1 INTRODUCTION

Optical coherence tomography (OCT) [1] is actively used in various medical diagnostic fields because it ensures the noninvasive study of the properties of transparent objects. A typical OCT system based on low-coherent (or spectral-domain (SD)) interferometry [2, 3] has the structure of a Michelson interferometer where one arm is reference and the second arm is formed by reflection and scattering of light in an object under study.

Optical coherence tomography images are formed from so-called A-scans obtained either by scanning the difference between the arm lengths at a small coherence length of a source (time-domain OCT) or by taking the Fourier transform of the measured dependence of the intensity of an interference signal on the wavelength of light (SD OCT). The authors of [4, 5] showed that SD OCT is more stable with respect to noise and more sensitive than time-domain OCT.

Since OCT measurements are based on the interference between the reference wave and the wave passed through the studied sample, the chromatic dispersion of the sample at a wide spectrum of the waves results in an inhomogeneous frequency dependence of the optical path difference, which reduces the longitudinal spatial resolution [6].

The quantum optical version of OCT was proposed and implemented for the first time in [7, 8] and attracted the wide attention of numerous research groups because of the improved resolution, compensation of the effect of chromatic dispersion of the sample under study, and a low intensity of probe radiation. Quantum OCT is based on the interference of biphotons [9–12], which is often called the Hong–Ou–Mandel effect and allows one to determine the optical path difference between two arms of the interferometer.

In the scheme of the experimental setup, the quantum OCT technique is equivalent to classical time-domain OCT because the position of the reflector is localized inside the sample by the mechanical scanning of the reference mirror. Since the coherence length of the biphoton is short, interference occurs only when the optical paths in the arms coincide with each other, which is manifested as a peak or a dip in the measured dependence of the joint photon detection probability (JPDP) on the difference of the arm lengths. The main demerit of this approach, as well as time-domain OCT, is the necessity of the introduction of mechanically scanned elements in the setup, which reduces the accuracy and speed of measurements.

This work is devoted to the theoretical justification and analysis of a new quantum OCT approach, which can be implemented without any moving parts and involves the spectral tuning of a source of biphotons. To this end, we derive an expression for the JPDP as a function of spectral properties of the biphoton under the condition that one of the photons passes through the studied sample placed in the signal arm of the interferometer, as shown in Fig. 1.

Fig. 1.
figure 1

(Color online) Layout of the experimental setup for quantum spectral-domain optical coherent tomography.

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2 BIPHOTON INTERFERENCE SIGNAL

The theoretical analysis in this section is performed by analogy with [13, 14]. According to the model used in those works, the joint spectral amplitude (JSA) of the frequency nondegenerate biphoton in the case of the Gaussian spectrum can be described by the expression

$$\begin{array}{*{20}{c}} {f({{\Omega }_{1}},{{\Omega }_{2}}) = \frac{2}{{\sqrt {2\pi {{\sigma }_{{\text{a}}}}{{\sigma }_{{\text{d}}}}} }}} \\ \begin{gathered} \times \;\exp \left[ { - {{{\left( {\frac{{({{\Omega }_{1}} - {{\omega }_{{01}}}) - ({{\Omega }_{2}} - {{\omega }_{{02}}})}}{{\sqrt 2 {{\sigma }_{{\text{a}}}}}}} \right)}}^{2}}} \right. \\ \left. { - \;{{{\left( {\frac{{({{\Omega }_{1}} - {{\omega }_{{01}}}) + ({{\Omega }_{2}} - {{\omega }_{{02}}})}}{{\sqrt 2 {{\sigma }_{{\text{d}}}}}}} \right)}}^{2}}} \right], \\ \end{gathered} \end{array}$$
(1)

where \({{\Omega }_{1}}\) and \({{\Omega }_{2}}\) are the variables responsible for the fluctuation of frequencies of photons from pair to pair, \({{\omega }_{{01}}}\) and \({{\omega }_{{02}}}\) are the central frequencies of the spectral distributions of entangled photons, and σd and σa are the diagonal and antidiagonal widths of the JSA, respectively. The Gaussian shape of the JSA corresponds well to experimental results for biphoton spectra [13–15].

