The high-purity alkaline earth halide crystals, especially CaF2, BaF2 or MgF2 crystals, have a diffuse scatter distribution function value of less than 7×10−7, an RMS uniformity of refractive index of less than 15×10−8 after subtraction of Zernike coefficients and an RMS value of birefringence in the (111) direction of less than 0.2 nm/cm. Preferably the crystals exhibit a loss coefficient of less than 5×10−4 cm−1 after irradiation with 10×109 laser pulses with an energy density of 10 mJ/cm2 at a wavelength of 193 nm. Also they have RMS birefringence in the (100) direction or the (111) direction that is less than 0.35 nm/cm.