Light is a transverse electromagnetic wave.  The electric field of an electromagnetic plane wave is of the form 
E(r,t) = E_maxcos(k∙r - ωt + φ),  E_max k. 
The electric field of an EM plane wave traveling in the x-direction therefore may be written as
E(x,t) = E_maxcos(kx - ωt + φ),  E_max i.
If we consider a linearly polarized plane wave and we orient our coordinate system so that E_max is directed along the y-axis then we can drop the vector notation and write
E(x,t) = E_maxcos(kx - ωt + φ), E = E_y.
A more convenient notation is complex notation.  For any angle θ, 
cosθ = Re(exp(iθ)), exp(iθ) = cosθ + i sinθ.
We therefore rewrite the above equation as
E(x,t) = E_max exp(i(kx - ωt + φ)), E = E_y.
When we use complex notation to specify the electric field of an electromagnetic wave, it is implied that only the real part of the equation describes the electric field.

Double or multiple slit interference (interference by division of wave front)

The coherent sources for division of wave front interference are obtained by dividing a wave front originating from a single source.  Double slit interference is an example of division of wave front interference.

Let us calculate the interference pattern in the Fraunhofer regime produced by monochromatic light after diffraction on multiple parallel slits.  Assume the slits have width w and are spaced by a distance d.

A single slit produces an electric field distribution

E_slit(θ) = E₀cos(kr₀ - ωt) sin(β)/β, with β = πw(sinθ)/λ.

The total electric field E_total(θ) from N slits is just the sum of the contributions from the individual slits.  Because the sources are coherent, the fields from the individual sources interfere.  We must take into account relative phases of the contributions from individual slits to a point on a far away screen.

Consider even number of slits, with a first pair of slits (a and b) at distances ±d/2 from the central line, the second pair at distances ±3d/2 and so on.  For an angle θ, the path difference to a far away point for successive pairs of slits is

±dsinθ/2,  ±3dsinθ/2, ..., ±(n-½)dsinθ/2.

Therefore the phase difference or successive pairs of slits is

Δφ₁ = ±φ/2, Δφ₂ = ±3φ/2, Δφ₃ = ±5φ/2 ,..., Δφ_n = ±(2n-1)2πdsinθ/2 = ±(n-½)φ,
where φ ≡ 2π(dsinθ/λ).

To find the total field at an angle θ we add the contributions of all pairs.

Using complex notation we have

E_total(θ) = E₀ [sin(β)/β] Σ_n=1^N/2[exp(i(kr₀ - ωt + Δφ_n)) + exp(i(kr₀ - ωt - Δφ_n))]
= 2E₀ [sin(β)/β] exp(i(kr₀ - ωt)) Σ_n=1^N/2cos(Δφ_n).

The intensity is proportional to the square of the field,  I(θ) Re(E_total(θ))².

For N evenly spaced slits
I(θ) = 4I₀ (sin(β)/β)² [ Σ_n=1^N/2cos(Δφ_n)]²,
I(θ) = 4I₀ [sin(πw(sinθ)/λ)/(πw(sinθ)/λ)]² [ Σ_n=1^N/2cos((n-½) 2π(dsinθ/λ))]²,

where I₀ is the intensity at the center at the center of the pattern,
I(θ = 0) = I₀ = N²I_{single slit}.

Evaluating the sum we can write
I(θ) = I₀[sin(πw(sinθ)/λ)/(πw(sinθ)/λ)]²*[sin(πNd(sinθ)/λ)/(sin(πd(sinθ)/λ)]².

The interference pattern of multiple slits is the product of the diffraction pattern from a single slit of width "w" and the interference pattern from multiple very narrow slits of slit spacing d.  For the double slit we obtain

I(θ) = 4I₀ [sin(πw(sinθ)/λ)/(πw(sinθ)/λ)]² cos²(π(dsinθ/λ))².

Double slit intensity pattern, d > a (on the plot the width a is marked as b). Single slit diffraction factor oscillates with lesser frequency and modulates the higher frequency multi-slit interference factor, which determines maxima and minima of intensity.

Double slit intensity pattern, d > w.

I(θ) ∝ [ Σ_n=1^N/2cos((2n-1)π(dsinθ/λ))]² for N evenly spaced slits.

This pattern has maxima where all cosine terms are either 1, or all are -1 .
All cosines are +1, when (2n-1)π(dsinθ/λ) = 2mπ, or dsinθ/λ = n, n odd integer.
All cosines are -1, when (2n-1)π(dsinθ/λ) π = (2m+1)π, or dsinθ/λ = n, n even integer.
So the angles θ of the directions of intensity maxima in diffraction pattern for N evenly spaced slit satisfy
dsinθ = nλ, the same as he directions of intensity maxima for double-slit interference pattern.
The value of n is called the order of the maximum .