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If photons come in from either side separated by a time delay τ , the corresponding state and its projection in time will be 

where ψ ( t ) is a gaussian wavepacket and ψ ( ω ) its Fourier transform. The output state will be 

For τ much larger than the wavepacket length the last two terms are completely independent; for τ = 0 they cancel each other and the photons bunch. The amplitude corresponding to photons exiting the beamsplitter from different outputs | Ψ$_{do}$ 〉 is 

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such that the total probability of photons not bunching is 

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If we define ψ ( t ) = 1 √ $_{σ}$√$_{2}$$_{π}$ e t 2 4 σ 2 $_{, such that 2}$√$_{2 ln 2}$$_{σ}$ is the pulse width at FWHM, we get 

The resulting visibility is 

If we rewrite this with the dimensionless ratio Δ = τ /σ 

and if we define a probability distribution for this ratio 

we can find the average visibility 

We also perform the same calculations using FWHMs instead of variances. We use the variables 

The final result takes the same form in new units: 

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