Position-Sensitive Measurements of a Single-Mode Laser Beam Spot Using the Dividing Plate Method

Abstract

New position-sensitive experimental results have been obtained using the Dividing Plates method to detect the displacement of a single-mode laser beam spot of various diameters. Measurements show that with this method it is possible to attain sensitivities down to 0.18 µm^–1 for a laser beam spot diameter of 8.8 µm displaced over 2.4 µm diameter zone, defined as the area where measurements non-linearities are within 1%. The achievable measurement accuracy is limited by ADC noise and corresponds to ±0.030 nm. Application of the Dividing Plates method for position-sensitive measurements allows to build a compact version of the Precision Laser Inclinometer, the Compact PLI, with reduced overall dimensions of 20 × 20 × 20 cm³ and weight of approximatively 10 kg.

Appendices

APPENDIX A

When dividing a laser beam with a Dividing Plate, it is necessary to determine the difference signal from two photodetectors, to which the transmitted and reflected parts of the laser beam are directed. Figure 29 shows the profile of a single-mode laser beam with a diameter of 200 µm, divided into two parts by a Dividing Plate.

Fig. 28.

Separation of a 200 μm diameter single-mode laser beam with a Dividing Plate.

Fig. 29.

Determination of the difference between the signal ΔK (n; n + 1) in the adjacent values of α_n and α_{n + 1}.

We can calculate the relative difference in the powers of the laser beams incident on the PhD₁ and PhD₂ photodetectors (Fig. 11).

The power P₁ of the laser beam transmitted by the Dividing Plate and reflected power P₂ are shown in Fig. 28

$$\begin{gathered} {{P}_{1}} = \frac{{2P}}{{\pi {{r}^{2}}}}{{P}_{y}}{{P}_{x}} = \frac{{2P}}{{\pi {{r}^{2}}}}\left( {\mathop \smallint \limits_{ - \infty }^\infty \exp \left( { - \frac{{2{{y}^{2}}}}{{{{r}^{2}}}}} \right)dy} \right) \\ \times \,\,\mathop \smallint \limits_{ - {\text{w}}}^a \exp \left( { - \frac{{2{{x}^{2}}}}{{{{r}^{2}}}}} \right)dx \\ = P\frac{1}{2}\left\{ {{\text{erf}}\left( {\frac{{w\sqrt 2 }}{r}} \right) + {\text{erf}}\left( {\frac{{\alpha \sqrt 2 }}{r}} \right)} \right\}, \\ {{P}_{2}} = \frac{{2P}}{{\pi {{r}^{2}}}}{{P}_{y}}{{P}_{x}} = \frac{{2P}}{{\pi {{r}^{2}}}}\left( {\mathop \smallint \limits_{ - \infty }^\infty \exp \left( { - \frac{{2{{y}^{2}}}}{{{{r}^{2}}}}} \right)dy} \right) \\ \times \,\,\mathop \smallint \limits_\alpha ^{\text{w}} \exp \left( { - \frac{{2{{x}^{2}}}}{{{{r}^{2}}}}} \right)dx \\ = P\frac{1}{2}\left\{ {{\text{erf}}\left( {\frac{{\alpha \sqrt 2 }}{r}} \right) - {\text{erf}}\left( {\frac{{w\sqrt 2 }}{r}} \right)} \right\}, \\ {{P}_{1}} - {{P}_{2}} = P{\text{erf}}\left( {\frac{{\alpha \sqrt 2 }}{r}} \right), \\ {{P}_{1}} + {{P}_{2}} = P{\text{erf}}\left( {\frac{{{\text{w}}\sqrt 2 }}{r}} \right), \\ \frac{{{{P}_{1}} - {{P}_{2}}}}{P} = \frac{{{\text{erf}}\left( {\frac{{\alpha \sqrt 2 }}{r}} \right)}}{{{\text{erf}}\left( {\frac{{{\text{w}}\sqrt 2 }}{r}} \right)}}, \\ \end{gathered} $$

(A1)

where P is the power of the laser beam, α is the distance between the center of the laser beam and the dividing line of the dividing plate, r is the radius of the single-mode laser beam; w. the quantity w = 300 µm was chosen as integration limit to be large enough to take into account the entire power of the single-mode laser beam for the numerical calculation of the integral.

