons. Next another twin-photon confocal case is assessed where I imply the condition of so = 1.5f and sI = 3f as above. One more condition of λs = 1595 nm and λi = 450 nm with the pump of 351 nm is considered where the ratio of signal and idler wavelengths is increased than the previous case. The comparison of standard confocal, degenerate twin-photon and non-degenerate twin-photon cases are shown in Fig. 4. In this case, the resolution is improved by 6% than the degenerate twin-photon case with the same condition. For all the above cases, the f-number of the objective lens of 0.5, i.e. a = f, is assumed. If we can use a smaller f-number objective lens, then the acceptance angle of the objective lens is also increased. Now I use the f-number of the objective lens of 0.25. As a consequence, the resolution, in this case, is improved by 10% than the standard degenerate twin-photon confocal case with the same 3
Optik - International Journal for Light and Electron Optics 198 (2019) 163209
S. Karmakar
Fig. 2. Comparison of resolution for three cases: (1) standard confocal microscope, (2) twin-photon confocal microscope with degenerate case and (3) twin-photon confocal microscope with non-degenerate case.
Fig. 3. Comparison of resolution for three cases with higher acceptance angle of objective lens by decreasing the object distance in twin-photon case.
4
Optik - International Journal for Light and Electron Optics 198 (2019) 163209
S. Karmakar
Fig. 4. Comparison of resolution for three cases with higher acceptance angle of objective lens by decreasing the object distance and higher ratio of signal and idler wavelength in twin-photon case.
Fig. 5. Comparison of resolution for three cases with higher acceptance angle of objective lens by decreasing both the object distance and the fnumber of the lens and also with higher ratio of signal and idler wavelength in twin-photon case. 5
Optik - International Journal for Light and Electron Optics 198 (2019) 163209
S. Karmakar
condition. This is shown in Fig. 5. From the above results of the non-degenerate case, one can say that the resolution can be improved further by increasing the signal and idler wavelength ratio and also by increasing the acceptance angle of the objective lens. The resolution shown in Fig. 2 in the degenerate case, 350 nm, is improved to the resolution of 315 nm described in the non-degenerate case of Fig. 5. The resolution is determined by measuring the full-width-half-maximum (FWHM) of the peaks presented in Figs. 2 and 5. Here the specimen is illuminated by signal idler photons which have very low intensity and also the signal and idler beams are either visible or infrared regime. Here, CW laser is used to generate signal and idler pair through SPDC process by using a nonlinear crystal. As a result, the specimen is illuminated by very low power. Hence the specimen is very safe during the process. In coincidence detection system, the simultaneously generated signal and idler pair are collected from the point of interest. That means the unwanted photons other than signal-idler photons are not collected. By discarding these noise producing unwanted photons in a coincidence system, the image contrast is improved. In addition, the image contrast is also improved by blocking unwanted noise producing photons using a pinhole in a confocal system. In this configuration, the confocal system and coincidence measurement together may provide very high image contrast. 4. Conclusion From this study, one may say that this proposed configuration promises to have a confocal microscope with three important features:(1) excellent resolution of approximately 315 nm, (2) noninvasive and (3) very high image contrast. By increasing the signal and idler wavelength ratio and also by increasing the acceptance angle of the objective lens, the resolution can be improved further. This proposed microscope might be very important to advance the progress of research and development as well as technology. Conflict of interest None. Acknowledgements The authors thank Y. Shih, R. Meyers, K. Deacon and A. Tunick for helpful discussions with them. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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