gates 40 and the cathode electrodes 10 whose wide surfaces facing each other formed with the carbon nanotubes 30 are supplied with the pulse wave by means of the pulse wave supplier 60 so that the gates 40 and the cathode electrodes 10 whose wide surfaces face each other alternately perform their roles so that the carbon nanotube field emitter is completed a metal matrix composite material layer 36 including the carbon nanotubes with a thickness of 10 u03bcm to several millimeters is film formed on the insulating substrate 15 for example the glass substrate so that a structure as shown in fig7 a is formed next as shown in fig7 b the composite material layer 36 including the carbon nanotubes is irradiated with a co 2 laser 38 beam having an output power of 1 to 5 w at a scanning speed of 0 1 to 100 mm per second so that the metal is selectively etched and the cathode electrodes 10 or gates 40 and the carbon nanotubes 30 remain by repeating such an etching the metal matrix composite material layer 36 is formed with stripe patterns having a width of 0 1 to 500 u03bcm at an interval of 0 1 to 500 u03bcm so that the carbon nanotube array structure with the three dimensional structure is completed as shown in fig7 c from after the carbon nanotube array structure with the three dimensional structure is completed the process of completing the entire structure of the carbon nanotube field emitter is the same as the embodiment 1 and the further description thereof will thus be omitted in the present.