For w = 300 µm and change r < 100 µm we get ${\text{erf}}\left( {\frac{{w\sqrt 2 }}{r}} \right) \approx 1$.

In this case, (A1) can be written as

$$\frac{{{{P}_{1}} - {{P}_{2}}}}{P} = {\text{erf}}\left( {\frac{{\alpha \sqrt 2 }}{r}} \right).~$$

(A2)

APPENDIX B

From the dependence of the dimensionless signal of the photodetectors (Fig. 8) on the transverse displacement α of the laser beam, we determine the differences in the signal ΔK (n; n + 1) in the adjacent intervals α_n and α_{n + 1}

To do this, we subtract the value of the signal calculated according to the formula in Appendix A (A1) and corresponding to the number n from the value of the signal corresponding to the next number n + 1. In Fig. 29 shows the dependence of the indicated differences in the signals on the displacement α for a single-mode laser beam with a diameter D_f = 60 μm.

The data obtained are approximated by a Gaussian distribution, as expected from the differentiation of a sigmoid, with a width of D_f = 60 µm corresponding to the initial conditions (Fig. 29).

Fig. 30.

Dependence of the difference in the signal ΔK_{n,n + 1}/(α_n – α_{(n + 1)}) on the displacement α.

APPENDIX C

DEPENDENCE OF THE SENSITIVITY OF THE DIVIDING PLATE METHOD ON THE DIAMETER OF THE FOCUSED SPOT OF THE LASER BEAM

Using the definition of the sensitivity of the dividing plate method, as the ratio of the change in the dimensionless signal ∆K to the displacement ∆α in the region of the point (0; 0) by the formula (3), we calculate the value of the sensitivity T for different diameters of the laser beam on the basis of the graph in Fig. 8. Figure 31 shows the calculated sensitivity T of the DP method depending on the diameter D_f of the laser beam.

Fig. 31.

Dependence of the calculated sensitivity T of the DP method from the diameter D_f  of the laser beam.

APPENDIX D

INFLUENCE OF THE DIVIDING LINE FABRICATION ERROR ON THE MEASUREMENT ACCURACY OF LASER BEAM OSCILLATIONS IN THE DIVIDING PLATE METHOD

As seen from Fig. 25, the dividing line of the dividing plate can deviate from straightness by 0.1 µm over a length of 1 µm. With a longitudinal displacement (Fig. 31) of a single-mode laser beam along the division line, the non-straightness of the division line will affect the signal K in the PSD using the dividing plate method. This is due to a change in the reflection conditions of a single-mode laser beam when it is displaced.

It is possible to estimate the magnitude of the change in the signal K with the longitudinal displacement of the laser beam.

In Fig. 31 shows the location of the spot of a single-mode laser beam on a dividing plate with a non-rectilinear dividing line.

The dividing strip is a zone of non-uniform metal coating with a vertical width η. For calculations, we define a dividing strip with a width η and a length L₀ = 2D_f, in which the dividing line is located.

Dividing the strip into squares η × η, we get $N = \frac{{2{{D}_{{\text{f}}}}}}{{{\eta }}}$ squares. Each square is partially coated with metal. Figure 32 shows a section of the dividing strip of the dividing plate with the deposited metal 0.1 µm wide. Using random numbers, a sequence from 0 to 0.1 µm in the amount of N can be defined, which determines the degree of metal coverage of the squares of the strip.

Fig. 32.

Location of the laser beam spot on the dividing plate.

When using a laser source with a rectangular power distribution, the reflection of the laser beam by the squares of the dividing strip will be proportional to the area of partial metal deposition in each square.

Figure 33 shows the change in the reflected power of the laser beam with a rectangular power distribution from the dividing strip.

Fig. 33.

Section of the dividing line of the dividing plate with a metal coating that changes randomly from 0 to η (η = 0.1 μm) over a length L.

When using a laser beam with a Gaussian power distribution in its cross section, the magnitude of the reflected power in the squares of the dividing strip will depend on their position in the cross-section of a single-mode beam and on the size of the metal coverage in squares.

