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Application of the ABCD ray matrix method for the design of a refractive 2.5× beam expander–focuser used for coupling laser beams into a 5 × 4 fiber-optic array: Comparison with Zemax Gaussian and geometrical optics calculations
G Model IJLEO-55565; No. of Pages 4
ARTICLE IN PRESS Optik xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Application of the ABCD ray matrix method for the design of a refractive 2.5× beam expander–focuser used for coupling laser beams into a 5 × 4 fiber-optic array: Comparison with Zemax Gaussian and geometrical optics calculations Sami D. Alaruri Independent Scholar, PO Box 152, Uniontown, OH 44685, USA
a r t i c l e
i n f o
Article history: Received 6 April 2014 Accepted 18 May 2015 Available online xxx Keywords: Galilean telescope Multimode optical fiber coupling efficiency Knife-edge Profilemeter Mathematica Galvanometer
a b s t r a c t In this article, an approach for the design and construction of a refractive 2.5× Galilean beam expander–focuser system which incorporates a two-axis galvanometer for coupling non-collinear laser beams into a 5×4200 m core diameter fiber-optic array is discussed. M2 Gaussian beam waist and beam waist location computations using the ABCD ray matrix and Zemax® paraxial Gaussian optics methods are given. Using the described optical system a 532 nm and 637 nm laser beams are coupled noncollinearly into 200 m core diameter fiber array by focusing down the laser beams to 71.26 ± 3.18 m and 90.32 ± 4.28 m diameter spots, respectively. Furthermore, an experimental validation for the focused beam size calculations using a knife-edge profilemeter is provided. The measured average transmission for the fibers in the fiber-optic array is 92.54 ± 3.78% (maximum FCE (fiber coupling efficiency): 99%; minimum FCE: 82%). Whereas the Zemax® calculated geometric ray-optics fiber coupling efficiency for the beam expander-focuser into a 200 m core diameter fiber using uncoated lenses and without accounting for Fresnel reflections and bulk absorption effects is 100%. Geometric ray-optics calculations indicate that Fresnel reflections and bulk absorption effects account for 7% of the 100% FCE value. The described optical switching scheme which incorporates a galvanometer is used for exciting asynchronously 20 PCR (Polymerase Chain Reaction) vessels in high throughput DNA/RNA amplification systems. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Often Galilean telescopes are used for expanding laser beams in order to increase the size of the beam incident on a focusing lens thereby allowing the beam to be focused to a very small spot [1–4] at a distance from the laser. Such optical systems are used in laser applications associated with welding, marking, cutting, material processing and surgery. Refracting laser beam expanders fall into two categories: Galilean and Keplerian. A Galilean beam expander consists of a negative lens (entry lens) and a positive lens (exit lens). Whereas a Keplerian beam expander consists of a positive short focal length lens (entry lens) and a long focal length lens (exit lens). In comparison with Keplerian beam expanders, Galilean beam expanders have shorter length, much smaller field of view (FOV) and form an erect image. Furthermore, Keplerian beam expanders are not
E-mail address: sami [email protected]
recommended for high power or energy laser applications because internal beam focusing can cause air to breakdown between the telescope two lenses. In this paper an approach for the design of 2.5× Galilean beam expander–focuser used for coupling 532 nm laser beam into a 5 × 4–200 m core diameter fiber array utilizing the ABCD ray matrix method [5–8] and Zemax paraxial Gaussian beam calculations [9] is described. The beam expander–focuser system which incorporates a galvanometer [10,11] for steering a focused 532 nm and 637 nm laser beams into a selected optical fiber in a 5 × 4 fiber array is part of the excitation optical system used in a high throughput DNA/RNA extraction, purification and amplification clinical system [12]. Additionally, beam waist and waist location calculations obtained using the ABCD ray matrix method and Zemax are compared and discussed. Finally, verification for the optical design and the beam waist calculations was performed using a spinning knife-edge beam profilemeter. Spot size measurements collected with the profilemeter are presented and discussed.
http://dx.doi.org/10.1016/j.ijleo.2015.05.076 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
Please cite this article in press as: S.D. Alaruri, Application of the ABCD ray matrix method for the design of a refractive 2.5× beam expander–focuser used for coupling laser beams into a 5 × 4 fiber-optic array: Comparison with Zemax Gaussian and geometrical optics calculations, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.076
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erroneous DNA amplification measurements collected by the system described in Ref. [12].