However, in the case of a significant difference between the central frequencies of biphotons \({\text{|}}{{\omega }_{{01}}} - {{\omega }_{{02}}}{\text{|}} = \delta \gg {{\sigma }_{{\text{a}}}}\) in the state described by Eq. (1), any uncertainty is absent (photons with higher and lower frequencies are present in certain optical modes), which makes the interference of such state impossible. The interference becomes possible after the symmetrization of the JSA with respect to the \({{\Omega }_{1}} = {{\Omega }_{2}}\) straight line; one of the variants of this state is described by the formula

$$g({{\Omega }_{1}},{{\Omega }_{2}}) = f({{\Omega }_{1}},{{\Omega }_{2}}){\text{/}}\sqrt 2 + f({{\Omega }_{2}},{{\Omega }_{1}}){\text{/}}\sqrt 2 .$$
(2)

Such states were implemented experimentally in [14, 15] using spontaneous parametric down-conversion and special polarization schemes.

The reflection properties of the sample strongly affect the interference signal in applications of OCT. The complex reflection coefficient for the sample with two reflecting interfaces between layers can be represented in the form

$$H(\omega ,T) = {{r}_{1}} + {{r}_{2}}\exp (2i\omega T),$$
(3)

where \({{r}_{1}}\) and \({{r}_{2}}\) are the reflection coefficients of the first and second layers of the sample for the electric field and \(T = nL{\text{/}}c\) is the time delay of light in the path between the interfaces (n and L are the refractive index and geometric thickness of the medium between the reflecting interfaces, respectively, and c is the speed of light in vacuum).

The joint photon detection probability in the experimental configuration shown in Fig. 1 is determined by the interference of the biphoton and, by analogy with the frequency degenerate case [13], can be represented in the form

$$\begin{gathered} C(\tau ,T) = \frac{1}{4}\int \int \left| {g({{\Omega }_{1}},{{\Omega }_{2}})H({{\Omega }_{2}},T)\exp \left( {i{{\Omega }_{1}}\tau } \right)} \right. \\ - \;{{\left. {g({{\Omega }_{2}},{{\Omega }_{1}})H({{\Omega }_{1}},T)\exp \left( {i{{\Omega }_{2}}\tau } \right)} \right|}^{2}}d{{\Omega }_{1}}d{{\Omega }_{2}}, \\ \end{gathered} $$
(4)

where τ is the difference between time delays of light in the arms in the absence of the sample in the signal arm of the interferometer.

The substitution of Eqs. (1)–(3) into Eq. (4) and integration yields

$$C(\tau ,T) = ({{r}_{1}}^{2} + {{r}_{2}}^{2})(1 + \exp ( - {{\delta }^{2}}{\text{/}}\sigma _{{\text{a}}}^{2})){\text{/}}2$$
$$ - \;{{r}_{1}}^{2}\exp \left[ { - \frac{{{{\sigma }_{{\text{a}}}}^{2}{{\tau }^{2}}}}{4}} \right]\cos \left( {\delta \tau } \right){\text{/}}2$$
$$ - \;r_{2}^{2}\exp \left[ { - \frac{{{{{(2T - \tau )}}^{2}}{{\sigma }_{{\text{a}}}}^{2}}}{4}} \right]\cos \left( {\delta \left( {2T - \tau } \right)} \right){\text{/}}2$$
$$ + \;{{r}_{1}}{{r}_{2}}\exp \left[ { - \frac{{{{T}^{2}}\left( {\sigma _{{\text{a}}}^{2} + \sigma _{{\text{d}}}^{2}} \right)}}{4}} \right]\cos \left( {2{{w}_{0}}T} \right)$$
$$ \times \;\left[ {\cos \left( {2\delta T} \right) + \exp \left( { - \frac{{{{\delta }^{2}}}}{{\sigma _{{\text{a}}}^{2}}}} \right)} \right]$$
$$ - \;{{r}_{1}}{{r}_{2}}\exp \left[ { - \frac{{{{{(T - \tau )}}^{2}}\sigma _{{\text{a}}}^{2}}}{4} - \frac{{{{T}^{2}}\sigma _{{\text{d}}}^{2}}}{4}} \right]\cos \left( {2{{w}_{0}}T} \right)$$
$$ \times \;\left[ {\cos \left[ {\delta \left( {T - \tau } \right)} \right] + \exp \left( { - \frac{{{{\delta }^{2}}}}{{\sigma _{{\text{a}}}^{2}}}} \right)} \right],$$
(5)

where \({{\omega }_{0}} = ({{\omega }_{{01}}} + {{\omega }_{{02}}}){\text{/}}2\).

