Maciej, for beam visible on photos it is good to have some fog/smoke as easiest solution (recommended for beginners).
Further it depeneds on camera settings (exposure time, aperture, ISO) - for best visibility -> (longer time, larger aperture /smaller f number, larger). However it is quite tricky, you "pay" for adjustment made by something: With high exposure time you need tripod or firm position of camera and beam. With larger aperture you loose depth of field (is issue for close range) -> picture becomes not focussed. With larger ISO you get more noise on photos. Try to experiment with settings, this is the best way how to learn it.
Also one note: Longer exposure times make beam visible, but "more solid", the dust/smoke/fog particles are blurred and not visible in the beam. You might check beamshots of my Sky beauty laser ("review" link in my signature), there is a one with dust in it visible.
With mobile camera your options will be limited, but if you are lucky, you might set up at least exposure time and ISO. I have Pro funcion and exposure time up to 10 seconds - enough for test shots when preparing artworks. For the artworks I use digital camera.
And tips: Use remote control or at least time delay (timer) to start exposure, otherwise your pics will be blurry as you move with camera a little when pushing the shutter button (to take the picture). Of course no flash (still might be exceptions, but it is more advanced technique). Do not forget to focus, you might use flashlight to illuminate the scene and to use autofocus function.
For my laserpaintings I often use exposure times like 10-20 or more minutes (but it is for more for complex painting instead of single beam photo), f number like 16-18 (but it depends also on further factors than depth of field only - interference effects, laser grain...), ISO up to 200 (low to reduce noise as much as possible).
Also you might experiment with this as advanced techique:
WARNING MIGHT BE VERY DANGEROUS! YOU NEED SOME EXPERIENCE WITH LASERS TO DO IT SAFE! The idea is to capture the beam with little angle towards laser aperture (
MAKE SURE LASER IS FIRMLY FIXED AND CANNOT MOVE, MAKE SURE YOU DO NOT HIT YOUR EYE OR CAMERA LENS - BOTH DEADLY - EYE PROTECTION IS A MUST HAVE!!!!). This setup makes the beam the most visible. The closer the camera and the lower the angle, the more visible the beam is.
In other words you do not capture the beam behind the laser, but in front of it (in the area where beam is - that is dangerous). Also the low angle (to beam) and close camera works behind the laser, but it is not as visible as in the first case desscribed.
And finally, please resize the pictures posted here to some reasonable resolution.
Enjoy and stay safe... 