2. System description As shown in Fig. 1, the beam emitted from 532 nm laser was expanded using a Galilean telescope and the 637 nm laser beam was reflected toward a beam splitter (>95% reflectance (average) at 532 nm; >95% transmittance (average) at 637 nm) using an aluminum coated front surface flat mirror. Both the mirror and the beam splitter were placed at 45◦ relative to the incident laser beams. Then, the two laser beams were directed toward a planoconvex focusing lens (FL = 60.00 mm, N-BK-7). The two laser beams were aligned non-collinearly to avoid crosstalk between the two laser excitation channels. Further, the two laser beams strike the focusing lens at approximately 1.0◦ from the optical axis. A twoaxis galvanometer driven x and y rotatable mirrors were used to steer either the green or the red laser beams into one of the 22 fused silica fibers in the excitation fiber-optic bundle (CeramOptec, fiber p/n Optran WF, Core Diameter = 200 m, Length = 1.5 m). The galvanometer mirrors can move 8.2 m per step and the reflectivity of the mirrors is >97% over the spectral range 450 nm to 650 nm at angles of incidence <65◦ . At the same time the galvanometer x and y rotatable mirrors direct the other laser beam outside the 22 fibers array to prevent cross-talk between the 532 nm and 637 nm excitation channels. The excitation fiber bundle consists of twenty two fibers (two spare fiber channels were added to the fiber-optic bundle later) arranged in 5 × 4 array with 0.425 mm spacing between the cores of the optical fibers. The core diameter and the numerical aperture of each fiber were 200 m and 0.12 ± 0.02, respectively. Here, it is worth noting that the measured average transmission of the fibers was 92.54 ± 3.78%. The transmission value is based on measurements collected from 22 fiber-optic bundles (i.e. 484 fibers). Light emerging from each excitation fiber was focused into a PCR (Polymerase Chain Reaction) vessel by the means of a short focal length lens (FL = 4.50 mm, N-BK-7). A full description for the excitation and collection optics is provided in Ref. [10]. The alignment of the two-axis galvanometer mirrors with the location of each of the excitation fibers was carried out by scanning separately the 532 nm and 637 nm laser beams across the excitation fibers and by recording the galvanometer mirrors position at which the maximum power was detected at the fiber output end. The Galilean telescope and the focusing lens (FL = 60 mm, NBK-7) were selected to focus the 532 nm and 640 nm laser beams into a spot less than 100 m in diameter at the input plane of the excitation fiber-optic bundle. The <100 m focused beam diameter limit was imposed to ensure that the focused laser beams were always coupled into the 200 m core diameter excitation fibers for the worst galvanometer drift, galvanometer repeatability, system operating temperature and laser beam pointing stability error conditions. Partially coupled laser beam into a fiber can lead to
FL = -50.00 mm
FL = 125.00 mm
FL = 60.00 mm
2-D Galvanometer Mirror (X-axis)
532 nm Laser Beam Splitter
84.24 mm 43.10 mm
Mirror (Y-axis)
5x4 FiberOptic Bundle Interface
3. Theoretical background (ABCD ray matrix method) [5–8] Utilizing the ABCD ray transfer matrix method, the laser beam expander–focuser system shown in Fig. 2 can be written as
A
B
C
D
×
=
1 L1 0
1 Z3 0
1
1
1
0
−1/FL3
1
1
0
1/FL1
1
1 Z0 0
1 L2 0
1
1
1
−1/FL2
1
(1)
1
The transformation matrix for the beam expander which is a subset of the optical system can be written as
A
B
C
D
=
1
1
−1/FL2
1
1 L1 0
1
1
0
1/FL1
1
(2)
By multiplying the matrices the beam expander transformation matrix can be reduced to
A
B
C
D
=
1 − L1/FL1
L1
− (FL1 + FL2 − L1) / (FL1 ∗ FL2)
1 − L1/FL2
(3)
The Galilean beam expander matrix represents afocal system, if the optical power of the optical system is zero, this leads to L1–FL1–FL2 = 0. Thus, the Galilean beam expander can be reduced to
A
B
C
D
=
−FL2/FL1
FL1 + FL2
0
−FL1/FL2
(4)
From Eq. (4) the Galilean beam expander waist magnification, ˇ, can be derived as
ˇ = ω1 /ω0 = FL2/FL1 > 1
(5)
A Gaussian laser beam can be characterized by a complex beam parameter q (z), ω and R(z). An expression for each of the terms is shown below 1 1 i = , − q (z) R (z) n ω2
and
R (z) = z 1 +
ω2 (z) = ω02 1 +
z 2 R
z
z 2 zR
(6)
where ω is the beam waist radius at any plane z along the beam propagation direction, R(z) is the radius of curvature of the spherical wave front at position z, is the wavelength, n is the medium refractive index, and z and zR (beam Rayleigh range) can be written as z=
ω2 n
M2
(7)
where M2 is the beam quality parameter. zR =
ω02 n
M2
193.00 mm
637 nm Laser
Mirror
Front View For the 5x4 Fiber-Optic Bundle Interface
200 μ Core Diameter Optical Fiber
Fig. 1. Schematic illustrating the Galilean beam expander–focuser and the 2-D galvanometer arrangement for coupling 532 ± 1 nm (20 mW) and 637 ± 3 nm (20 mW) laser beams into the excitation fiber optic bundle interface which consist of twenty two 200 m core diameter step-index multimode fibers arranged in a 5 × 4 array (two spare fibers are placed on the array two sides).