If the sample has chromatic dispersion, Eq. (3) can be written in the form

$$H(\omega ,T) = {{r}_{1}} + {{r}_{2}}\exp (2i\omega T + i{{\omega }^{2}}t{\text{/}}2),$$
(6)

where \(t = \beta {\kern 1pt} ''2L\) is the group delay dispersion determined by the thickness of the considered layer L and the group velocity dispersion \(\beta {\kern 1pt} ''\) in this layer. After the substitution of Eq. (6) into (4), it is seen that inte-rference components include the products \(H({{\Omega }_{{1,2}}},T)H{\text{*}}({{\Omega }_{{1,2}}},T)\), which can be represented in the  form \(H(\omega + {{\Omega }_{{\text{a}}}},T)H{\text{*}}(\omega - {{\Omega }_{{\text{a}}}},T)\), where Ωa = \(({{\Omega }_{1}} - {{\Omega }_{2}}){\text{/}}\sqrt 2 \), taking into account the anticorrelation between the frequencies of entangled photons (caused by the energy conservation law at their generation). Similar to [7], after the corresponding transformations in Eq. (6), it is seen that the term in the integrand in Eq. (4) corresponding to the interference of biphoton has the form

$$\begin{gathered} S = r_{1}^{2}\cos \left( {{{\Omega }_{{\text{a}}}}\tau } \right) \\ + \;r_{2}^{2}\cos \left[ {{{\Omega }_{{\text{a}}}}(2T - \tau ) + {{\Omega }_{{\text{a}}}}({{\omega }_{0}} + {{\Omega }_{{\text{d}}}})t{\text{/}}2} \right], \\ \end{gathered} $$
(7)

where \({{\Omega }_{{\text{d}}}} = ({{\Omega }_{1}} + {{\Omega }_{2}}){\text{/}}\sqrt 2 \). The integral of the product of S and the JSA g specified by Eq. (2) is given by the expression

$$S = r_{1}^{2}{{A}_{1}}\cos \left( {\delta \tau } \right) + r_{2}^{2}{{A}_{2}}\cos \left[ {\frac{{4\delta \left( {2T - \tau + {{\omega }_{0}}t{\text{/}}2} \right)}}{{\sigma _{{\text{a}}}^{2}\sigma _{{\text{d}}}^{2}{{t}^{2}} + 4}}} \right],$$
(8)

where \({{A}_{1}}\) and \({{A}_{2}}\) are the factors that determine the contrast of the interference signal and depend on the spectral properties of the biphoton. Since the interference components do not contain harmonic functions whose arguments include \({{\delta }^{2}}\), parasitic frequency modulation does not occur and the spatial resolution is not reduced. At the same time, the parasitic amplitude modulation of the interference components can also worsen the components of the A-scan and, thereby, reduce the spatial resolution. The factor \({{A}_{2}}\) is given by the expression

$${{A}_{2}} = \frac{{\exp \left[ { - \sigma _{{\text{a}}}^{2}{{{(2T - \tau + {{\omega }_{0}}t{\text{/}}2)}}^{2}}} \right]}}{{\sqrt {\sigma _{{\text{a}}}^{2}\sigma _{{\text{d}}}^{2}{{t}^{2}} + 4} }}\exp \left[ {\frac{{ - {{\delta }^{2}}\sigma _{{\text{d}}}^{2}{{t}^{2}}{\text{/}}2}}{{\sigma _{{\text{a}}}^{2}\sigma _{{\text{d}}}^{2}{{t}^{2}} + 4}}} \right].$$
(9)

According to Eq. (9), chromatic dispersion can reduce the resolution of quantum SD OCT because of the amplitude modulation of the interference components, particularly at large Δ and σd values. A decrease in the efficiency of dispersion compensation in traditional quantum OCT in the case of a wide spectrum of the biphoton was discussed in [16]. We consider below conditions under which dispersion is compensated in the proposed quantum SD OCT approach.

