Fresnel diffraction theory is a scalar diffraction theory.  It ignores the vector nature of the electromagnetic field.

Assume a scalar field y(x,y,z,t) = y(x,y,z)exp(-iwt) is a solution to the scalar wave equation

Ѳy(x,y,z,t) - (1/v²)¶²y(x,y,z,t) /¶t² = 0.

Ѳy(x,y,z) + k²y(x,y,z) = 0,       k²= w²/v².

Assume space is divided into two regions separated by a surface S.  Region 1 contains the sources of the field y(x,y,z).  These sources produce a wave incident on the surface S, which separates region 1 from region 2.  The field everywhere in region 2 is a solutions of Ѳy(x,y,z) + k²y(x,y,z) = 0 and can be found if the field y and its normal derivative ¶y/¶n on the surface S are known.  The assumptions that are usually made are that y(x,y,z) and ¶y/¶n vanish everywhere on S except where there is an opening.  The values of y(x,y,z) and ¶y/¶n in the opening are equal to the values of the incident wave in the absence of any obstacle or screen.

The standard calculations of classical optics are all based on this approximation.  They have only limited validity.  They are valid in the paraxial approximation.  (In the paraxial approximation the direction of the electromagnetic fields is constant to first order, so ignoring the vector nature can be justified.)

So let us keep on working in the paraxial approximation.  Assume the surface S is located in the z = 0 plane.  Region 1 is the z < 0 region.  In the paraxial approximation the wave vectors of all plane waves make small angles with respect to the z-axis, so that we can write

k_z = (k² – k_x² – k_y²)^½ » k – (k_x² + k_y²)/2k.

Any y(x,y,z) may be expanded in terms of plane waves.  The general solution to Ѳy(x,y,z) + k²y(x,y,z) = 0 in region 2 is

,

where u₀(k_x,k_y) is the amplitude of the plane wave solution with frequency w and the particular transverse components k_x and k_y of the wave vector.  In the paraxial approximation we may write

.

If in region 2 we write y(x,y,z) = u(x,y,z) exp(ikz), then

.

In order for the paraxial approximation to be valid, u₀(k_x,k_y) must vanish unless (k_x² + k_y²)/2k << 1.
If we set z = 0 then we see that u₀(k_x,k_y) is the inverse Fourier transform of u(x, y, z=0).

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[Fourier transform:

Let f(x) = ò_-¥^+¥f(k)exp(ikx)dk. Then f(k) =(1/2p) ò_-¥^+¥f(x)exp(-ikx)dk.

Define u(x₀,y₀) = u(x, y, z=0).
u(x₀,y₀) = ò_-¥^+¥ò_-¥^+¥u₀(k_x,k_y)exp(i(k_xx₀ + k_yy₀))dk_xdk_y.  Then
u₀(k_x,k_y) = (1/2p)²ò_-¥^+¥ò_-¥^+¥u(x₀,y₀)exp(-i(k_xx₀ + k_yy₀))dx₀dy₀.]

Therefore the field in a plane at z is related to the field in the z = 0 plane through

We can evaluate k-space integral by completing the square in the exponent.

,

where u₀(x₀,y₀) is the known distribution at z = 0.  This is called the Fresnel diffraction integral.

The Fresnel diffraction integral in the paraxial approximation is the convolution of u(x₀,y₀) and the Fresnel kernel h(x,y,z).

u(x,y,z) = h * u₀.

Now consider the Fraunhofer approximation.  Assume u(x₀,y₀) = 1 if |x₀| < a_x/2, |y₀| < a_y/2 and zero otherwise, and let z/k >> a_xa_y.

Then we may approximate

.

We then can evaluate the integral for u(x,y,z).  We find

,

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The intensity distribution is proportional to |y|².

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Convolution:

The convolution of two functions u(x) and v(x) is defined through

u(x')_*v(x') = ò_-¥^+¥u(x'')v(x'-x'')dx''.

The convolution integral has the following properties:

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