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We proposed a novel FM sideband technique to reduce drastically the double resonance linewidth in a Rb atomic clock pumped by a stabilized diode laser. This could be achieved by the use of the nonlinear susceptibility of three-level Rb atoms when the incident microwave signals is modulated. By using resultant FM sidebands of the laser and synchronous detection, high-order derivative shapes of absorptive and dispersive parts of double-resonance spectral line were measured. The linewidth as narrow as 20 Hz was obtained which was 1/18 times that of the rf-excited 87.Rb atomic clock. Optimum values of modulation parameters were found by the computer simulation. Furthermore, contributions of the laser frequency detuning to the light shifts were also estimated to attain the microwave frequency accuracy as high as 10-12.