3 ANALYSIS OF THE TARGET AND PARASITIC COMPONENTS OF THE BIPHOTON INTERFERENCE SIGNAL

As seen in Eq. (5), the interference signal in quantum SD OCT includes several terms with different physical meanings. The first term corresponds to the constant component of the signal and does not carry any information on the structure of the sample. The second and third terms are interference and correspond to the reflection of the signal photon from the first and second interfaces of the sample, respectively. These terms as functions of the difference between the central frequencies of photons δ are harmonic oscillations with frequencies directly related to the interferometer optical phase difference. Consequently, these two terms carry information on the internal structure of the sample.

The last two composite terms are also related to the structure of the sample, but they are due to the possibility of the reflection of one photon from different interfaces because the amplitudes of these terms include the product of the reflection coefficients of the layer \({{r}_{1}}{{r}_{2}}\). Thus, these terms are due to the interference of the photon signal in the interferometer formed by the layer of the sample. The same with parasitic interference components occurs in quantum time-domain OCT [9, 13]. Therefore, for the optimal application of the proposed quantum SD OCT approach, it is necessary to examine conditions under which the amplitudes of the second and third terms are maximal, whereas the amplitudes of the fourth and fifth terms are minimal.

The amplitudes of the target (second and third) terms include the reflection coefficients of the corresponding layers and exponential factors \(\exp ( - {{\tau }^{2}}\sigma _{{\text{a}}}^{2}{\text{/}}4)\). Thus, for the correct detection of the biphoton interference signal, the absolute values of the exponents should be small. This requirement imposes the following constraint on the maximum optical thickness of the sample:

$${{T}_{{{\text{max}}}}} < 2{\text{/}}{{\sigma }_{{\text{a}}}}.$$
(10)

According to Eq. (10), the antidiagonal width of the JSA determines the maximum depth of the resulting OCT image. Thus, the second-order coherence length of the biphoton is determined by the antidiagonal width of its JSA.

The parasitic (fourth and fifth) terms in Eq. (5) contain the main factor \({{r}_{1}}{{r}_{2}}\) and the exponential factors \(\exp ( - {{T}^{2}}\sigma _{{\text{d}}}^{2}{\text{/}}4)\). Consequently, a sufficient condition for the suppression of parasitic component is a relatively large negative exponent. As a result, the minimum distance between the layers of the sample at which parasitic interference components do not appear is limited from below by the diagonal width of the JSA and is given by the expression

$${{T}_{{{\text{min}}}}} > 2{\text{/}}{{\sigma }_{{\text{d}}}}.$$
(11)

At the same time, according to Eq. (9), the efficiency of compensation of chromatic dispersion decreases with an increase in the diagonal width of the JSA. The term \({{\omega }_{0}}t{\text{/}}2\) in the argument of the harmonic function in Eq. (8) and the first exponential factor in Eq. (9) only shifts the corresponding components of the A-scan, similar to classical interferometry [17], and does not change the constraint (10) or (11). To compensate dispersion, the term \(\sigma _{{\text{a}}}^{2}\sigma _{{\text{d}}}^{2}{{t}^{2}}\) in the radicand in Eq. (9), as well as the numerator \({{\delta }^{2}}\sigma _{{\text{d}}}^{2}{{t}^{2}}{\text{/}}2\) in the second exponent in Eq. (9), should be small. In the proposed quantum SD OCT approach, δ can be much larger than σa; therefore, the latter condition is more stringent than the former and is also responsible for the absence of the broadening of the component of the A-scan because of the parasitic amplitude modulation. Simplifying the second exponent in Eq. (9) to the form \( - {{\delta }^{2}}\sigma _{{\text{d}}}^{2}{{t}^{2}}{\text{/}}8\) and setting the absolute value of its maximum allowed value to 1, we obtain the constraint on the group delay in the form

$${\text{|}}t{\text{|}} < \frac{{2\sqrt 2 }}{{{{\sigma }_{{\text{d}}}}{{\Delta }_{{\text{M}}}}}},$$
(12)

where ΔM is the maximum difference between the central frequencies of the photons.

