Depth of focus simplified for a laser-optical system James T. Luxon

GMI Engineering & Management Institute, Science-Mathematics Department, Flint, Michigan 48504. Received 26 August 1987. 0003-6935/88/101910-02$02.00/0. © 1988 Optical Society of America. Depth of focus (sometimes incorrectly referred to as depth

of field) is the allowable distance from the point of best focus (smallest spot size), which produces an acceptable spot size. Depth of focus is, therefore, a tolerance range established by the requirements for a particular application. In material processing the tolerance is determined by a threshold power density (irradiance), which if not exceeded results in poor quality or even cessation of the process.

Clearly, there is no unique definition of depth of focus, but useful formulas have been presented for TEM00 and higher-order mode beams. These equations are generally given in terms of focused spot size or ƒ/No.1 A general expression for depth of focus is2

In Eq. (1) ρ gives the allowable spot size variation (e.g., ρ = 1.1 for a 10% increase), W0 is the focused spot radius for circular modes or the spot center to outer peak center dis­tance for rectangular modes, and C is a factor that accounts for higher-order modes. For a circular (TEMpq) mode C = 2P + q + 1.3 For a rectangular mode C = (l + ½)/kl where l is either mode number, generally the largest one, and kl is a function of mode number given approximately by kl = 1 + 0.73l-0·78.

If w2 is the beam size entering the lens, the focused spot size is given by

where s' ≈ ƒ is the lens to waist location distance. For welding and material removal applications the difference between s' and ƒ is <1%. Hence Eq. (1) becomes

Equation (3) illustrates the danger of expressing a quantity such as depth of focus as an explicit function of spot size. High-power CO2 laser systems have ƒ/Nos. comparable with lower-power Nd:YAG and Nd:glass laser systems and, there­fore, actually may benefit from operating at a longer wave­length.

A third, simple but useful, expression for depth of focus can be written by realizing that depth of focus is a function of Rayleigh range. Equation (1) becomes

where Z R = πw2/(Cλ) is the Rayleigh range produced by the lens (or mirror).

Self4 has presented an equation which relates the Rayleigh range produced by an equivalent thin lens to the Rayleigh range of the incident laser beam. This equation is

where Z′R is the Rayleigh range produced by the lens, Z R is the Rayleigh range of the incident beam, and m is magnifica­tion given by

1910 APPLIED OPTICS / Vol. 27, No. 10 / 15 May 1988

In Eq. (6) s is the distance from the waist of the incident beam to the lens.

If there is more than one lens in the system or reference is made back to the waist produced by the laser and the output coupler acts as a lens, then

where there are as many magnifications terms as there are lenses.

Since the Rayleigh range is a geometrical quantity, the depth of focus can be expressed as

independent of mode and wavelength. All that is required is knowledge of the resonator geometry and focusing element focal lengths. This is particularly useful in applications where a variable beam path is encountered, such as with large gantry-robot systems. In a system where only one lens or mirror is used to focus the beam m2 will be the only variable, since it is the only quantity that is a function of distance from the laser to the focusing element.

References 1. D. C. O'Shea, W. R. Callen, and W. T. Rhodes, Introduction to

Lasers and Their Applications (Addison-Wesley, Reading, MA, 1978), Chap. 9.

2. J. T. Luxon and D. E. Parker, "Higher-Order CO2 Laser Beam Spot Size and Depth of Focus Determination," Appl. Opt. 20, 1933 (1981).

3. R. L. Phillips and L. C. Andrews, "Spot Size and Divergence for Laguerre Gaussian Beams of Any Order," Appl. Opt. 22, 643 (1983).

4. S. A. Self, "Focusing of Spherical Gaussian Beams," Appl. Opt. 22, 658 (1983).

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