Reduced breath because of breathing meditation is distracting.

[inwentor]

I am practicing meditation with the focus on my breath. Recently, I am experiencing that my mind is going deeper and the breath is becoming reduced during meditation. I believe that the breath is associated with the depth in meditation.
My problem is that whenever I get deeper into meditation my breath is "Naturally" reduced and at a point, it creates a little bit of uncomfortableness. So I am forced to draw some more air into my lungs and again start focusing on my breathing sensation. This creates a distraction. Every time I go deeper, my breath is reduced and I consciously take more air.
This problem might be because I unconsciously believe that "Low breath=depth in meditation". So I unconsciously focus on lowering my breath as I stay longer. But when I notice my breathing pattern, its "Naturally" reduced and as I effortlessly try to breathe while focusing on my breath, its naturally reducing.
So, how should I overcome this problem?

[Vivek]

My problem is that whenever I get deeper into meditation my breath is "Naturally" reduced and at a point, it creates a little bit of uncomfortableness.

Breath getting shorter and softer is quite natural a we go deep in meditation. I would suggest to watch the discomfort you feel when the breath gets shorter. The discomfort should pass away sooner or later which could then lead to sustained meditation.

[inwentor]

Uncomfortable in the sense of suffocation.

[Goofaholix]

You are not going to suffocate, it's just that the breath has become more subtle but the mind has not become sensitive enough to keep up with it.

Just relax and keep observing where you last could last feel the breath, try to feel the subtlest evidence that you are still breathing.

[inwentor]

What if I breathe subtly at the beginning? will that get me to the depth from the start? if No, why?
I am asking with the intention to know the indepth relation between Meditative depth and breathing.

[Vivek]

What if I breathe subtly at the beginning? will that get me to the depth from the start? if No, why?

Meditation is about unconditionally observing what is here and now, not manipulating states etc. As you sit observing the breath, eventually you will go deep. So, there is no need to deliberately make the breath soft or hard.

[rogp99]

What if I breathe subtly at the beginning? will that get me to the depth from the start? if No, why?
I am asking with the intention to know the indepth relation between Meditative depth and breathing.

Nope. The natural course is your breath will be coarse as first, then when your mind becomes calmer, your breath will be more subtler. I'd prefer belly breathing though.

[Goofaholix]

The point is to bring the mind from coarse to subtle awareness, the breath is irrelevant and only an aid to help you do that but generally you can't use the breath to change the mind.

[inwentor]

Distraction by thoughts, is that a common thing even for experienced meditators? Because for now as a beginner, I am just focusing on the breath naturally and I am getting distracted by thoughts now and then. Do I need to put some efforts and try not to get distracted and focus solely on breathing (by constant reinforcements)? Or do I need to have relaxed awareness on the breath?

[Vivek]

Distraction by thoughts, is that a common thing even for experienced meditators?

Yes, it does happen from time to time.

Or do I need to have relaxed awareness on the breath?

Maintain relaxed awareness of breath.

[smritiyoga]

Yes its happen frequently   

[maybeimeditate]

Fear is the obstacle.

Let go of the breath. Let go of your body. Mind. Ego. Thoughts. Let go of everything.

In letting go, fear will go.

Then you'll be able to advance further along your path.