Rapid communication: $$K_{S}^{0}$$ Production from beryllium target using 120 $$\hbox {GeV}/\hbox {c}$$GeV/c protons beam interactions at the MIPP experiment

We have measured the cross-section for the \(K_{S}^{0}\) production from beryllium target using 120 \(\hbox {GeV}/\hbox {c}\) protons beam interactions at the main injector particle production (MIPP) experiment at Fermilab. The data were collected with target having a thickness of 0.94% of the nuclear interaction length. The \(K_{S}^{0}\) inclusive differential cross-section in bins of momenta is presented covering momentum range from \(0.4\,\hbox {GeV}/\hbox {c}\) to \(30\,\hbox {GeV}/\hbox {c}\). The measured inclusive \(K_{S}^{0}\) production cross-section amounts to \(39.54\pm 1.46\delta _{\mathrm {stat}}\pm 6.97\delta _{\mathrm {syst}}\) mb and the value is compared with the prediction of FLUKA hadron production model.     

Spin-label Order Parameter Calibrations for Slow Motion

Calibrations are given to extract orientation order parameters from pseudo-powder electron paramagnetic resonance line shapes of ¹⁴N-nitroxide spin labels undergoing slow rotational diffusion. The nitroxide z-axis is assumed parallel to the long molecular axis. Stochastic-Liouville simulations of slow-motion 9.4-GHz spectra for molecular ordering with a Maier–Saupe orientation potential reveal a linear dependence of the splittings, \(2A_{\hbox{max} }\) and \(2A_{\hbox{min} }\), of the outer and inner peaks on order parameter \(S_{zz}\) that depends on the diffusion coefficient \(D_{{{\text{R}} \bot }}\) which characterizes fluctuations of the long molecular axis. This results in empirical expressions for order parameter and isotropic hyperfine coupling: \(S_{zz} = s_{1} \times \left( {A_{\hbox{max} } - A_{\hbox{min} } } \right) - s_{o}\) and \(a_{o}^{{}} = \tfrac{1}{3}\left( {f_{\hbox{max} } A_{\hbox{max} } + f_{\hbox{min} } A_{\hbox{min} } } \right) + \delta a_{o}\), respectively. Values of the calibration constants \(s_{1}\), \(s_{\text{o}}\), \(f_{\hbox{max} }\), \(f_{\hbox{min} }\) and \(\delta a_{o}\) are given for different values of \(D_{{{\text{R}} \bot }}\) in fast and slow motional regimes. The calibrations are relatively insensitive to anisotropy of rotational diffusion \((D_{{{\text{R}}//}} \ge D_{{{\text{R}} \bot }} )\), and corrections are less significant for the isotropic hyperfine coupling than for the order parameter.     

First-principle calculations of structural,electronic, optical,elastic and thermal properties of $$\hbox {MgXAs}_{2}$$ ($$\hbox {X}=\hbox {Si}, \hbox {Ge}$$X=Si,Ge) compounds

First-principle calculations on the structural, electronic, optical, elastic and thermal properties of the chalcopyrite \(\hbox {MgXAs}_{2}\) (\(\hbox {X}=\hbox {Si}, \hbox {Ge}\)) have been performed within the density functional theory (DFT) using the full-potential linearized augmented plane wave (FP-LAPW) method. The obtained equilibrium structural parameters are in good agreement with the available experimental data and theoretical results. The calculated band structures reveal a direct energy band gap for the interested compounds. The predicted band gaps using the modified Becke–Johnson (mBJ) exchange approximation are in fairly good agreement with the experimental data. The optical constants such as the dielectric function, refractive index, and the extinction coefficient are calculated and analysed. The independent elastic parameters namely, \(C_{11}\), \(C_{12}\), \(C_{13}\), \(C_{33}\), \(C_{44}\) and \(C_{66 }\) are evaluated. The effects of temperature and pressure on some macroscopic properties of \(\hbox {MgSiAs}_{2}\) and \(\hbox {MgGeAs}_{2}\) are predicted using the quasiharmonic Debye model in which the lattice vibrations are taken into account.     