Fig. 2. Schematic depicting the Galilean laser beam expander–focuser optical components and the input and output laser beam symbols.
Please cite this article in press as: S.D. Alaruri, Application of the ABCD ray matrix method for the design of a refractive 2.5× beam expander–focuser used for coupling laser beams into a 5 × 4 fiber-optic array: Comparison with Zemax Gaussian and geometrical optics calculations, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.076
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Beam Radius (mm)
1 0.8 0.6
FL= 60.00 mm FL= 50.00 mm
0.4
FL= 40.00 mm
0.2 0
0 10 20 30 40 50 60 70 80 90 100 110 120 Beam Posion (mm)
Fig. 3. Plot showing the focused beam radius size, ω2, as a function of the beam radius location, Z3, calculated for three focusing lens focal lengths, namely, 40 mm, 50 mm and 60 mm.
qout =
(A ∗ qin + B) (C ∗ qin + D)
8
Setting the real part of qout = 0 and the imaginary part of qout = Z3, one can calculate the size of the beam waist radius, ω2, and the location of ω2. Here, it is worth noting that the collimation distance, L1, between the negative and the positive lenses can be calculated for different ω0 and z0 values by setting the D term in Eq. (3) equal to zero [7]. 4. Results and discussion The first step in the design of the beam expander–focuser was to select the focusing lens focal length. As shown in Fig. 3, 40 mm, 50 mm and 60 mm plano-convex focal length lenses were examined and a 60 mm focal length focusing lens was selected due to the small variation in the size of the focused laser beam as a function of beam position. Consequently, error contributions due to the size of the focused beam as function of distance such as opto-mechanical mounts (decenter, tilt and de-space), laser beam pointing stability, galvanometer drift and galvanometer stability can be minimized during assembly and the focused laser beam remains contained within the fiber 200 m core diameter under all system operating conditions. Furthermore, the plano-convex lens aberrations were minimized by orienting the lens flat surface toward the input end of the fiber-optic bundle. The second step in the design procedure was to determine the minimum beam diameter entering the focusing lens, 2*ω1, which produces a focused beam diameter 2*ω2 < 100 m in diameter. The system design budget allocated 100 m for the focused beam diameter, 20 m as a safety margin and 80 m for the optical system errors (i.e., galvanometer drift and repeatability, thermal system effects, laser beam pointing stability and opto-mechanical components stackup errors). This allocation was intended to contain the laser beam within the core diameter of the 200 m excitation fiber for the worst condition of beam drift with respect to the center of the fiber core. Fig. 4 is a schematic depicting the focused 532 nm spot at the center of the 200 m core diameter fiber and the shift in the center of the focused spot due to error contributions caused by the laser beam pointing stability, galvanometer repeatability and galvanometer drift. The third step in the design procedure was to calculate the beam diameter, 2*ω1, needed to achieve a focused beam diameter, 2*ω2, less than 100 m. Using the ABCD ray trace method and Zemax the 2*ω2 values were calculated as a function of 2*ω1 in the range between 0.20 mm to 2.00 mm for a 60 mm FL lens. A summary for
Fig. 4. Schematic depicting the beam size error budget which includes error contributions due to the laser beam pointing stability, and galvanometer drift and stability for a focused 532 nm laser beam coupled into a 200 m core diameter fiber using a 60 mm FL focusing lens. In the case of the 637 nm laser the calculate focused beam diameter is ∼44.73 m (worst case), laser beam pointing stability, galvanometer repeatability and galvanometer drift error contributions are ∼28.0 m, 6.0 m and 10.5 m, respectively.
