Theoretical Framework of Gaussian Beam Propagation
~ Analytical Methods Using the q-Parameter and ABCD Matrices ~
table of contents:
1. Shape of a Gaussian beam
First of all, what is a Gaussian beam? A Gaussian beam is a light wave whose intensity distribution follows a normal distribution (Gaussian distribution) even in a cross-section perpendicular to the direction of light propagation. Laser light is an example of this. It can be represented in the following diagram.
Figure 1-1
Here, each variable is defined as follows:
z: Position in the direction of light propagation
w(z): Beam radius at position z
r: Distance from the central axis
I(r,z): Light intensity distribution in the r direction at position z
I0 (z): Light intensity at the central axis at position z
R(z): Radius of curvature of the wavefront at position z
In a light intensity distribution I(r,z), the value of r at position z that results in a light intensity of 1/(e^2) times that of the light intensity I0(z) at the central axis (r=0 ) is defined as the beam radius w(z). In this case, I(r,z) can be expressed as follows.
Furthermore, each parameter is defined as follows:
λ : wavelength
L: Beam waist position
wo : Beam waist radius
ZR : Rayleigh length
θ: Divergence angle
Regarding the color coding of each parameter, the values given in the settings (in the recurrence relation model shown later) are shown in blue, and the values given in the calculation formula are shown in black.
The point where the beam diameter is narrowest is called the beam waist. The distance from the beam waist to the point where the beam diameter widens to √2 times its minimum value is called the Rayleigh length, and the region between them is called the Rayleigh region. This is an important indicator of how far a laser beam can maintain a constant diameter without widening. The Rayleigh length ZR is expressed by the following equation, using the beam waist radius w0 and the wavelength λ .
The beam radius w(z) at position z is given by the beam waist radius w0, beam waist position L, and Rayleigh length ZR. It can be expressed using the following equation.
Substituting z = L ± ZR into (1)-③ and calculating, we can confirm that, as defined, w(z) = √2 w0 .
For z = ±∞, (1)-③ converges to the following equation.
If we define the slope of this asymptote as the beam divergence angle θ, then θ can be expressed by the following equation.
Therefore, from (1)-② and ⑤, the following equation is obtained.
In other words, the value obtained by dividing the square of the beam waist radius w₀ by the Rayleigh length ZR , and the product of the beam waist radius w₀ and the divergence angle θ, are both constants λ / π, which depend only on the wavelength λ . Since the wavelength λ is known, if the beam waist radius w₀ is given, the divergence angle θ and the Rayleigh length ZR are also uniquely determined.
Furthermore, the radius of curvature of the beam wavefront at position z is expressed by the following equation.
2. Gaussian beam propagation in an ideal lens system
2-1. Recurrence Relation Model
Next, we consider a recurrence relation model of beam propagation when a Gaussian beam is passed through optical elements arranged in sequence (assuming the optical elements are ideal lenses without aberrations). The parameters representing the state and positional relationship of the beam before and after the i-th optical element are illustrated as follows.
Figure 3-1
*For dimensional notation, see Appendix: Direction and sign of one-dimensional vectors and angles
Here, the subscript i for each parameter represents a physical quantity relating to the i-th optical element from the beam incidence position. Furthermore, for each parameter, the quantity given in the settings is shown in blue, and the quantity given in the calculation formula is shown in black.
For each parameter in the above diagram, those that are the same as those shown in Figure 1-1 will be retained, while all other new ones will be defined as follows.
fi : Focal length of an optical element (ideal lens)
HH'i : Distance between the principal points of the optical element
Ri : Radius of curvature of the wavefront just before it enters the optical element .
R'i : Radius of curvature of the wavefront immediately after the optical element is emitted .
si : Distance from the optical element to the beam waist (before passing through the optical element)
s'i : Distance from the optical element to the beam waist (after passing through the optical element)
di : Distance from the i-th optical element to the (i+1)th optical element
The following are the formulas for calculating each parameter that represents the state of the beam before and after incidence to the i-th optical element.
-
The parameters shown in the previous chapter before the light enters the i-th optical element can be expressed as follows:
-
The parameters shown in the previous chapter after passing through the i-th optical element can be expressed as follows:
-
From (2)-①-1 and (2)-②-1, the following relationship can be obtained.
-
Define Ri(z) and wi(z) at position Li - si as Rp,i and wp,i respectively, and from (2)-①-2,3, we can express it as follows.
-
Define R'i(z) and w'i(z) at position L'i - s'i as R'p,i and w'p,i respectively, and from (2)-②-2,3, we can express it as follows.
