2.1.

yybar Diagram

The yybar diagram^7,8 was developed as a graphical representation of geometrical paraxial optics by Delano. One plots the paraxial chief ($\overline{y}$) and marginal ($y$) ray heights at each surface as they are sequentially encountered through the optical system. From the plot, one can visually infer much information about the system and the individual components that form it, and there are simple algebraic equations that allow for precise numerical evaluations of the system (Fig. 1).

Fig. 1

(a) Marginal and chief rays given a simple lens with the stop near the FFP. (b) The yybar diagram for the same lens system.

2.2.

Modeling Gaussian Beams with the yybar Diagram

The yybar method was first applied to laser beam propagation by Kessler and Shack.⁹ To make the correspondence, consider that a Gaussian beam can be represented by two rays; a divergence ray and a waist ray (Fig. 2), both of which follow all the standard paraxial ray optics relations for refraction and translation.

Fig. 2

Gaussian beam $1/e^2$ point curves shown near waist, with divergence and waist rays indicated.

We arbitrarily define the height of the waist ray as $y$ and the height of the divergence ray as $\overline{y}$ and plot the result (Fig. 3).

Fig. 3

A yybar diagram, red dashed line, for the propagating beam in Fig. 2, Note that line is tangent to blue circle of radius wo.

In the same way that paraxial optics parameters can be inferred graphically from the yybar diagram so can Gaussian beam parameters. A primary one is the location of the beam waist. In Fig. 3, the red dashed line represents the propagating Gaussian beam. The beam waist is located where $\overline{y}=0$, which is also where the line is closest to the yybar origin. The beam waist size, wo, is defined by the smallest distance from the line to the yybar diagram origin. Or, the beam waist is located where the line is tangent to a circle centered on the origin, and its size is the radius of the circle.

There are also a number of simple relations that can be used to compute various parameters for the propagating beam. First, the beam size can be computed at any point along the propagating beam using the following equation:

Distances between two points, $(\overline{y}o,yo)$ and $(\overline{y}1,y1)$, along a segment defining the propagating beam is calculated using the following equation:

Note that Eqs. (2) and (3) are identical and similar, respectively, compared to those used in the yybar diagram for paraxial optics. As an example, $R(\overline{y})$ and $w(\overline{y})$ for a propagating Gaussian beam in vacuum of $\lambda =632.8\mathrm{nm}$, $wo=0.0031\mathrm{mm}$, and represented as in Fig. 3 are calculated and plotted in Fig. 4. To compute, $R(\overline{y})$ requires $\overline{u}$ and $u$. We choose to define the beam from the transmission sphere (TS) to have $y=wo$ and $u=0$ (Fig. 3) so from Eq. (3), $\overline{u}=\frac{L}{y}=\frac{\lambda }{\pi wo}$. The plots for the Gaussian beam size and radius of curvature, Fig. 4, should be recognizable.¹⁰ $\overline{y}$ is directly related to propagation distance, $z$, through Eq. (2) yielding $z=\frac{\pi wo\overline{y}}{\lambda }$. We note here that, throughout this paper, examples are based on a red HeNe laser system with a 100-mm diameter $F/7.6$ TS, which we define to have $wo=0.0031\mathrm{mm}$ at its focus.

Fig. 4

(a) The beam width, $w(\overline{y})$, through the waist and (b) radius of curvature, $R(\overline{y})$, up to the waist for the Gaussian beam in Fig. 3.

One final critical difference between the yybar representation of propagating laser beams and paraxial optics is that the meaning of the yybar diagram for laser beam propagation is unchanged when the traces are rotated about the origin. Figure 5 shows two representations of the same propagating beam with a beam waist wo. Numerically, they only differ in how the two rays are defined; Fig. 5(a) flips the ray definition set in Fig. 3, and in Fig. 5(b), neither $y$ nor $\overline{y}$ is the waist or divergence ray.

Fig. 5

Different yybar representations of the same Gaussian beam of waist radius of $wo$.

3.1.

Geometry of the Interferometric Test

The interferometer configuration analyzed in this paper consists of a Fizeau system from TS to the reference sphere under test, a small reflective sphere of radius rb, where it reflects and then returns to the TS (Fig. 6). Position of the ball, $z$, is defined by the location of the convex ball surface relative to the beam waist produced by the TS, where $z=0$ when they are coincident. Since the ball surface has power, the reflected Gaussian beam may have a different divergence, waist size, and waist location.

Fig. 6

The interferometer system being modeled, from TS to test ball back to TS.

3.2.