0.250 0.200 2*ω2 (mm)
Defining the q-parameter in terms of the beam input (output) plane and the A, B, C and D elements of the transformation ray matrix representing the optical system given in Eq. (1), yields
532 nm-ABCD
0.150
0.051x-0.85
y= R² = 0.979
0.100
532 nm-Zemax 637 nm-Zemax
0.050 0.000 0.000
637 nm-ABCD
0.500
1.000
1.500
2.000
2*ω1 (mm) Fig. 5. Graph showing the variation in the focused beam diameter, 2*ω2, as a function of the incident beam diameter, 2*ω1, for the 60 mm FL focusing lens. The calculations were performed for the 532 nm and 637 nm wavelengths using the ABCD ray matrix method and Zemax (Physical Optics/Paraxial Gaussian Beam Feature).
the obtained results is provided in Fig. 5. From Fig. 5 it can be concluded that in order to achieve 2*ω2 ≤ 0.10 mm the beam diameter incident on the focusing lens, 2*ω1, needs to be >0.454 mm. Further, from Fig. 5 it can be seen that the diameter of the focused laser spot is inversely proportional to the diameter of the laser beam entering the focusing lens and an increase in the size of 2*ω1 leads to a decrease in the beam diameter 2*ω2. The fourth design step was to select the focal lengths for the Galilean telescope negative and positive lenses based on the beam diameter incident on the focusing lens, 2*ω1, which leads to a beam spot size 2*ω2 < 100 m in diameter. Since the beam diameter produced by the 637 nm laser, 2*ω0 , is greater than the calculated beam diameter incident on the focusing lens, 2*ω1, (i.e., 1.250 mm > 0.454 mm), then a Galilean telescope is not required for expanding the 637 nm laser beam. Utilizing Eq. (5), one can calculate the minimum magnification required for the 532 nm laser beam: ˇ = 2*ω1 /2*ω0 = 0.454 mm/0.320 mm = 1.42. Thus, the minimum telescope magnification needed to achieve a focused laser beam diameter (2*ω2) < 100 m is 1.42. This result means that the needed ratio of FL2/FL1 should be >1.42. Due to commercial availability of lenses, focal lengths equal to −50 mm (planoconcave) and 125 mm (planoconvex) were selected for the Galilean telescope negative and positive lenses, respectively, (|FL2/FL1| = |125/−50| = 2.5). In the fifth step of the design the ABCD ray matrix method and Zemax were utilized for calculating the focused laser beam
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Table 1 Key laser beam specifications for the 532 nm and 637 nm lasers provided by the lasers manufacturer. Wavelength (nm)
Beam waist diameter 2ω0 (mm) (at 1/e2 )
Beam waist location with respect to laser front (mm)
Beam quality factor M2
Beam divergence (full angle at 1/e2 diameter) (mrad)
532.0 ± 1.0 637.0 ± 3.0
0.320 ± 0.048 <1.25
42.0 ± 75.0 0 ± 1000
<1.1 <1.2
<3 <1
Table 2 Summary for the average beam spot diameter measurements obtained with the profilemeter for the 532 nm and 637 nm lasers. nm
Measured average incident beam dia. on focusing lens (m)
Max calculated focused beam dia. at 61.2 mm (m)
Measured average focused beam dia. at 61.2 mm (m)
Measured max/min focused beam dia. at 61.2 mm
Measured beam ellipticity
532 637
829.01 ± 15.42 1374.37 ± 20.2
73.3 44.73
71.26 ± 3.18 90.32 ± 4.28
74.20/63.50 95.00/85.60
0.85 ± 0.03 0.89 ± 0.06
diameter using the 532 nm and 637 nm laser beam specifications listed in Table 1. Based upon the obtained results from both methods, the maximum calculated focused beam diameter for the 532 nm and 637 nm laser beams were 73.3 m and 44.73 m, respectively, at 61.2 mm from the focusing lens back surface. Here, it is worth noting that the beam diameters were calculated for all possible input beam waist diameters and locations entering the optical system. The location of the focused beam spot size ranged between 56.58 mm and 60.00 mm for all possible iterations of input laser beam parameters. Utilizing Eq. (3) the Galilean telescope collimation distance was calculated for the 532 nm input laser beam parameters listed in Table 1. The ABCD ray matrix method calculations were performed using Mathematica [13]. An average telescope collimation distance, L1, equal to 77.10 mm was obtained for all possible input beam waist sizes and waist locations. The maximum and minimum calculated telescope