Ven. Ajahn Brahm is widely recognized as an expert of samatha meditation leading to jhana from mindfulness of breathing, with his book, "Mindfulness, Bliss and Beyond".
I quote a part of this book below.
Here, he explains the transition from focusing and watching the breath, to the entry point into jhana.
There's a point where the mind simply lets go of everything and simply stays with the peace and bliss of the breath, and finally drops the focus on the breath, and stays with the peace and bliss, which is at the "center of the experience of breath", like the stillness at the center of the eye of a cyclone.
To get there, ironically, the mind has to let go, and not grasp anything. It must simply be, and not try to become.
In other words, you start with mindfulness of breathing, and then reach the stage of "just sitting" (sounds familiar?) and then from there, could progress into jhana (a deeper "just sitting").
And the terms jhana, dhyana, ch'an and zen all mean the same thing in different languages.
So, Shikantaza ("just sitting") is not really a totally different and unrelated meditation, in my opinion, based on what I have heard of it.
You experience every part of each in-breath and out-breath
continuously for many hundred breaths in a row. That is why this stage
is called full sustained attention on the breath. You cannot reach this
stage through force, through holding or gripping. You can attain this
degree of stillness only by letting go of everything in the entire
universe except for this momentary experience of the breath happening
silently. Actually “you” do not reach this stage, the mind does. The
mind does the work itself. The mind recognizes this stage to be a very
peaceful and pleasant place to abide, just being alone with the
breath. This is where the doer, the major part of one’s ego, starts to
disappear.
One finds that progress happens effortlessly at this stage of
meditation. We just have to get out of the way, let go, and watch it
all happen. The mind will automatically incline, if we only let it,
toward this very simple, peaceful, and delicious unity of being alone
with one thing, just being with the breath in each and every moment.
This is the unity of mind, the unity in the moment, the unity in
stillness.
The fourth stage is what I call the “springboard” of meditation,
because from it one may dive into the blissful states. When we simply
maintain this unity of consciousness by not interfering, the breath
will begin to disappear. The breath appears to fade away as the mind
focuses instead on what is at the center of the experience of breath,
which is awesome peace, freedom, and bliss.
At this stage I introduce the term “beautiful breath.” Here the mind
recognizes that this peaceful breath is extraordinarily beautiful. We
are aware of this beautiful breath continuously, moment after moment,
with no break in the chain of experience. We are aware only of the
beautiful breath, without effort and for a very long time.
Now as I will explain further in the next chapter, when the breath
disappears, all that is left is “the beautiful.” Disembodied beauty
becomes the sole object of the mind. The mind is now taking the mind
as its own object. We are no longer aware of the breath, body,
thought, sound, or outside world. All that we are aware of is beauty,
peace, bliss, light, or whatever our perception will later call it. We
are experiencing only beauty, continuously, effortlessly, with nothing
being beautiful! We have long ago let go of chatter, let go of
descriptions and assessments. Here the mind is so still that it cannot
say anything. One is just beginning to experience the first flowering
of bliss in the mind. That bliss will develop, grow, and become very
firm and strong. And then one may enter into those states of
meditation called the jhanas.