As a practitioner, my expertise lays in the area of training and operational guidelines. And along this route I shall direct my writing below.
Everybody is familiar with image of hypnotist using a slowly swinging pendulum with the eyes falling the swinging ball. With the mind conditioned to follow the instructions of the hypnotist and the eyes "faithfully" following the swinging ball, a person can get hypnotized. The scientific fact is, slow movement of the eye muscles can induce one towards a sleepy state. This is the only relevant fact on hypnosis for our explanatory purpose here.
Eye muscles are important route to the inner world. One major limitation of the eye-muscles is that they are weak muscles, and as such, cannot induce a massive amount of chi (which is essential for deep meditation). Deep meditation necessarily involves our strong breathing muscles. For deep meditation, such energy has to be guided to our brain through our third eye. To open the third eye (or using the point called third eye [making a triangle with our physical eyes] as focused point) has a prior pre-requisite: the ability to control the muscles of our physical eyes.
In classical Taoist meditation (Neidan), there is a training method called "incense gazing". It is simple, gazing continually at a burning incense. Tears will flow, eyes will be blurred. This method is also mentioned in the Classic text Taoist Yoga translated into English by Zen master Charles Luk. A contemporary method is gazing a far-away trees. Master Wang Xianzahi also used tree gazing in healing impaired eye-sights for his students (per reference from his student Madam Zhuang JingKe)..."until the tree tops become blurred", as reportedly said by Master Wang.
Now this is the first step. With some sound prior practice in chi-related disciplines or meditation, the following steps will be easy. Step one: visualize connecting your eyes with broad area of your abdomen with rubber bands, which means your eye movements will trigger synchronically reactions in your abdomen. The eye movements can be left/right, up/down, doing circles etc. With good training in this direction, you are ready to move on. Final step: visualize connecting your third eyes to your Dantian (i.e. doing navel gazing). The eye muscles will act as balancing of chi-power while the movement of the third eye will become almost stationary or nano in nature.
You are onto the road to deep meditation.