Republication of: Mean density of radiation in the Metagalaxy and certain problems in relativistic cosmology

This paper appeared in Russian in Doklady AN SSSR and in English in the translation journal Soviet Physics–Doklady. In it the cosmic radiation from galaxies in the frequency range \(10^8 \hbox {Hz}\) to \(10^{15} \hbox {Hz}\) was estimated and shown to agree with data. It was then shown, following earlier ideas of Gamow, that the black-body cosmic microwave background from a big bang would significantly exceed the galactic radiation in the range \(10^9 \hbox {Hz}\) to \(10^{11} \hbox {Hz}\). This was the first accurate prediction of that background, which now plays a key role in cosmology. The first announcement of its detection came in the following year. The paper has been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series of the journal. This republication is accompanied by an editorial note written by R. Durrer and a brief autobiography by A. G. Doroshkevich: an autobiography of I. D. Novikov appeared with a previous Golden Oldie.     

Design of offset core photonic crystal fiber filter based on surface plasmon resonance

The novel offset core photonic crystal fiber filter is designed and analyzed, whose dispersion relations and polarization characteristics are simulated by finite element method using COMSOL Multiphysics software. The filter structure is optimized by changing diameter of air holes and the thickness of Au layer. Simulation results show that loss of y-polarized mode reaches 657 dB/cm while the loss of x-polarized mode is very low at the communication window (\(1.55\,\upmu \hbox {m}\)). The crosstalk of filter reaches 56.2 dB at \(1.55\,\upmu \hbox {m}\) wavelength and the 20 dB band width of the filter is 100 nm when the propagation distance is \(1\,\upmu \hbox {m}\). Not only the filter shows good performance but also the proposed photonic crystal fiber can be applied to other fields.     

Applicability of strange nonchaotic Wien-bridge oscillators for secure communication

A numerical analysis of ion cyclotron resonance heating scenarios in two species of low ion temperature plasma has been done to elucidate the physics and possibility to achieve H-mode in tokamak plasma. The analysis is done in the steady-state superconducting tokamak, SST-1, using phase-I plasma parameters which is basically L-mode plasma parameters having low ion temperature and magnetic field with the help of the ion cyclotron heating code TORIC combined with ‘steady state Fokker–Planck quasilinear’ (SSFPQL) solver. As a minority species hydrogen has been used in \(^3\hbox {He}\) and \(^4\hbox {He}\) plasmas to make two species \(^3\hbox {He(H)}\) and \(^4\hbox {He(H)}\) plasmas to study the ion cyclotron wave absorption scenarios. The minority heating is predominant in \(^3\hbox {He(H)}\) and \(^4\hbox {He(H)}\) plasmas as minority resonance layers are not shielded by ion–ion resonance and cut-off layers in both cases, and it is better in \(^4\hbox {He(H)}\) plasma due to the smooth penetration of wave through plasma–vacuum surface. In minority concentration up to 15%, it has been observed that minority ion heating is the principal heating mechanism compared to electron heating and heating due to mode conversion phenomena. Numerical analysis with the help of SSFPQL solver shows that the tail of the distribution function of the minority ion is more energetic than that of the majority ion and therefore, more anisotropic. Due to good coupling of the wave and predominance of the minority heating regime, producing energetic ions in the tail region of the distribution function, the \(^4\hbox {He(H)}\) and \(^3\hbox {He(H)}\) plasmas could be studied in-depth to achieve H-mode in two species of low-temperature plasma.     

Strange quark star with Tolman IV background

In this paper, using the optical emission spectroscopy (OES) technique, the optical characteristics of a radiofrequency (RF) plasma jet are examined. The \(\hbox {Ar}/\hbox {O}_{2}\) mixture is taken as the operational gas and, the Ar percentage in the \(\hbox {Ar}/\hbox {O}_{2}\) mixture is varied from 70% to 95%. Using the optical emission spectrum analysis of the RF plasma jet, the excitation temperature is determined based on the Boltzmann plot method. The electron density in the plasma medium of the RF plasma jet is obtained by the Stark broadening of the hydrogen Balmer \(H_{\beta }\). It is mostly seen that, the radiation intensity of Ar 4p\(\rightarrow \)4s transitions at higher argon contributions in \(\hbox {Ar}/\hbox {O}_{2}\) mixture is higher. It is found that, at higher Ar percentages, the emission intensities from atomic oxygen (O) are higher and, the line intensities from the argon atoms and ions including O atoms linearly increase. It is observed that the quenching of \(\hbox {Ar}^{*}\) with \(\hbox {O}_{2}\) results in higher O species with respect to \(\hbox {O}_{2}\) molecules. In addition, at higher percentages of Ar in the \(\hbox {Ar}/\hbox {O}_{2}\) mixture, while the excitation temperature is decreased, the electron density is increased.     

