Linear electro optic effect in a non linear material

hi,
Help me to understand is there any relation between Linear electro optic coefficient (rij Pockel's coefficient) & χ2 (chi2) coefficient for a second order non-linear materials? Are they same or physically completely different?

In linear electro-optic effect (Pockel's effect) the change in refractive index is directly proportional to the applied DC or low frequency Electric field, and when an optical wave travel through medium it will a see a completely different medium as compared to when there is no field. and
Δn = c r E ........... (i)
where Δn is change in r.i. n "r" is an electro optic coefficient & E is applied DC electric field

For a material to show pockel's effect should not posses any inversion symmetry (non centrosymmetric). i.e. Lithium Niobate. which is basically a non linear material & have second order non linear coefficient (chi2).

The polarization of a non linear material in the presence of an E-field can be written as
P = ε0 χ1 E + ε0 χ2 EE + ε0 χ3 EEE+.....
P = ε0(χ1 + χ2 E + χ3 EE+...................)E

where χ1 , χ2 and χ3 are linear susceptibility, second order non linear susceptibility and third order non linear susceptibility respectively. If we consider only χ1 and χ2 and leaving the other higher order term then Polarization is P = ε0(χ1 + χ2 E)E

Now the relative permittivity of the medium can be given as εr(E) = 1 + χ1 + χ2 E
since 1 + χ1 = εr (original relative permittivity when there was no field)
thus Δεr = εr(E) - εr
Δεr = χ2 E

since Δεr = 2nΔn from εr = n2 , therefore Δn = (χ2/2n) E..........(ii)
Thus, change in r.i. is directly proportional to the applied electric field.

Are eq(i) & eq(ii) are both physically same?

Thank you