We discuss the recent insights into the spatial-mode structure of traveling-wave optical parametric amplifiers, with focus on optical processing of quantum images. OCIS codes: (190.4970) Parametric oscillators and amplifiers; (999.9999) Optical image amplification.Optical parametric amplifiers (OPAs) are important tools of both classical and quantum information processing. Phase-sensitive optical parametric amplifiers (PSAs), unlike any other optical amplifiers, can increase the magnitude of the signal without adding any noise [1]. This property, coupled with the wide temporal bandwidth of fiber-based parametric amplifiers, makes them attractive as nearly noiseless inline amplifiers for optical communication systems [2][3][4]. Similarly, the spatially-broadband PSAs can generate multimode squeezed vacuum for parallel continuousvariable quantum information protocols [5] and can noiselessly amplify faint images [6][7][8][9]. The signal-to-noise ratio improvement provided by the image pre-amplifiers can translate into increased resolution [10][11][12]. Precise knowledge of the spatial quantum correlations in OPAs, however, is difficult to obtain because the traveling-wave OPAs use tightly focused pump beams; the resulting spatially-varying gain, together with the limited spatial bandwidth, couples and mixes up the modes of the quantum image [13]. These spatial mode-mixing effects, known as gain-induced diffraction [14], also make it difficult to detect the lowest-noise mode of a traveling-wave squeezer [15], because a properly mode-matched homodyne detector needs exact knowledge of this mode's spatial profile.We have recently found the orthogonal set of independently squeezed eigenmodes [16] of a traveling-wave PSA with spatially inhomogeneous (circular or elliptical Gaussian) pump by extending the mode-expansion method [17,18] to elliptical Hermite-Gaussian (HG) basis. Such expansion reduces the PSA's partial differential equation to a system of coupled ordinary differential equations for the mode amplitudes in the HG basis. The conventional HG expansion basis has the signal waist 2 1/2 times wider than the pump waist to match the signal and pump beam curvatures. The eigenmodes are found by singular-value decomposition [19,20] of the numerically computed Green's function [20]. This method is the spatial version of the quantum Karhunen-Loève expansion [21] previously used in the temporal domain to study quantum soliton propagation [22] and four-wave-mixing processes [23] in optical fiber. While a rough estimate of the number of PSA modes is (pump waist × spatial bandwidth) 1/2 , our rigorous method determines the exact number, gains, and the shapes of the PSA modes supported by the pump beam of a given spot size. These eigenmodes are the spatial analogs of the temporal "supermodes" of optical parametric oscillators [24]. The PSA eigenmodes are also closely related to the Schmidt modes of spontaneous parametric down conversion: at very low PSA gains, the eigenmodes correspond to frequency-degenerate transverse Schmid...