The rapid developments of ultra-intense and ultra-short laser offer the possibility to study laser driven ion acceleration with using solid density target. However, the prepulse and amplified spontaneous emission generated in the amplification can create preplasma at the target front by heating, melting and evaporating a portion of a solid density. The main pulse then interacts with the preplasma, which would be harmful to laser ion acceleration. Therefore, many methods have been developed to enhance the temporal contrast of high power laser system, such as saturable absorber, cross polarized wave generation (XPW) and plasma mirror. With many advantages, such as high conversion efficiency, introducing neither spatial nor spectral distortions, and easy setup compared with other mechanisms, XPW has been used to clean the femtosecond laser system. Besides that, the spectrum of the XPW pulse could be broadened by 3 times under the best condition compared with the initial spectrum. It can solve the spectrum narrowing problem during the laser amplification to obtain ultra-short femtosecond laser pulse. Here, we experimentally investigate the output power, spectrum bandwidth and center wavelength shift of the generated cross-polarized wave according to the input pulse quadratic spectral phase. The femtosecond laser pulse in compact laser plasma accelerator system at Peking University is used to investigate the role of quadratic spectral phase in characterizing the two crystal cross-polarized generation. The Ti:Sapphire-based laser system has a central wavelength of 798 nm and bandwidth of 35.5 nm which allows the pulse to be compressed down to 40 fs duration (FWHM). Typical the input pulse energy of XPW is 150 upJ and the laser system operates well at 1 kHz repetition rate. The quadratic spectral phase can be increased by changing the position of compressor grating. The conversion efficiency, spectrum bandwidth and the central wavelength shift by changing the quadratic spectral phase are measured. The conversion efficiency is 17% when quadratic spectral phase 2=0, and decreases as quadratic spectral phase increases. The rapid decrease is caused by negative quadratic spectral phase. The spectrum bandwidth is 62 nm under the optimum condition, and the broadening effect exists when quadratic spectral phase is in a range of -280 fs2 2 1400 fs2. It is slowly blue-shifted when 20 and stays at 772 nm when 21000 fs2. It starts to be red-shifted when 20 and stays at 806 nm finally. In conclusion, with the increase of quadratic spectral phase, we observe the effects of conversion efficiency and spectrum bandwidth and the shift of central wavelength. Moreover, the influences of positive and negative quadratic spectral phase on characteristics of XPW are different. Our result shows that the negative quadratic spectral phaseis more effective at reducing the conversion efficiency and spectrum bandwidth than the positive one.