Analysis of Interferometric Test

The goal of this model is to analyze where the test ball must be to yield null fringes. Logically, this will occur if the reflected beam re-entering the TS meets two constraints: it has the same beam size and the same radius of curvature that the beam had when it initially exited the TS, $w_{\mathrm{TS}}={w_{\mathrm{TS}}}^{\prime }$ and $R(z_{\mathrm{TS}})=R{(z_{\mathrm{TS}})}^{\prime }$ where the primes indicate the beam after reflection from the reference ball. If this condition is met, one can also state that the reflected beam waist size and position must be identical to the initial beam waist size and position for the beam from the TS, $wo=w{o}^{\prime }$ and ${w(0)}^{\prime }=wo$. For simplicity, we again define the beam from the TS as a horizontal line of height wo. The TS beam, red lines in Fig. 6, interacts with the test surface at some position $z$ measured from the incident beam waist, which is located at $(\overline{y},y)=(0,wo)$. Since the divergence and waist rays still follow the laws of paraxial optics, the waist ray, $u=0$, will virtually go to a height of zero at the front focal point (FFP) of the convex ball a distance $\frac{-rb}{2}$ from the reflecting surface. Since we also know $\overline{u}$, the value of $\overline{y}$ at the FFP is

To ensure the first constraint, the reflected beam must be on a line that is tangent to the centered circle of radius wo. After some algebra, this results in the following equations for deriving two $\overline{y_p}$, values:

Fig. 7

The yybar diagram for both the null and cat’s eye positions, from TS to test ball back to TS’.

These are the well-known null and cat’s eye positions—$({\overline{y}}_{pn},y_{pn})$ and $({\overline{y}}_{pc},y_{pc})$.¹¹ But at this point, we have only met one of the requirements stated above: $wo=w{o}^{\prime }$. We prove that the other condition is automatically met based on the properties of the yybar diagram. Since the reflected and incident beams are both represented by lines tangent to the same circle, and the test ball position is located at the two lines’ intersection, and the initial and reflected waists are located at the tangent points of the lines, the area of triangle [(0, 0), $(0,wo)$, $({\overline{y}}_p,wo)$] is identical to the area of triangle [(0, 0), $({\overline{y}}_{wo},y_{w0})$, $({\overline{y}}_p,wo)$] (Fig. 8). In a yybar diagram, this means the distances are identical and therefore the incident and reflected beam waists are located at the same position; both requirements for null fringes are met. The identical waist positions can also be derived algebraically using Eq. (2), computing the distance from the TS to the ball, and then from the ball back to the TS, or the distances from the ball to the incident and reflected waists.

Fig. 8

Area is proportional to distance in a yybar diagram, so the triangles show the distance from the ball surface to the incident or reflected waist is identical for the (a) null or (b) cat’s eye position; the reflected and incident waists are coincident.

Also note that Eqs. (6) and (7) reveal that when $\overline{y}F=wo$, there is only one solution. We define this as the minimum test ball size, using Eq. (5) and $\overline{y}F=wo$,

When $\overline{y}F<wo$, Eqs. (6) and (7) yield imaginary values for ${\overline{y}}_p$; there is no solution for this case. This is readily observed from the yybar diagram since any beam represented by a line that passes through a point inside the circle of radius wo cannot have a beam waist size of wo, thus the two constraints cannot be met. We must redefine what constraints should be met to achieve a null fringe when the ball radius is below the minimum defined by Eq. (8). One approach is that only the radius of curvature of the reflected beam at the TS needs to match the radius of curvature of the initial beam at the TS. By equating the distances from the TS to the ball, and from the ball back to the TS, and then also making Eq. (4) match for the beam first leaving and then entering the TS after reflection, we find that the solution is that the reflected beam is represented by a ray parallel to the $y$ axis, or $\overline{y}=\frac{-rb\min }{2}$. Note that now, the divergence and waist rays are defined by $y$ and $\overline{y}$, respectively. The location of the reflected TS is such that the beam size ($w^{\prime }$) is far larger than when the beam exited the TS, so the contrast of the fringes will be lower. This is again evident based on the area of triangles; the distance from the TS to the ball is measured along the line $y=wo$, but back to ${\mathrm{TS}}^{\prime }$ along $\overline{y}=x$, where $x<wo$. To match areas, ${{\overline{y}}_{ts}}^{\prime }\sim {\overline{y}}_{ts}\times \frac{wo}{x}$, and since both ${{\overline{y}}_{ts}}^{\prime }$ and ${\overline{y}}_{ts}$ are far larger than $y_{ts}=wo$ and ${{\overline{y}}_{ts}}^{\prime }=x$, respectively, $w_{ts}\sim {\overline{y}}_{ts}$ and ${w_{ts}}^{\prime }\sim {{\overline{y}}_{ts}}^{\prime }$. So, if $\overline{y}F=\frac{wo}{2}$, ${w_{ts}}^{\prime }\sim 2w_{ts}$. One final point to note is that this analysis shows how the accuracy of the standard method for determining the radius of curvature of a surface, measuring the distance between the null and cat’s eye positions,¹² rapidly degrades as the radius approaches the minimum. For example, Fig. 9 is a plot of the absolute percent error in logarithmic scale $(100[\frac{\text{actual}-\text{measured}}{\text{actual}}])$ given the $F/7.6$ TS used throughout this paper.

Fig. 9

The percent error in logarithmic scale of the measured radius of curvature using the null to cat’s eye test.

For reference, radii of 2, 0.6, 0.2, and 0.093 mm yield percent errors of 0.1%, 1%, 10%, and 100%, respectively.