-
The relationship between each parameter before and after passing through the i-th optical element is expressed as follows:
From (2)-⑤-1, it can be seen that the relationship between Ri and R'i has the same form as Gauss's imaging formula . That is, the relationship between the spherical center position of Ri and the spherical center position of R'i is the same as the relationship between the object plane and the image plane in geometric ray tracing. On the other hand, (2)-⑤-2, which shows the relationship between the beam waist positions si and s'i , has a similar but slightly different form to Gauss's imaging formula. This is a correction made to the geometric model of Gauss's imaging formula to take into account the effect of light diffraction, and was proposed by Sidney Self et al. in 1983 [1] .
From (2)-⑤-2, the relationship between si / fi and s'i / fi , with ZR,i / fi as parameters, can be illustrated as follows.
Figure 2-2
The condition ZR,i / fi = 0 is equivalent to Gauss's imaging formula in geometrical optics. In this case , s'i / fi diverges to ±∞ around si / f i = -1. As the value of Z R,i / fi increases, s'i / fi takes finite upper and lower bounds , and when si / fi = -1, s'i / fi = 1. Further increasing the value of ZR,i / fi narrows the range of possible values for s'i / fi . This means that the distance over which the beam waist can propagate through the lens is finite, and in order to make this distance as far as possible, the Rayleigh length ZR,i (before propagation) must be as short as possible, and the focal length fi must be as long as possible .
Here, we will show why (2)-⑤-5 holds true.
From (2)-⑤-2,3 and (2)-③, s'i can be expressed as follows.
From (2)-⑤-2',3, (2)-④-2b can be transformed as follows.
Therefore, it has been shown that the beam diameter is the same immediately before and after the i-th optical element is incident on it, i.e., (2)-⑤-5 holds true.
Next, we show the relationships between the parameters that hold true between the beam emitted from the i-th optical element and the beam incident on the (i+1)-th optical element. These relationships are illustrated in Figure 2-3.
Figure 2-2
Figure 2-3 shows that the following relationships hold between each parameter. Furthermore, it can be seen that all parameters except (2)-⑥-1 are passed on to the next optical element without changing their values.
Next, let's consider the initial term condition of the recurrence relation. This can be illustrated as follows.
Figure 3-2
Here again, for each parameter, the amount given by the setting is shown in blue, and the amount given by the calculation formula is shown in black.
As shown in Figure 3-2, the beam's starting point is considered to be the exit position of the virtual 0th optical element, and s'0 , L'0 , and w'o,0 are given as initial values.
Generally, the starting point of the beam is often set at the beam waist position. In this case, s'0 = 0, R'p,0 = ∞, and w'p,0 = w'o,0 .
2-2. Introduction of Complex Beam Parameters
The beam radius, wavefront curvature radius, Rayleigh length, and divergence angle derived so far are important parameters that characterize the propagation of a Gaussian beam. However, describing beam propagation using these parameters individually is not always efficient. Therefore, we introduce complex beam parameters (q-parameters) that express the beam radius and wavefront curvature radius as a single complex number. By using complex beam parameters, the propagation of a Gaussian beam can be described more concisely.
The complex beam parameters, which represent the state of the beam at position z before it is incident on the optical element i, are defined as follows:
Taking the reciprocal of this, we get the following expression:
Extract the real and imaginary parts.
Taking the reciprocals of these, we get (2)-①, which is as follows.
Therefore, (2)-⑪' can be expressed as follows.
Similarly, the following parameters are defined to represent the state of the beam after it has been emitted from the optical element i.
Here, we define the following parameters that represent qi(z) and q'i(z) at positions Li - si and L'i - s'i .
At this time, from (2)-④-1a, 2a, it can be expressed as follows.
Here, we will derive the relationship between qp,i and q'p,i .
From (2)-⑬a,b and (2)-⑤-1,5,
Here, in order to represent this transformation in matrix form, we introduce the matrix Si representing the lensing effect, which is defined by the following equation.
At this point, equation (2)-⑬' can be rewritten in the form of the following fractional linear transformation using each component of matrix Si .
Next, we derive the relationship between q'p,i and qp,i+1 .
From (2)-⑪ and ⑫, the following equation holds true.
Applying (2)-⑥-1,4 to (2)-⑮a,b yields the following equation.
Here, in order to represent this transformation in matrix form, we introduce a matrix Ti representing free-space propagation, defined by the following equation.
In this case, (2)-⑮' can be expressed as the following fractional linear transformation using each component of matrix Ti .
Using these results, we derive the relationship between qp,i and qp,i+1 .
The matrix Mi is defined as the product of the matrix Si representing lens action and the matrix Ti representing free-space propagation, as follows:
At this point, by the composition rule of the Möbius transform, the relationship between qp,i+1 and qp,i can be expressed by the following equation.
Next, we will show the conditions for the first term.
