Short pulsed fiber lasers have been widely made using single-walled carbon nanotubes as a saturable absorber (SA). However, most of the currently used devices can only operate in one determined operation state with an unchangeable modulation SA depth in the cavity, which significantly limits their application in photonic devices. In this paper, well-aligned carbon nanotube arrays are synthesized using zeolite AlPO4-5 as a template, which features anisotropic optical absorption. The linear optical absorption of the as-synthesized carbon nanotube arrays can easily be tuned by adjusting a polarization controller, thus providing a tunable modulation depth for the carbon nanotube SA. By exploiting this SA in an erbium-doped fiber laser cavity, both Q-switched and mode-locked pulsed lasers are achieved by simply adjusting a polarization controller under a fixed pump power of 330 mW. In addition, the repetition rate of the Q-switching and pulse duration of the mode-locking can be tuned according to the variation of modulation depth. Moreover, soliton molecules can be obtained when the modulation depth of the SA is 4.5%.