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Recurrence formula model for Gaussian beam propagation through ideal lenses

table of contents:

・Gaussian beam shape

・Recurrence formula model for ideal Gaussian beam propagation

・Recurrence formula model for real Gaussian beam propagation

・Excel calculation format

  • 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.

This shows the beam shape of a Gaussian beam. Any cross section has a normal distribution. In the Z direction, the area within the Rayleigh region is called the beam waist.

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.

A recurrence model for Gaussian beam propagation through an ideal lens is described.

*For dimensional notation, see Appendix: Direction and sign of one-dimensional vectors and angles

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.

The formulas for calculating each parameter for the i-th optical element are shown below.

The formulas for calculating each parameter of an ideal Gaussian beam 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].

from (1)-①-6:

gaussian-beam-propagation_058.png

The relationship between si / fi and s'i / fi can be illustrated as follows, with ZRi / fi as a parameter .

Graph of the relational expression for propagation through a lens at the beam waist position

ZRi / fi = 0 represents the same state as Gauss's imaging formula in geometric optics. In this case, s'i / fi diverges to ±∞ when si / fi = -1. As the value of ZRi / fi is increased from here, s'i / fi begins to take on finite upper and lower limits, and when si / fi = -1, s'i / fi = 0. As the value of ZRi / fi is further increased, the range of possible values for s'i / fi narrows. This means that there is a finite distance that the beam waist can travel through the lens, and to make this distance as long as possible, the Rayleigh length ZRi needs to be as short as possible and the focal length fi needs to be as long as possible, which is before propagation.

Next, the recurrence formula that holds between the i-th optical element and the (i+1)-th optical element is as follows:

gaussian-beam-propagation_019.PNG

Here, the beam diameters just before and just after entering the i-th optical element can be calculated as follows:

Calculations show that the diameter of a Gaussian beam is the same before and after passing through an optical element.
Calculations show that the diameter of a Gaussian beam is the same before and after passing through an optical element.
gaussian-beam-propagation_022.PNG

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:

gaussian-beam-propagation_023.PNG

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 .

gaussian-beam-propagation_024.PNG

  • 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:

gaussian-beam-propagation_026.PNG

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.

gaussian-beam-propagation_027.PNG

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:

gaussian-beam-propagation_029.PNG

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.

A recursive model of Gaussian beam propagation through an ideal lens is presented, taking into account M-square.

*For dimensional notation, see Appendix: Direction and sign of one-dimensional vectors and angles

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.

The formulas for calculating each parameter of an actual Gaussian beam 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:

gaussian-beam-propagation_033.PNG

There are several cases for what can be considered a known quantity in the first term of the recurrence formula, but here, the first term wo1_real of woi_real and the first term θ1_real of θi_real are given as set values. These are given by actual measurements, or in the case of fiber input, the mode field diameter of the fiber corresponds to 2 x wo1_real. The conditions for the first term can be illustrated as follows:

gaussian-beam-propagation_034.PNG

In this case, M2 is given by:

gaussian-beam-propagation_043.png

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.

gaussian-beam-propagation_044.png

  • 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)

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