At the end of "2-1. Recurrence Relation Model," the starting point of the beam was defined as the exit position of the hypothetical 0th optical element as the initial term condition of the recurrence relation. Therefore, the initial value of the complex beam parameter is q'p,0 .
At this point, from (2)-⑬b, the general formula for q'p,0 is given by the following equation.
As mentioned above, the starting point of the beam is generally set at the beam waist position. In this case, s'0 = 0, R'p,0 = ∞, and w'p,0 = w'o,0 , so (2)-⑰ is simplified to the following equation.
Using the results derived so far, we will show a method to directly determine the complex beam parameters at the exit position of the i-th optical element from the initial conditions given at the beam's starting point.
If we define Msys,i as the composite matrix representing all optical element actions and free-space propagation from the beam's starting point to the exit position of the i-th optical element, then Msys,i can be expressed by the following equation.
However, if i=1, the following equation holds.
In this case, using the composition rule of the Möbius transform, we can directly determine q'p,i from q'p,0 , and the relationship between the two is expressed by the following equation.
Finally, we will show the relationship between q'p,i and q'i (z).
The matrix Tz,i(z) representing the free-space propagation from the exit position to position z of the i-th optical element is defined by the following equation.
In this case, using the matrix Tz,i(z), the relationship between q'p,i and q'i(z) can be expressed by the following equation.
Therefore, by sequentially applying (2)-⑱ and (2)-⑲, the complex beam parameter q'i(z) at any position z after passing the i-th optical element can be determined from the complex beam parameter q'p,0 at the beam starting point.
Finally, from (2)-⑪'a',b', the wavefront curvature radius R'i(z) and beam radius w'i(z) at position z can be found by the following equations.
3. Actual Gaussian beam propagation (introduction of M-square)
From (2)-①-1, the product of the beam waist radius wo,i and the divergence angle θi is a constant that does not depend on i. This value is defined as the beam parameter product (BPP) .
On the other hand, for a real beam, if the beam radius is wo,i,r and the beam divergence angle is θi,r , then their product is also a constant that does not depend on i. This is defined as the parameter product BPPr for a real beam.
By taking the ratio of (3)-①a and (3)-①b, the M^2 (M square) index, which represents beam quality, is defined by the following equation.
Since (3)-①a and (3)-①b are both independent of i, M^2 defined in (3)-①c is also an invariant that does not depend on i. Therefore, as long as there is no degradation of beam quality due to losses or aberrations caused by optical elements, M^2 remains constant throughout the entire optical system.
Therefore, M^2 can be determined by measuring the beam waist radius wo,i,r and beam divergence angle θi,r at any position within the optical system. In the case of fiber output, the mode field diameter of the fiber corresponds to the beam waist, so the beam waist radius wo,i,r is given as half of the mode field diameter. In this case, M^2 can be calculated by measuring the beam divergence angle θi,r immediately after fiber exit .
The value of M^2 is greater than or equal to 1, and the closer the value is to 1, the closer it is to an ideal Gaussian beam. Here, on this page, an ideal Gaussian beam is treated as a beam that maintains a normally distributed (Gaussian) intensity distribution during propagation. It should be noted that since M^2 is an index that represents the degree of deviation from an ideal Gaussian beam, its evaluation accuracy may decrease for beams with a highly distorted intensity distribution.
Here, by transforming (3)-①c, we obtain the following equation.
Comparing this with equation (3)-①a, we can see that the relationship for a real beam is expressed in the same form as for an ideal Gaussian beam, except that the wavelength λ is replaced by λM^2 . Therefore, we define the effective wavelength as follows.
In this case, by replacing λ with λeff in each relation derived for an ideal Gaussian beam , those equations can be directly applied to real beams as well.
-
Excel calculation format
Using the calculation model explained so far, we have created an Excel simulation format that actually simulates the propagation of a Gaussian beam. Please feel free to use it.
-
Related Topics
We have shown the propagation of a Gaussian beam in an ideal lens system, but the geometric model of light propagation is different. This is shown below.
While we have shown the ideal lens system up to this point, actual lenses are affected by aberrations and effective diameter. It is recommended to ensure an effective diameter at least twice the beam diameter, and some of the reasons for this are explained below.
Up to this point, we have treated optical elements as ideal lenses, but the relationship between the shape of an actual lens (single lens) and its focal length is shown in (5)-②b' below (the Lens-Maker's Formula).
-
References
-
Related literature
[A] Kishikawa, Toshiro. Yūzā Enjinia no Tame no Kōgaku Nyūmon (An Introduction to Optics for User Engineers). Optronics Co.
-
Update History
-
2026-10 Major Revision and Introduction of the q-Parameter
-
2026-10 Revised the equation for R to fix an error.
-
2025-11 Completely revised
-
2025-07 Newly released