Conventional and microwave-assisted synthesis of new indole-tethered benzimidazole-based 1,2,3-triazoles and evaluation of their antimycobacterial,antioxidant and antimicrobial activities

A new series of triheterocycles containing indole–benzimidazole-based 1,2,3-triazole hybrids have been synthesized in good yields via a microwave-assisted click reaction. All the compounds were characterized by IR, \(^{1}\hbox {H}\) NMR, \(^{13}\hbox {C}\) NMR and mass spectroscopy and were evaluated for their in vitro antitubercular activity against the Mycobacterium tuberculosis H37Rv strain. Compounds 4b, 4h and 4i displayed highly potent antitubercular activity with MIC 3.125–6.25 \(\upmu \hbox {g}/\hbox {mL}\). The antioxidant potential was evaluated using 2,2-diphenyl-1-picryl hydrazine and \(\hbox {H}_{2}\hbox {O}_{2}\) radical scavenging activity, and compounds 4e,4f and 4g showed excellent radical scavenging activity with \(\hbox {IC}_{50}\) values in the range of 08.50–10.05 \(\upmu \hbox {g}/\hbox {mL}\). Furthermore, the compounds were evaluated for antimicrobial activity against numerous bacterial and fungal strains, and compounds 4b, 4c and 4h were found to be the most promising potential antimicrobial molecules with MIC 3.125–6.25 \(\upmu \hbox {g}/\hbox {mL}\).     

The envelope travelling wave solutions to the Gerdjikov–Ivanov model

Valence universal multireference coupled cluster (VUMRCC) method via eigenvalue independent partitioning has been applied to estimate the effect of three-body transformed Hamiltonian (\(\widetilde{{H}}_3\)) on ionisation potentials through full connected triple excitations \(S_{3}^{(1,0) }\). \(\widetilde{H}_3 \) is constructed using CCSDT1-A model of Bartlett et al for the ground-state calculation. Contribution of transformed Hamiltonian through full connected triples \(\overline{\widetilde{H}_3 S_3^{\left( {1,0} \right) }}\) involves huge amount of computational operations that is time-consuming. Investigation on \(\hbox {Cl}_{2}\) and \(\hbox {F}_{2}\) molecules using cc-pVDZ and cc-pVTZ basis sets shows that the above effect varies from 0.001 eV to around 0.5 eV, suggesting that inclusion of \(\overline{\widetilde{H} _3 S_3^{\left( {1,0} \right) } }\) is essential for highly accurate calculations.     

Ionic liquid-functionalized magnetic nanostructures as an efficient catalyst for the synthesis of 6H-chromeno[4,3-b]quinolin-6-ones

A novel imidazole ionic liquid (IL)-functionalized \(\hbox {silica}@\gamma \)-\(\hbox {Fe}_{2}\hbox {O}_{3}\) (\(\hbox {IL-SiO}_{2}@\hbox {MNP}\)) is prepared by the functionalization of \(\hbox {SiO}_{2}@\hbox {MNP}\) by 1-butyl-3-(3-trimethoxypropyl)-1H-imidazol-3-ium chloride as the IL moiety. The catalyst is characterized by transmission electron microscopy, scanning electron microscope, vibrating sample magnetometer, dynamic light scattering and Fourier transform infrared spectroscopy. \(\hbox {IL-SiO}_{2}@\hbox {MNP}\) showed good activity in the synthesis of 6H-chromeno[4,3-b]quinolin-6-one derivatives via multicomponent reaction of 4-hydroxycoumarin, anilines and benzaldehydes. The nanocatalyst is magnetically separable and easily recoverable and showed successful activity up to 10 runs.     

On quantum analogue of dynamical stabilisation of inverted harmonic oscillator by time periodical uniform field