collimation distance was 84.04 mm and 71.27 mm, respectively, for all possible laser beam input parameters. As such, L1 was set to 84.20 mm to ensure that the 532 nm beam is expanded for all possible scenarios of input beam parameters. Lastly, the optical system shown in Fig. 1 was constructed and the beam spot size calculations were validated using a spinning knife-edge beam profilemeter (Thorlabs, Model BP 104-VIS) with ten randomly selected 532 nm and 637 nm lasers. The tilt, de-space and de-centering opto-mechanical tolerances for all components were maintained at ±0.5◦ , ±0.25 mm and ±0.1 mm, respectively. To ensure that the 532 nm laser beam coming out of the Galilean telescope is expanded for each selected laser, the beam diameter was measured at 0.4 m and 0.6 m from the back surface of the telescope positive lens. The measured average beam diameters at 0.4 m and 0.6 m were 890.75 ± 28.94 m and 965.51 ± 29.22 m. Table 2 summarizes the average measurements obtained for the beam diameter in the focusing lens plane, the focused beam diameter at 61.2 mm from the focusing lens back surface, the maximum and minimum focused beam diameters at 61.2 mm from the focusing lens back surface and the measured focused beam ellipticity for both lasers. Finally, the geometric ray-trace fiber coupling efficiency [13] for the laser beam expander focuser into a 200 m core diameter multimode fiber was calculated using Zemax for uncoated lenses. The calculated fiber coupling efficiency was 100% when Fresnel reflections and bulk absorption effects were not accounted for;
and 92.599% when Fresnel and material absorption effects were accounted for. Thus, the estimated contribution due to reflections and bulk absorption is approximately 7%. Further, the geometric ray-trace Zemax fiber coupling efficiency calculations for the multimode fiber are in good agreement with the experiment measurements reported earlier in this paper (i.e., 92.54 + 3.78%). 5. Conclusions A 2.5× Galilean beam expander–focuser for coupling a 532 nm laser beam into 200 m core diameter 5 × 4 fiber array has been design, constructed and validated. Experimental measurements collected with the profilemeter indicate that the 532 nm laser beam can be focused down to an average spot of 71.26 ± 3.18 m in diameter. Furthermore, the 60 mm focal length focusing lens which is part of the optical system is used to focus a 637 nm laser beam into an average spot 90.32 ± 4.28 m in diameter. The optical system incorporates a galvanometer which can be used for coupling light into the fibers asynchronously for 50 ms periods of time. Such an optical switching scheme is used for exciting multiple PCR cells in a DNA/RNA amplification system. References [1] L. Lévesque, Divergence of far-infrared laser beam and collimation for Galilean and Keplerian system designs, Opt. Laser Technol. 41 (2009) 557–561. [2] Robb, Paul N., Laser Beam Expanders, US Patent No. 5532,880, July 2, 1996. [3] Laikin Milton, Lens Design, fourth ed., CRC, Press, Boca Raton, FL, 2006. [4] Kingslake Rudolf, R. Barry Johnson, Lens Design Fundamentals, second ed., Elsevier, Inc, Burlington, MA, 2010. [5] Amnon Yariv, Optical Electronics, third ed., Holt, Rinehart and Winston, New York, NY, 1985. [6] A. Gerrard, J.M. Burch, Introduction to Matrix Methods in Optics, John Wiley and Sons, Ltd, New York, NY, 1975. [7] Gerhard Kloos, Martix Methods for Optical Layout, vol. TT77, SPIE Press, Bellingham, WA USA, 2007. [8] Alexander Hornberg, Propagation of Gaussian Beams, Laser Tech. J. 2 (2) (2007) 75–80. [9] Radiant Zemax, Version 12, Redmond, WA, 2013, https://www.radiantzemax. com/en. [10] Wang, Jianhua, Zuyun Fang, Jian J. Chen, Fiber Optic Switch Using Galvanometer-Driven X–Y Scanning, US Patent No. US 6721,474 B2, April 13, 2004. [11] Chande, Tushar S., Marshall G. Jones, Angel L. Ortiz, John L. August, Laser Beam Directing System, US Patent No. US 4,838,631, June 13, 1989. [12] S. Alaruri, et al. System and Method Including Thermal Cycler Modules, US patent US8962308 B2, Feburary 24, 2015. [13] Wolfram Research, Mathematica 2015, version 9.0, Champaign, IL. http://www.wolfram.com/mathematica/.
