Many connections can be found between the praṇava oṃ and the mantra “haṃsa” (sohaṃ). The symbol of the swan, like a bird, is found in the Nādabindūpaniṣad. This image and the connection with praṇava are described there as following:
अकारो दक्षिणः पक्ष उकारस्तूत्तरः स्मृतः।
मकारं पुच्छमित्याहुरर्धमात्रा तु मस्तकम् ॥मस्तकम् १॥
akāro dakṣiṇaḥ pakṣa ukārastūttaraḥ smṛtaḥ।
makāraṃ pucchamityāhurardhamātrā tu mastakam ॥mastakam 1॥
It is believed (smṛtaḥ) the mātrā “A” (akāraḥ) is in the right wing (dakṣinaḥ-pakṣaḥ), the mātrā “U”(ukāraḥ) in the north (uttaraḥ – “north” is “in the left” wing). Thus (iti), in his tail (puccham) is the mātrā “Ma” (makāraṃ), also (tu) they say (āhuḥ), ia half the mātrā, (ardhamātrā) is in his head (mastakam), i.e. nasalized vibration.
I heard from my Guru that the so-ham mantra is connected with the experience related to the sahasrāra–cakra, when kuṇḍalinī-śakti connects jīva with Śiiva. The haṃsa-mantra (inverted “so-ham”) is located in the six-cakras and symbolises the jīvātman (the eternal soul embodied in the body). Both mantras, which are essentially one mantra, have many meanings. I suppose, haṃsa, which is formally translated as “a swan”, requires special explanations of what is actually meant by a swan. You can find the essence of the haṃsa concept in one of the ślokas from the Rigveda, which is later repeated in the Yajurveda, the Taittirīya-āraṇyaka and the Kaṭha Upaniṣad (one of the earliest Upaniṣadas), the Mahānārāyaṇa Upaniṣad and others. This mantra is also found in tantric sādhana. It refers to the haṃsa – a swan, generally symbolising the sun. The sun is the source of prāṇa (the symbol of the jīva respectively), the sun – like a swan, rises on the horizon, moves in the sky and goes back to the earth in the evening. Thus, it is present both in the sky and on the earth, is also able to be under water and on its surface. It is something omnipresent, like prāṇa that pervades the entire universe. Here is the mantra from the Rig Veda dedicated to the Swan Sun (haṃsa):
हं॒सः शु॑चि॒षद्वसु॑रंतरिक्ष॒सद्धोता॑ वेदि॒षदति॑थिर्दुरोण॒सत् ।
नृ॒षद्व॑र॒सदृ॑त॒सद्व्यो॑म॒सद॒ब्जा गो॒जा ऋ॑त॒जा अ॑द्रि॒जा ऋ॒तं ॥ ०४।०४०।०५
The sun (swan – haṃsaḥ) located in the clear sky (śuciṣat), in the atmosphere (antarikṣasat), moving through the air (vasu), it is a priest (hotar – Agni himself) staying in a hollow (the center of the vedic altar – vediṣat), the guest (atithi) abiding in the house (duroṇasat). Dwelling in people (nṛṣat), being in the best (varasat) (that can be deities or siddhas), in the world order (ṛtasat), located in space (vyomasat), born in water (abjā, the fire inside the ocean – vaḍavānala), born of “the cows” (gojā – the sun rays), born of a world order (ṛtajā), appeared from the mountain (adrijā) – the great truth (ṛtaṃ bṛhat) (the rising sun).
The sun (savitā) is one of the main worshiping aim in the Vedas. Sāvitrī in the Vedas is a prototype of Śakti, after which it became known as kuṇḍalinī, which ascends through suṣumṇā to Śiva in the form of the light of prākāśa. The kuṇḍalinī raises the jīva (haṃsa) through the cakras and is absorbed back into itself in the sahasrāra, turning into the realisation of so-ham (I am him) in the form of Śiva.