Quantum analogue of stabilised forced oscillations around an unstable equilibrium position is explored by solving the non-stationary Schrödinger equation (NSE) of the inverted harmonic oscillator (IHO) driven periodically by spatial uniform field of frequency \(\Omega \), amplitude \(F_{0}\) and phase \(\phi \), i.e. the system with the Hamiltonian of \(\hat{{H}}=(\hat{{p}}^{2}/2m)-(m\omega ^{2}x^{2}/2)-F_0 x\sin \) \(\left( {\Omega t+\phi } \right) \). The NSE has been solved both analytically and numerically by Maple 15 in dimensionless variables \(\xi = x\sqrt{m\omega /\hbar }\hbox {, }f_0 =F_0 /\omega \sqrt{\hbar m\omega }\) and \(\tau =\omega t\). The initial condition (IC) has been specified by the wave function (w.f.) of a generalised Gaussian type which suits well the corresponding quantum IC operator. The solution obtained demonstrates the non-monotonous behaviour of the coordinate spreading \(\sigma \left( \tau \right) \hbox { =}\sqrt{\big ( {\overline{\Delta \xi ^{2}\big ( \tau \big )} } \big )}\) which decreases first from quite macroscopic values of \(\sigma _{0} =2^{12,\ldots ,25}\) to minimal one of \(\sim \!(1/\sqrt{2})\) at times \(\tau <\tau _0 =0.125\ln \!\left( {16\sigma _0^4 +1} \right) \) and then grows back unlimitedly. For certain phases \(\phi \) depending on the \(\Omega /\omega \) ratio and \(n=\log _2\!\sigma _0 \), the mass centre of the packet \(\xi _{\mathrm {av}}( \tau )= \overline{\hat{{x}}(\tau )} \cdot \sqrt{m\omega /\hbar }\) delays approximately two natural ‘periods’ \(\sim \!(4\pi /\omega )\) in the area of the stationary point and then escapes to ‘\(+\)’ or ‘?’ infinity in a bifurcating way.  For ‘resonant’ \(\Omega =\omega \), the bifurcation phases \(\phi \) fit well with the regression formula of Fermi–Dirac type of argument n with their asymptotic \(\phi ( {\Omega ,n\rightarrow \infty } )\) obeying the classical formula \(\phi _{\mathrm {cl}} ( \Omega )=-\hbox {arctg} \, \Omega \) for initial energy \(E = 0\) in the wide range of \(\Omega =2^{-4},...,2^{7}\).     

Design and study on square lattice-based photonic crystal fibre under different air holes for supercontinuum generation

In the present paper, the first results concerning the theoretical analysis of the \(^{16}\hbox {C} + \hbox {p}\) reaction by investigating two elastic scattering angular distributions measured at high energy compared to low energy for this system are reported. Several models for the real part of the nuclear potential are tested within the optical model formalism. The imaginary potential has a Woods–Saxon shape with three free parameters. Two types of density distribution and three different cluster structures for \(^{16}\hbox {C}\) are assumed in the analysis. The results are compared with each other as well as with the experimental data to give evidence of the importance of these studied items.     

Design,synthesis, in silico and biological evaluation of novel 2-(4-(4-substituted piperazin-1-yl)benzylidene)hydrazine carboxamides

A series of novel 2-(4-(4-substituted piperazin-1-yl)benzylidene)hydrazinecarboxamide derivatives has been successfully designed and synthesized to evaluate their potential as carbonic anhydrase (CA) inhibitors. The inhibitory potential of synthesized compounds against human CAI and CAII was evaluated. Compounds 3a–n exhibited \(\hbox {IC}_{50}\) values between \(1.89{-}415.1\,\upmu \hbox {M}\) against CAI and \(0.62{-}66.9\,\upmu \hbox {M}\) against CAII. Compound 3g was the most active inhibitor, with an \(\hbox {IC}_{50}\) value of \(0.62\,\upmu \hbox {M}\) against CAII. Molecular docking studies of compound 3g with CAII showed this compound fits nicely in the active site of CAII and it interacts with the zinc ion (\(\hbox {Zn}^{2+}\)) along with three histidine residues in the active site. Molecular dynamics simulation studies of compound 3g complexed with CAII also showed essential interactions which were maintained up to 40 ns of simulation. In vivo sub-acute toxicity study using 3g (300 mg/kg) was found non-toxic in adult Wistar rats.     

Decay widths of the spin-2 partners of the <Emphasis Type="Italic">X</Emphasis>(3872)

We consider the X(3872) resonance as a \(J^\mathrm{{PC}}=1^{++}\) \(D\bar{D}^*\) hadronic molecule. According to heavy quark spin symmetry, there will exist a partner with quantum numbers \(2^{++}\), \(X_{2}\), which would be a \(D^*\bar{D}^*\) loosely bound state. The \(X_{2}\) is expected to decay dominantly into \(D\bar{D}\), \(D\bar{D}^*\) and \(\bar{D} D^*\) in d-wave. In this work, we calculate the decay widths of the \(X_{2}\) resonance into the above channels, as well as those of its bottom partner, \(X_{b2}\), the mass of which comes from assuming heavy flavor symmetry for the contact terms. We find partial widths of the \(X_{2}\) and \(X_{b2}\) of the order of a few MeV. Finally, we also study the radiative \(X_2\rightarrow D\bar{D}^{*}\gamma \) and \(X_{b2} \rightarrow \bar{B} B^{*}\gamma \) decays. These decay modes are more sensitive to the long-distance structure of the resonances and to the \(D\bar{D}^{*}\) or \(B\bar{B}^{*}\) final state interaction.     