Please cite this article in press as: S.D. Alaruri, Application of the ABCD ray matrix method for the design of a refractive 2.5× beam expander–focuser used for coupling laser beams into a 5 × 4 fiber-optic array: Comparison with Zemax Gaussian and geometrical optics calculations, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.076
ARTICLE IN PRESS Optik xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Application of the ABCD ray matrix method for the design of a refractive 2.5× beam expander–focuser used for coupling laser beams into a 5 × 4 fiber-optic array: Comparison with Zemax Gaussian and geometrical optics calculations Sami D. Alaruri Independent Scholar, PO Box 152, Uniontown, OH 44685, USA
a r t i c l e
i n f o
Article history: Received 6 April 2014 Accepted 18 May 2015 Available online xxx Keywords: Galilean telescope Multimode optical fiber coupling efficiency Knife-edge Profilemeter Mathematica Galvanometer
a b s t r a c t In this article, an approach for the design and construction of a refractive 2.5× Galilean beam expander–focuser system which incorporates a two-axis galvanometer for coupling non-collinear laser beams into a 5×4200 m core diameter fiber-optic array is discussed. M2 Gaussian beam waist and beam waist location computations using the ABCD ray matrix and Zemax® paraxial Gaussian optics methods are given. Using the described optical system a 532 nm and 637 nm laser beams are coupled noncollinearly into 200 m core diameter fiber array by focusing down the laser beams to 71.26 ± 3.18 m and 90.32 ± 4.28 m diameter spots, respectively. Furthermore, an experimental validation for the focused beam size calculations using a knife-edge profilemeter is provided. The measured average transmission for the fibers in the fiber-optic array is 92.54 ± 3.78% (maximum FCE (fiber coupling efficiency): 99%; minimum FCE: 82%). Whereas the Zemax® calculated geometric ray-optics fiber coupling efficiency for the beam expander-focuser into a 200 m core diameter fiber using uncoated lenses and without accounting for Fresnel reflections and bulk absorption effects is 100%. Geometric ray-optics calculations indicate that Fresnel reflections and bulk absorption effects account for 7% of the 100% FCE value. The described optical switching scheme which incorporates a galvanometer is used for exciting asynchronously 20 PCR (Polymerase Chain Reaction) vessels in high throughput DNA/RNA amplification systems. © 2015 Elsevier GmbH. All rights reserved.
1. Introduction Often Galilean telescopes are used for expanding laser beams in order to increase the size of the beam incident on a focusing lens thereby allowing the beam to be focused to a very small spot [1–4] at a distance from the laser. Such optical systems are used in laser applications associated with welding, marking, cutting, material processing and surgery. Refracting laser beam expanders fall into two categories: Galilean and Keplerian. A Galilean beam expander consists of a negative lens (entry lens) and a positive lens (exit lens). Whereas a Keplerian beam expander consists of a positive short focal length lens (entry lens) and a long focal length lens (exit lens). In comparison with Keplerian beam expanders, Galilean beam expanders have shorter length, much smaller field of view (FOV) and form an erect image. Furthermore, Keplerian beam expanders are not
E-mail address: sami [email protected]
recommended for high power or energy laser applications because internal beam focusing can cause air to breakdown between the telescope two lenses. In this paper an approach for the design of 2.5× Galilean beam expander–focuser used for coupling 532 nm laser beam into a 5 × 4–200 m core diameter fiber array utilizing the ABCD ray matrix method [5–8] and Zemax paraxial Gaussian beam calculations [9] is described. The beam expander–focuser system which incorporates a galvanometer [10,11] for steering a focused 532 nm and 637 nm laser beams into a selected optical fiber in a 5 × 4 fiber array is part of the excitation optical system used in a high throughput DNA/RNA extraction, purification and amplification clinical system [12]. Additionally, beam waist and waist location calculations obtained using the ABCD ray matrix method and Zemax are compared and discussed. Finally, verification for the optical design and the beam waist calculations was performed using a spinning knife-edge beam profilemeter. Spot size measurements collected with the profilemeter are presented and discussed.