4 ADDITIONAL REQUIREMENTS ON SCANNING OF THE DIFFERENCE BETWEEN THE FREQUENCIES OF THE PHOTONS

The parameters of the frequency scanning of the JSA also affect the characteristics of the measured biphoton interference signal. To implement the proposed OCT approach, the difference of the central frequencies of the photons δ should be controllably varied in a certain range \([{{\delta }_{0}},{{\delta }_{0}} + \Delta ]\). The [–ΔM, ΔM] scanning regime is the most favorable for the minimization of the maximum difference of the central frequencies of the photons. According to the general theory of OCT following from the theory of spectral estimates, the minimum resolvable time delay is related to the spectral width of the source used [6] and is expressed in terms of the FWHM of the A-scan component for the proposed approach as

$${{T}_{{{\text{min}}}}} > 2\pi \times 1.2{\text{/}}\Delta = 1.2\pi {\text{/}}{{\Delta }_{{\text{M}}}}.$$
(13)

Thus, taking into account that layers separated by the minimum resolvable distance should not generate parasitic interference components, the relation between the diagonal width of the JSA and the width of the scanning range of the difference of the frequencies of the photons is obtained from Eqs. (11) and (13) in the form \({{\Delta }_{{\text{M}}}} \lesssim {{\sigma }_{{\text{d}}}}\).

At the same time, since information on the optical thicknesses of the layers of the sample is carried in frequencies of oscillatory components of the signal (5), the conditions of the measurement of the JPDP should allow one to detect the frequencies of these oscillatory components. To this end, Kotel’nikov conditions relating the maximum frequency of oscillations to the discretization interval should be satisfied:

$${{\delta }_{{\text{S}}}} < \pi {\text{/}}{{T}_{{{\text{max}}}}}.$$
(14)

We emphasize that a particular \({{\delta }_{{\text{S}}}}\) value should be chosen such that an instrument should not detect all i-nterference components that do not meet the condition (14) because they will demonstrate the aliasing effect. Correspondingly, the condition (10) should also be not satisfied for these components. Consequently, the antidiagonal width of the JSA and the discretization interval of the difference of the central frequencies of the photons are related as \({{\sigma }_{a}} \gg {{\delta }_{{\text{S}}}}\).

5 LIMITS OF THE SPATIAL RESOLUTION

Expressions (11)–(13) relate the achievable spatial resolution, the diagonal width of the JSA, and the variation range of the central frequencies. The relation \({{\sigma }_{{\text{d}}}} = {{\Delta }_{{\text{M}}}}2{\text{/}}\pi \) between the minimum values of the last two parameters follows from Eq. (11) and (12). As a result, \({{\Delta }_{{\text{M}}}} < \sqrt {\sqrt 2 \pi /{\text{|}}t{\text{|}}} \), and the dependence of the maximum achievable spatial resolution on the dispersion of the group delay is obtained in the form

$$\Delta L = c{{T}_{{{\text{min}}}}} = c\sqrt {(1.2\pi {\text{/}}\sqrt 2 ){\text{|}}t{\text{|}}} .$$
(15)

We compare this relation with constraints on the resolution of traditional quantum OCT and classical OCT. The authors of [13, 18] presented the following relations between the spatial resolution and the parameters of the sample and biphoton:

$$\Delta L = c{{T}_{{{\text{min}}}}} > 2c{\text{/}}{{\sigma }_{{\text{d}}}},$$
(16)
$$\Delta L > 2\sqrt {2\ln (2)} c{\text{/}}{{\sigma }_{{\text{d}}}}\sqrt {1 + {{t}^{2}}\sigma _{{\text{d}}}^{2}{\text{/}}6} ,$$
(17)

where it is assumed in Eq. (17) that the JSA is axisymmetric and its width is equal to the half-width of the pump spectrum and a factor of \(2\sqrt {2\ln (2)} \) is added in order to obtain the FWHM instead of the rms width. The following constraint on the maximum achievable spatial resolution is obtained from Eqs. (16) and (17):

$$\Delta L = c{{T}_{{{\text{min}}}}} = c\sqrt {16\ln (2){\text{/}}\sqrt 6 {\text{|}}t{\text{|}}} .$$
(18)

The spatial resolution for the sample with dispersion in classical OCT is specified by the expression [6]

$$\Delta L = \sqrt {{{{\left( {\frac{{2\ln (2)\lambda _{0}^{2}}}{{\pi \Delta \lambda }}} \right)}}^{2}} + {{{(2\pi {{c}^{2}}{\text{|}}t{\text{|}}\Delta \lambda {\text{/}}\lambda _{0}^{2})}}^{2}}} .$$
(19)