Two Types of Maximally Entangled Bases and Their Mutually Unbiased Property in ℂd⊗ℂd′

We first construct a new maximally entangled basis in bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{kd}\ (k\in Z^{+})\) which is diffrent from the one in Tao et al. (Quantum Inf. Process. 14, 2291 (2015)), then we generalize such maximally entangled basis into arbitrary bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{d^{\prime }}\). We also study the mutual unbiased property of the two types of maximally entangled bases in bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{kd}\). In particular, explicit examples in \(\mathbb {C}^{2} \otimes \mathbb {C}^{4}\), \(\mathbb {C}^{2} \otimes \mathbb {C}^{8}\) and \(\mathbb {C}^{3} \otimes \mathbb {C}^{3}\) are presented.     

Theoretical analysis of somatostatin receptor 5 with antagonists and agonists for the treatment of neuroendocrine tumors

We report on SSTR5 receptor modeling and its interaction with reported antagonist and agonist molecules. Modeling of the SSTR5 receptor was carried out using multiple templates with the aim of improving the precision of the generated models. The selective SSTR5 antagonists, agonists and native somatostatin SRIF-14 were employed to propose the binding site of SSTR5 and to identify the critical residues involved in the interaction of the receptor with other molecules. Residues Q2.63, D3.32, Q3.36, C186, Y7.34 and Y7.42 were found to be highly significant for their strong interaction with the receptor. SSTR5 antagonists were utilized to perform a 3D quantitative structure–activity relationship study. A comparative molecular field analysis (CoMFA) was conducted using two different alignment schemes, namely the ligand-based and receptor-based alignment methods. The best statistical results were obtained for ligand-based (\({q}^{2} = 0.454\), \({r}^{2}\) = 0.988, noc = 4) and receptor-guided methods (docked mode 1:\({q}^{2} = 0.530\), \({r}^{2} = 0.916\), noc = 5), (docked mode 2:\({q}^{2}\) = 0.555, \({r}^{2 }= 0.957\), noc = 5). Based on CoMFA contour maps, an electropositive substitution at \(\hbox {R}^{1}\), \(\hbox {R}^{2}\) and \(\hbox {R}^{4}\) position and bulky group at \(\hbox {R}^{4}\) position are important in enhancing molecular activity.     

Uncertainties of the $$B\rightarrow D$$B→ D transition form factor from the D-meson leading-twist distribution amplitude

The \(B\rightarrow D\) transition form factor (TFF) \(f^{B\rightarrow D}_+(q^2)\) is determined mainly by the D-meson leading-twist distribution amplitude (DA) , \(\phi _{2;D}\), if the proper chiral current correlation function is adopted within the light-cone QCD sum rules. It is therefore significant to make a comprehensive study of DA \(\phi _{2;D}\) and its impact on \(f^{B\rightarrow D}_+(q^2)\). In this paper, we calculate the moments of \(\phi _{2;D}\) with the QCD sum rules under the framework of the background field theory. New sum rules for the leading-twist DA moments \(\left\langle \xi ^n\right\rangle _D\) up to fourth order and up to dimension-six condensates are presented. At the scale \(\mu = 2 \,\mathrm{GeV}\), the values of the first four moments are: \(\left\langle \xi ^1\right\rangle _D = -0.418^{+0.021}_{-0.022}\), \(\left\langle \xi ^2\right\rangle _D = 0.289^{+0.023}_{-0.022}\), \(\left\langle \xi ^3\right\rangle _D = -0.178 \pm 0.010\) and \(\left\langle \xi ^4\right\rangle _D = 0.142^{+0.013}_{-0.012}\). Basing on the values of \(\left\langle \xi ^n\right\rangle _D(n=1,2,3,4)\), a better model of \(\phi _{2;D}\) is constructed. Applying this model for the TFF \(f^{B\rightarrow D}_+(q^2)\) under the light cone sum rules, we obtain \(f^{B\rightarrow D}_+(0) = 0.673^{+0.038}_{-0.041}\) and \(f^{B\rightarrow D}_+(q^2_{\mathrm{max}}) = 1.117^{+0.051}_{-0.054}\). The uncertainty of \(f^{B\rightarrow D}_+(q^2)\) from \(\phi _{2;D}\) is estimated and we find its impact should be taken into account, especially in low and central energy region. The branching ratio \(\mathcal {B}(B\rightarrow Dl\bar{\nu }_l)\) is calculated, which is consistent with experimental data.     

