Recurrence formula model for Gaussian beam propagation through ideal lenses
table of contents:
・Recurrence formula model for ideal Gaussian beam propagation
・Recurrence formula model for real Gaussian beam propagation
*For more information on geometric ray propagation, please refer to the "Recurrence formula model for geometrical ray propagation through ideal lenses" at the following link.
-
Gaussian beam shape
First of all, what is a Gaussian beam? A Gaussian beam is a light wave whose intensity distribution is a normal distribution (Gaussian distribution) at any cross section perpendicular to the direction of light propagation, and laser light corresponds to this. It can be represented as shown in the diagram below.
Here, each variable is defined as follows:
z: Position in the direction of light travel
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 on the central axis at position z
The beam radius w(z) at position z is defined as the value of r when the light intensity I(r,z) at the central axis (r=0 ) becomes 1/(e^2) with respect to the light intensity I0(z) in the light intensity distribution I(r,z).
Moreover, each parameter is defined as follows.
λ: wavelength
L: Beam waist position
wo : Beam waist radius
ZR : Rayleigh length
θ: Divergence angle
The range below the Rayleigh length is called the Rayleigh region, and is an important indicator of the extent to which a laser beam can maintain a constant diameter without spreading.
-
Recurrence formula model for ideal Gaussian beam propagation
Next, let us consider the beam propagation when a Gaussian beam passes through an optical element with focal length fi arranged in order. The diagram is as follows.
The parameters in the above diagram are defined as follows:
fi : Focal length of optical element (ideal lens)
di : Distance to the next optical element
Li : Distance from the z origin to the beam waist (before passing through the optical element)
HH'i : Distance between principal points
Ri : Radius of curvature of the wavefront immediately before it enters the optical element
R'i : Radius of curvature of the wavefront immediately after it leaves the optical element
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)
Here, the subscript i of each parameter indicates the physical quantity related to the i-th optical element from the beam incidence position. The set value of each parameter is shown in blue, and the calculated value is shown in black.
Also, arrows on only one side of each dimension are one-dimensional vector notations. Their definitions are explained as follows.
・ The scalar quantity of a vector pointing to the right is a positive value, and the scalar quantity of a vector pointing to the left is a negative value.
・Multiplying the scalar quantity of each vector by -1 means swapping the start and end points of the vector and reversing its direction.
If the scalar quantity of a vector is A, and A (+) > 0 and A (-) < 0, the relationship between the scalar quantity and the vector is shown below.
The formulas for calculating each parameter for the i-th optical element are shown below.
From (1)-①-5, we can see that the relationship between Ri and R'i has the same form as Gaussian lens formula. In other words, 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 surface and the image surface in geometric ray tracing. On the other hand, (1)-①-6, which shows the relationship between the beam waist positions si and s'i, is similar to Gaussian lens formula , but has a slightly different form. This is a correction to the geometric model Gaussian lens formula that takes into account the effects of light diffraction, and is a formula proposed by Sidney Self et al. in 1983 [1].
In addition, the recurrence formula that holds between the i-th optical element and the (i+1)-th optical element is as follows:
Here, the beam diameters just before and just after entering the i-th optical element can be calculated as follows:
Therefore, it can be said that the beam diameter is equal immediately before and after it enters the i-th optical element.
Next, we will show the first term of the recurrence formula. The diagram is as follows:
As shown in the above figure, wo1, which is the first term of woi, is given as a set value. In addition, L1, which is the first term of Li, and s1, which is the first term of si, are expressed as follows using the set values s'0 and d0 .
-
Recurrence formula model for real Gaussian beam propagation
We define the product of woi and θi as BBPi, which is a constant independent of i and can be expressed as follows:
Also, when the real beam radius and the real beam divergence are given as woi_real and θi_real, respectively, and the product of woi_real and θi_real is given as BBPi_real, then the relationship between these is expressed as follows.
Although woi_real and θi_real are values obtained by actual measurement, BBPi_real is an amount independent of i. Therefore, when calculating BBPi_real, it can be measured between any optical elements as long as there is no effect of aberration of the optical elements.
By taking the ratio of (2)-①-1 to (2)-①-2, we obtain the following quantity, which represents the beam quality:
The value of M2 is 1 or more, and the closer the value is to 1, the closer it is to an ideal Gaussian beam.
Here, regarding the assumption of an actual beam, it is assumed that the beam spreads while maintaining a normal distribution (Gaussian distribution) compared to an ideal Gaussian beam, and M2 represents the degree of this spread. Therefore, it should be noted that this is not a sufficient approximation for beams where the normal distribution itself is significantly broken.
Similar to the ideal Gaussian beam in the previous chapter, we will consider the beam propagation when a Gaussian beam passes through an array of optical elements (ideal lenses) in order for an actual beam. Here, the actual measurement of the beam is performed at a position before it enters the optical system (i.e., the first term of the recurrence formula). The diagram is as follows.
The parameters in the above diagram are defined as follows:
fi : Focal length of optical element (ideal lens)
di : Distance to the next optical element
Li : Distance from the z origin to the beam waist (before passing through the optical element)
HH'i : Distance between principal points
Ri_real : Radius of curvature of the wavefront just before it enters the optical element
R'i_real : Radius of curvature of the wavefront immediately after it leaves the optical element
si_real : Distance from the optical element to the beam waist (before passing through the optical element)
s'i_real : Distance from the optical element to the beam waist (after passing through the optical element)
Here, the subscript i of each parameter indicates the physical quantity related to the i-th optical element from the beam incidence position. The set value of each parameter is shown in blue, and the calculated quantity is shown in black. Since all calculated quantities are different from the ideal Gaussian beam, the subscript "real" is used to distinguish them.
The formulas for calculating each parameter for the i-th optical element are shown below.
In addition, the recurrence formula that holds between the i-th optical element and the (i+1)-th optical element is as follows:
There are several cases for what is the known quantities in the first term of the recurrence formula, here, wo1_real, which is the first term of woi_real, and θ1_real, which is the first term of θi_real, are given as the setting values. The diagram is as shown below.
In this case, M2 is given by:
L1_real, which is the first term of Li_real, and s1_real, which is the first term of si_real, are expressed as follows using the setting values of s'0 and d0.
-
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.
・Gaussian beam propagation (ideal)
・Gaussian beam propagation (real)
-
References
[1] Sidney A. Self, "Focusing of spherical Gaussian beams," Appl.Opt. 22, 658-661 (1983).