For the optimal width of the spectrum [6]

$$\Delta \lambda = \lambda _{0}^{2}{\text{/}}(\pi c)\sqrt {\ln (2){\text{/|}}t{\text{|}}} ,$$
(20)

Eq. (19) takes the form

$$\Delta L = c\sqrt {8\ln (2){\text{|}}t{\text{|}}} .$$
(21)

The analytical expressions for the spatial resolution have the same common form \(\Delta L = kc\sqrt {{\text{|}}t{\text{|}}} \) for all co-nsidered approaches and differ only in numerical factor k: k ≈ 2.35, 2.13, and 1.63 for classical OCT, quantum time-domain OCT, and proposed quantum SD OCT, respectively. The advantage of the proposed quantum SD OCT approach is that the effective width of the spectrum, which is used to estimate the difference of optical paths, is doubled because of the scanning of the difference of the central frequencies of the photons in both directions with respect to the frequency degenerate state.

6 EXAMPLES OF BIPHOTON INTERFERENCE SIGNALS

We consider the effect of conditions (10) and (11) on the resulting interference signal with the reflection coefficients of the layers \({{r}_{1}} = 0.4\) and \({{r}_{2}} = 0.4\), time delays of light \(T = 7\) ps and \(\tau = - 10\) ps (i.e., the re-ference arm is shorter than the signal), ω0 = \(2\pi \times 2.3 \times {{10}^{{14}}}\) rad/s (corresponding to a wavelength of 1.3 μm often used in OCT), \(\Delta = 2\pi \times 5 \times {{10}^{{11}}}\) rad/s, and \({{\delta }_{{\text{S}}}} = 2\pi \times 5 \times {{10}^{9}}\) rad/s.

Figure 2 shows the (a) interference signal, (b) corresponding A-scan, and (c) JSA in the case where Eqs. (10) and (11) are violated. This case corresponds to the coinciding diagonal and antidiagonal widths of the JSA σa = σd = 2π × 2 × 1010 rad/s. The substitution of these parameters and delays \(2T\) and \(2T - \tau \) into Eqs. (10) and (11) indicates that both conditions are not satisfied, which is manifested both in the appearance of parasitic components corresponding to delays of 14 and 17 ps and in the noticeable decrease in the amplitude of the second target component.

Fig. 2.
figure 2

(Color online) (a) Biphoton interference signal, (b) the A-scan (squares mark target components), and (c) the corresponding joint spectral amplitude when conditions (10) and (11) are not satisfied.

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Figure 3 shows the (a) interference signal, (b) A‑scan, and (c) JSA in the case where Eqs. (10) and (11) are met. This case corresponds to σa = 2π × 5 × 109 rad/s and σd = 2π × 2 × 1011 rad/s. It is seen that the calculated A-scan certainly corresponds to the model structure of the sample.

Fig. 3.
figure 3

(Color online) (a) Biphoton interference signal, (b) the A-scan, and (c) the corresponding joint spectral amplitude under conditions (10) and (11).

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Biphoton interference signals and corresponding A-scans were also calculated for the case of the chromatic dispersion of the optical delay both by Eq. (8) with the parameters σa = 2π × 2 × 1010 rad/s and σd = 2π × 6 × 1012 rad/s and by numerical calculation of the integral in Eq. (4) with the parameters \({{\omega }_{0}} = 2\pi \times 2.3 \times {{10}^{{14}}}\) rad/s, \({{\Delta }_{{\text{M}}}} = 2\pi \times 10 \times {{10}^{{12}}}\) rad/s, \({{\delta }_{{\text{S}}}} = 2\pi \times 5 \times {{10}^{{10}}}\) rad/s, and \(\delta \omega = 2\pi \times 2 \times {{10}^{9}}\) rad/s (step of photon frequency grids).

The first layer of the sample under study was simulated by a water layer 25 mm in thickness, which approximately corresponds to the depth of a human eye; the next two layers had a thickness of 26 μm and the refractive index \({{n}_{{2,3}}} = 1.46\); the reflection coefficients of the interfaces between the layers were \({{r}_{1}} = {{r}_{2}} = {{r}_{3}} = 0.3\); and the reference arm of the interferometer (air) had a length of 46.2 mm. The group velocity dispersion in the considered spectral range for water is about 100 fs2/mm, and the group delay dispersion in the sample is \(t = 2500\) fs2.

