Context :

Aims : We study the magnetic flux emergence and formation of a sunspot penumbra in the active region NOAA 11024 .

Methods : We simultaneously observed the Stokes parameters of the photospheric iron lines at 1089.6 nm with the TIP and 617.3 nm with the GFPI spectropolarimeters along with broad-band images using G-band and Ca ii K filters at the German VTT .
The photospheric magnetic field vector was reconstructed from an inversion of the measured Stokes profiles .
Using the AZAM code , we converted the inclination from line-of-sight ( LOS ) to the local reference frame ( LRF ) .

Results : Individual filaments are resolved in maps of magnetic parameters .
The formation of the penumbra is intimately related to the inclined magnetic field .
No penumbra forms in areas with strong magnetic field strength and small inclination .
Within 4.5 h observing time , the LRF magnetic flux of the penumbra increases from 9.7 \times 10 ^ { 20 } to 18.2 \times 10 ^ { 20 } Mx , while the magnetic flux of the umbra remains constant at \sim 3.8 \times 10 ^ { 20 } Mx .
Magnetic flux in the immediate surroundings is incorporated into the spot , and new flux is supplied via small flux patches ( SFPs ) , which on average have a flux of 2–3 \times 10 ^ { 18 } Mx .
The spot ’ s flux increase rate of 4.2 \times 10 ^ { 16 } Mx s ^ { -1 } corresponds to the merging of one SFP per minute .
We also find that during the formation of the spot penumbra : a ) the maximum magnetic field strength of the umbra does not change , b ) the magnetic neutral line keeps the same position relative to the umbra , c ) the new flux arrives on the emergence side of the spot while the penumbra forms on the opposite side , d ) the average LRF inclination of the light bridges decreases from 50 ^ { \circ } to 37 ^ { \circ } , and e ) as the penumbra develops , the mean magnetic field strength at the spot border decreases from 1.0 to 0.8 kG .

Conclusions : The SFPs associated with elongated granules are the building blocks of structure formation in active regions .
During the sunspot formation , their contribution is comparable to the coalescence of pores .
Besides a set of critical parameters for the magnetic field , a quiet environment in the surroundings is important for penumbral formation .
As remnants of trapped granulation between merging pores , the light bridges are found to play a crucial role in the formation process .
They seem to channel the magnetic flux through the spot during its formation .
Light bridges are also the locations where the first penumbral filaments form .