We study the spatial properties of high-gain parametric down-conversion (PDC). From the Hamiltonian we find the Schmidt modes, apply the Bloch-Messiah reduction, and calculate analytically the measurable quantities, such as the angular distributions of photon numbers and photon-number correlations. Our approach shows that the Schmidt modes of PDC radiation can be considered the same as for the low-gain (biphoton) case while the Schmidt eigenvalues strongly depend on the parametric gain. Its validity is confirmed by comparison with several experimental results, obtained by us and by other groups.PACS numbers: 42.65. Lm, 42.65.Yj, 42.50.Lc Introduction. Currently, much interest is attracted to bright squeezed vacuum (BSV), a macroscopic nonclassical state of light that can be obtained via high-gain parametric down conversion (PDC). This is because BSV contains huge photon numbers and at the same time is strongly nonclassical, manifesting entanglement [1] and noise reduction below the standard quantum limit [2]. These features make it interesting for applications such as quantum imaging [3,4] and metrology [5], quantum optomechanics [6] and nonlinear optics with quantum light [7]. Besides, large photon-number correlations arising in BSV are richer than entanglement in two-photon light emitted via low-gain PDC [8].At the same time, theoretical description of BSV presents certain difficulties. In contrast to low-gain PDC, BSV generation cannot be described in the framework of the perturbation theory. The quantum state contains not only two-photon terms but also higher-order Fock components, and its calculation in the Schrödinger picture is difficult. Many recent theoretical investigations of such strong pumping regime are based on the concept of collective input/output modes introduced mostly to describe the spectral properties in the frequency domain [9][10][11][12][13]. In a high-gain regime it is convenient to calculate the observables in the Heisenberg picture. In this case the Schmidt-mode formalism used in the Schroedinger picture for the description of multimode two-photon light [14] is replaced by a similar procedure, called Bloch-Messiah reduction [15]. However beyond the perturbation approach the solution for the field operators was up to now obtained only numerically, usually through a set of integro-differential equations [12,13,[16][17][18].Here we present a fully analytical description of the angular properties of BSV. Our approach is based on the Bloch-Messiah reduction and allows one to obtain the evolution of the photon creation (annihilation) operators both for the Schmidt modes and for the plane-wave modes. After obtaining the evolution of these, we calculate analytically the angular distributions of the inten-

Bright squeezed vacuum, a macroscopic nonclassical state of light, can be obtained at the output of a strongly pumped nonseeded traveling-wave optical parametric amplifier (OPA). By constructing the OPA of two consecutive crystals separated by a large distance, we make the squeezed vacuum spatially single-mode without a significant decrease in the brightness or squeezing.

Bright squeezed vacuum (BSV) is a non-classical macroscopic state of light, which can be generated through high-gain parametric down-conversion or four-wave mixing. Although BSV is an important tool in quantum optics and has a lot of applications, its theoretical description is still not complete. In particular, the existing description in terms of Schmidt modes fails to explain the spectral broadening observed in experiment as the mean number of photons increases. On the other hand, the semi-classical description accounting for the broadening does not allow to decouple the intermodal photon-number correlations. In this work, we present a new generalized theoretical approach to describe the spatial properties of BSV. This approach is based on exchanging the (k, t) and (ω, z) representations and solving a system of integro-differential equations. Our approach predicts correctly the dynamics of the Schmidt modes and the broadening of the spectrum with the increase in the BSV mean photon number due to a stronger pumping. Moreover, the model succesfully describes various properties of a widely used experimental configuration with two crystals and an air gap between them, namely an SU(1,1) interferometer. In particular, it predicts the narrowing of the intensity distribution, the reduction and shift of the side lobes, and the decline in the interference visibility as the mean photon number increases due to stronger pumping. The presented experimental results confirm the validity of the new approach. The model can be easily extended to the case of frequency spectrum, frequency Schmidt modes and other experimental configurations.