In the following, we consider only one-dimensionally incommensurate modulated crystals. The structure of the object is represented by the potential distribution function, which is now aperiodic and can be considered as a 3-dimensional hypersection j(x ₁, x₂, x₃, 0) of the 4-dimensional periodic function j( x₁, x₂, x₃, x₄). The position vector in the 4-dimensional direct space is

x_s= x₁ a₁ +  x₂ a₂ + x₃a₃ + x₄a ₄ ,     (1)

where a₁ , a₂, a₃, and a₄ are the four vectors defining the 4-dimensional unit cell, which is related to the 3-dimensional unit cell of the basic structure by

a ₁=a -q ₁d ,
a ₂= b -q ₂d ,
a ₃= c- q ₃d ,
a ₄= (0, d) .     (2)

Where a, b and c are the three vectors defining the 3-dimensional unit cell of the basic structure. d is the unit vector along the extra dimension, which is perpendicular to the 3-dimensional physical space, i.e. simultaneously perpendicular to a, b and c. q₁, q ₂andq₃ are components of the modulation wave vector q = q₁ a* + q₂b* + q ₃c* . a*, b* and c* are the three vectors defining the reciprocal unit cell of the basic structure.

In the 4-dimensional reciprocal space, a position vector is expressed as

h_s= h₁ b₁ +  h₂ b₂ + h₃b₃ + h₄b ₄  .     (3)

Where b₁ , b₂ , b₃and b₄ are the four vectors defining the 4-dimensional reciprocal unit cell. a_i and b _i are related by

a_i ^.b _j = d_ij      (i, j =1, 2, 3, 4) .     (4)

Hence

b₁ = (a*, 0) ,
b₂ = (b*, 0) ,
b₃ = (c*, 0) ,
b₄ = (q , d) .      (5)

Suppose that the incident electron beam is parallel to a₁ and that there is no modulation along this direction, i.e. q₁ = 0 and thus a₁= a. According to the multislice method, the dynamical electron diffraction wave function of the object can be expressed as

.     (6)

Where

,     (7)

s = p / lU  ,Uis the accelerating voltage,

is the potential distribution function of the jth slice projected along a₁ and cut perpendicular to a₄ at x₄ = 0 ,
F_j(0, k, l, m) is the structure factor of the jth slice on the section (h = 0) of a 3-dimensional hyperplane projected from the 4-dimensional reciprocal space, .     (8)

Since  j _j(x₂, x₃, 0)  is an aperiodic function, the multislice calculation will be much more complicated than that for conventional crystalline samples. To avoid the difficulty, We calculate Q(h'_s) instead of Q(h) .

.     (9)

Where

,     (10)

  is periodic and is the 4-dimensional potential distribution function of the jth slice projected along a₁ , the direction of the incident electron beam,
F_j(0, h₂, h₃, h₄)  is the structure factor of the jth slice on a 3-dimensional hyper section at h ₁= 0 in the 4-dimensional reciprocal space. .     (11)

Now Q(h'_s) can be calculated similar to that for conventional structures and then Q(h) can be obtained by projecting Q(h'_s) along a₄ , the direction of the extra dimension. The validity of doing this is proven below

For the reason of simplicity, we will prove only

.     (12)

Notice that the Fourier transform of a function's projection will be the section of the function's Fourier transform through the origin and perpendicular to the projecting direction. Since

,     (13)

we have from the left-hand side of equation (12)

.     (14)

Hence

     .     (15)