Using the formula (A1), we determine the ratio K of the power P₁–P₂ incident on the strip η × 2D_f to the total power P of the laser beam

$$K = \frac{{{{P}_{1}} - {{P}_{2}}}}{P} = \frac{{{\text{erf}}\left( {\frac{{{{\mu }}\sqrt 2 }}{r}} \right)}}{{{\text{erf}}\left( {2\sqrt 2 } \right)}}.{{\;\;\;}}$$

(A3)

From the formula (A3) we determine the relative power reflected from the metal-covered dividing strip. Figure 34 shows the reflection of a single-mode laser beam with a diameter D_f = 10 μm from the dividing strip covered with metal, provided that the reflected relative total power is K. The dividing strip has dimensions L₀ = 20 µm (Fig. 1(4)) and η = 1 µm and is divided into squares 1 × 1 µm.

Fig. 34.

Change in the reflected power of the laser beam with a rectangular power distribution from the dividing strip.

Each dividing strip element is partially covered with metal, its reflectivity depends on the area of the metal coverage. Using random numbers, it is possible to define with a sequence from 0 to 1 the degree of metal coverage of the squares of the strip. This approximation makes it possible to evaluate the effect of the reflection of the laser beam from the non-rectilinear dividing line. Multiplying the sequence of random numbers with the magnitude of the reflection of the single-mode laser beam from the strip covered with metal, the reflection of the beam from the non-rectilinear dividing lines obtained. Figure 5(4) shows the reflection power of a single-mode laser beam from a non-linear dividing line.

Fig. 35.

Reflected power of a single-mode laser beam with a diameter of D_f = 10 μm from a dividing strip dimensions 20 μm by 1 μm with a division into squares 1 × 1 μm.

Shifting the single-mode laser beam at a distance of 1 μm along the dividing strip determine the type of reflection of the displaced laser beam Fig. 37.

Fig. 36.

Reflected power of a single-mode laser beam from a non-linear dividing line.

Fig. 37.

Reflected power of a 1 μm offset single-mode laser beam from a non-linear dividing line.

The reflected total power before the shift of the laser beam K₁ and after K₂ can be introduced. Subtracting the power K₁ (Fig. 36) before the shift of the single-mode laser beam and after K₂, the change in the relative power ΔK = K₁ – K₂ due to the non-straightness of the dividing line of the dividing plate can be obtained.

The analysis was carried out for two cases:

(1) For non-straightness η = 1 µm and a shift of the laser beam at a distance of 1 µm, the change in the relative power will be ΔK = 1.3 × 10^–3.

(2) For the non-straightness η = 0.1 µm (corresponding to the non-straightness of the razor blade, Fig. 25) and the shift of the laser beam at a distance of 0.1 µm, the change in the relative power will be ΔK = 5.8 × 10^–7.

GLOSSARY

γ	angle of the earth’s surface
1, 2, 3, 4,	photodetectors in a quadrant photodetector
D	diameter of the laser beam
d	is the width of the dielectric gap in a quadrant photodetector
L	width of position sensing sensor
I1, I2	currents with PSD
Y	is the distance between the first electrode and the center of the laser beam spot on the PSD
a	pixel size in CCD matrix
PhD1, PhD2	photodetectors in the dividing plate method
P	is the power of the laser beam
Α	is the distance between the center of the laser beam and the dividing line of the separating plate
r	is the radius of the single-mode laser beam
w = 300 µm	limits of integration
${{P}_{1}}$	is the power of the laser beam passed by the separating plate
${{P}_{2}}$	the power of the laser beam reflected from the separating plate
PSD	Position sensitive photodetector
DP	Dividing plate
${{U}_{1}},{{U}_{2}}$	signals from photodetectors PhD1, PhD2
$K = \frac{{{{U}_{1}} - {{U}_{2}}}}{{~{{U}_{1}} + {{U}_{2}}}}$	dimensionless signal
β	non-linearity of the dimensionless signal K of the dividing plate
$\Delta U = {{U}_{2}} - {{U}_{1}}$	offset signal
δ	angle of the tangent to the sigmoid at the point with α = 0
∆K	change of the dimensionless signal
∆α	shift of the laser beam around the point [0; 0]
T	sensitivity of the dividing plate method
A	zone of linear displacements of the laser beam spot on the most separating plate
D f	is the diameter of the focused laser beam
F	lens focus length
η	non-straightness of the blade point
S 0	area arising from the displacement α of the laser beam on the dividing line of the dividing plate
n	displacement of the laser beam spot on the dividing plate in the transverse direction