A single external cavity laser-based sensor for simultaneous detection and quantification of atmospheric methane and water vapor

A single external cavity diode laser (ECDL)-based sensor system was developed for simultaneous measurements of methane and water vapor in the atmospheric air. A home-made continuous wave (CW) ECDL with a wavelength of \(\sim\) 1325 nm was employed to cover two absorption lines for different gas species. A high-finesse optical cavity composed of two high-reflectivity mirrors (\(R=0.99927\)) was used to enhance the gas absorption in a space of 0.5 l. An isolated methane absorption line at \(7552.74 \hbox { cm}^{-1}\) and a common (non-isolated) absorption line of water vapor and methane \(7552.81\hbox {cm}^{-1}\) were utilized for simultaneous detection. A method was developed to calculate the water vapor concentration from these lines. To our knowledge, the method and the absorption lines used in the present paper were used for the first time for simultaneous measurement of the concentrations of atmospheric methane and water vapor.     

Time Delay for the Dirac Equation

We consider time delay for the Dirac equation. A new method to calculate the asymptotics of the expectation values of the operator \({\int\limits_{0} ^{\infty}{\rm e}^{iH_{0}t}\zeta(\frac{\vert x\vert }{R}) {\rm e}^{-iH_{0}t}{\rm d}t}\), as \({R \rightarrow \infty}\), is presented. Here, H₀ is the free Dirac operator and \({\zeta\left(t\right)}\) is such that \({\zeta\left(t\right) = 1}\) for \({0 \leq t \leq 1}\) and \({\zeta\left(t\right) = 0}\) for \({t > 1}\). This approach allows us to obtain the time delay operator \({\delta \mathcal{T}\left(f\right)}\) for initial states f in \({\mathcal{H} _{2}^{3/2+\varepsilon}(\mathbb{R}^{3};\mathbb{C}^{4})}\), \({\varepsilon > 0}\), the Sobolev space of order \({3/2+\varepsilon}\) and weight 2. The relation between the time delay operator \({\delta\mathcal{T}\left(f\right)}\) and the Eisenbud–Wigner time delay operator is given. In addition, the relation between the averaged time delay and the spectral shift function is presented.     

Light deflection and Gauss–Bonnet theorem: definition of total deflection angle and its applications

In this paper, we re-examine the light deflection in the Schwarzschild and the Schwarzschild–de Sitter spacetime. First, supposing a static and spherically symmetric spacetime, we propose the definition of the total deflection angle \(\alpha \) of the light ray by constructing a quadrilateral \(\varSigma ^4\) on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) determined by the optical metric \(\bar{g}_{ij}\). On the basis of the definition of the total deflection angle \(\alpha \) and the Gauss–Bonnet theorem, we derive two formulas to calculate the total deflection angle \(\alpha \); (1) the angular formula that uses four angles determined on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) or the curved \((r, \phi )\) subspace \({\mathscr {M}}^\mathrm{sub}\) being a slice of constant time t and (2) the integral formula on the optical reference geometry \({\mathscr {M}}^\mathrm{opt}\) which is the areal integral of the Gaussian curvature K in the area of a quadrilateral \(\varSigma ^4\) and the line integral of the geodesic curvature \(\kappa _g\) along the curve \(C_{\varGamma }\). As the curve \(C_{\varGamma }\), we introduce the unperturbed reference line that is the null geodesic \(\varGamma \) on the background spacetime such as the Minkowski or the de Sitter spacetime, and is obtained by projecting \(\varGamma \) vertically onto the curved \((r, \phi )\) subspace \({\mathscr {M}}^\mathrm{sub}\). We demonstrate that the two formulas give the same total deflection angle \(\alpha \) for the Schwarzschild and the Schwarzschild–de Sitter spacetime. In particular, in the Schwarzschild case, the result coincides with Epstein–Shapiro’s formula when the source S and the receiver R of the light ray are located at infinity. In addition, in the Schwarzschild–de Sitter case, there appear order \({\mathscr {O}}(\varLambda m)\) terms in addition to the Schwarzschild-like part, while order \({\mathscr {O}}(\varLambda )\) terms disappear.