To demonstrate the effect of dispersion, the interference signals were simulated both with the real frequency dependence of the refractive index of water \(n(\omega )\) and with \(n({{\omega }_{0}})\). The interference signals for the case of classical interference with the same parameters of the sample were calculated by the formula \(S = {\text{|}}\exp (i\omega \tau ) + H(\omega ,T){{{\text{|}}}^{2}}\). Since the spectrum of the source was assumed uniform rather than Gaussian as    in [6], \(\omega \) was varied in the range \([{{\omega }_{0}} - {{\Delta }_{{\text{M}}}}{\text{/}}2,{{\omega }_{0}} + {{\Delta }_{{\text{M}}}}{\text{/}}2]\), whose effective width is close to the optimal width of the Gaussian spectrum calculated by Eq. (20).

The calculations by Eq. (8) and the numerical integration of Eq. (4) for biphoton interference gave identical results. When calculating A-scans, the interference signal was tenfold zero-padded for a clearer estimate of the width of A-scan components.

According to Eq. (15), the highest longitudinal spatial resolution for the chosen sample is 24 μm with the corresponding optimal parameters ΔM and σd. As seen in Fig. 4, reflectors in the sample can be resolved using the proposed quantum SD OCT approach and cannot be resolved with classical OCT, whose highest longitudinal spatial resolution is about 39 μm.

Fig. 4.
figure 4

(Color online) Effect of chromatic dispersion on A-scans in the case of classical and quantum optical coherence tomography.

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As seen in Fig. 4, under the formulated conditions on the parameters of the sample under study and the JSA, the proposed quantum SD OCT provides a higher spatial resolution compared to classical OCT. The shift of A-scans for the case with dispersion is due to the term \({{\omega }_{0}}t{\text{/}}2\) in the argument of interference components. The presented A-scans also demonstrate that biphoton interferometry makes it possible to suppress parasitic interference components caused by interference of photons reflected from the layers of the sample under study (marked by the cross in Fig. 4).

7 DISCUSSION AND CONCLUSIONS

The proposed quantum SD OCT approach is equivalent to alternative biphoton interferometry and OCT approaches, where the dependence of the JPDP on the difference of the frequencies of the photons is determined by measuring these frequencies [19–22]. The dependence of the JPDP on the difference of their central frequencies was also reported in [23], but it was discussed only theoretically. The maximum difference between the frequencies of photons in the case of a constant shape of the JSA (which is usually considered axisymmetric unlike this work) is determined by its width \({{\sigma }_{\omega }}\), the minimum frequency difference can be taken equal to zero, and the frequencies of both photons are measured with a certain error \(\delta \omega \). Thus, clear correspondence exists between the proposed scheme with frequency scanning and the existing schemes with the measurement of the frequencies of photons. Theoretical results reported in this work using the approach with frequency scanning can also be applied to the approach with the measurement of frequencies with the substitution of \({{\sigma }_{\omega }}\) for \(\Delta \), \({{\delta }_{0}} = 0\), and \(\delta \omega \) for σa and σd.

Nevertheless, an equivalent of the JSA with different diagonal and antidiagonal widths cannot be implemented in practical measurements of frequencies of photons, which limits the applicability of the approach [20] for OCT. At the same time, to reach spectral correlations between entangled photons required in our approach, femtosecond pump pulses should be used and the phase matching conditions between the pump pulses and signal and idler photons in a narrow frequency range are necessary. The possibility of the experimental implementation of these conditions and the generation of biphotons with a significantly axi-asymmetric JSA was demonstrated in [24, 25].

Owing to the possibility of the independent control of the first- and second-order coherence in the biphoton, the proposed quantum SD OCT approach with frequency scanning can be effectively used instead of the existing OCT techniques in applications requiring a high spatial resolution, maximum suppression of parasitic interference components, compensation of dispersion, and a low intensity of probe light waves. It has been shown theoretically that the proposed approach can ensure the best longitudinal spatial resolution among the known OCT techniques in the absence of the compensation of dispersion by additional elements in the reference arm.