http://dx.doi.org/10.1016/j.ijleo.2015.05.076 0030-4026/© 2015 Elsevier GmbH. All rights reserved.
Please cite this article in press as: S.D. Alaruri, Application of the ABCD ray matrix method for the design of a refractive 2.5× beam expander–focuser used for coupling laser beams into a 5 × 4 fiber-optic array: Comparison with Zemax Gaussian and geometrical optics calculations, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.076
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ARTICLE IN PRESS
IJLEO-55565; No. of Pages 4
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2
erroneous DNA amplification measurements collected by the system described in Ref. [12].
2. System description As shown in Fig. 1, the beam emitted from 532 nm laser was expanded using a Galilean telescope and the 637 nm laser beam was reflected toward a beam splitter (>95% reflectance (average) at 532 nm; >95% transmittance (average) at 637 nm) using an aluminum coated front surface flat mirror. Both the mirror and the beam splitter were placed at 45◦ relative to the incident laser beams. Then, the two laser beams were directed toward a planoconvex focusing lens (FL = 60.00 mm, N-BK-7). The two laser beams were aligned non-collinearly to avoid crosstalk between the two laser excitation channels. Further, the two laser beams strike the focusing lens at approximately 1.0◦ from the optical axis. A twoaxis galvanometer driven x and y rotatable mirrors were used to steer either the green or the red laser beams into one of the 22 fused silica fibers in the excitation fiber-optic bundle (CeramOptec, fiber p/n Optran WF, Core Diameter = 200 m, Length = 1.5 m). The galvanometer mirrors can move 8.2 m per step and the reflectivity of the mirrors is >97% over the spectral range 450 nm to 650 nm at angles of incidence <65◦ . At the same time the galvanometer x and y rotatable mirrors direct the other laser beam outside the 22 fibers array to prevent cross-talk between the 532 nm and 637 nm excitation channels. The excitation fiber bundle consists of twenty two fibers (two spare fiber channels were added to the fiber-optic bundle later) arranged in 5 × 4 array with 0.425 mm spacing between the cores of the optical fibers. The core diameter and the numerical aperture of each fiber were 200 m and 0.12 ± 0.02, respectively. Here, it is worth noting that the measured average transmission of the fibers was 92.54 ± 3.78%. The transmission value is based on measurements collected from 22 fiber-optic bundles (i.e. 484 fibers). Light emerging from each excitation fiber was focused into a PCR (Polymerase Chain Reaction) vessel by the means of a short focal length lens (FL = 4.50 mm, N-BK-7). A full description for the excitation and collection optics is provided in Ref. [10]. The alignment of the two-axis galvanometer mirrors with the location of each of the excitation fibers was carried out by scanning separately the 532 nm and 637 nm laser beams across the excitation fibers and by recording the galvanometer mirrors position at which the maximum power was detected at the fiber output end. The Galilean telescope and the focusing lens (FL = 60 mm, NBK-7) were selected to focus the 532 nm and 640 nm laser beams into a spot less than 100 m in diameter at the input plane of the excitation fiber-optic bundle. The <100 m focused beam diameter limit was imposed to ensure that the focused laser beams were always coupled into the 200 m core diameter excitation fibers for the worst galvanometer drift, galvanometer repeatability, system operating temperature and laser beam pointing stability error conditions. Partially coupled laser beam into a fiber can lead to
FL = -50.00 mm
FL = 125.00 mm
FL = 60.00 mm
2-D Galvanometer Mirror (X-axis)
532 nm Laser Beam Splitter
84.24 mm 43.10 mm
Mirror (Y-axis)
5x4 FiberOptic Bundle Interface
3. Theoretical background (ABCD ray matrix method) [5–8] Utilizing the ABCD ray transfer matrix method, the laser beam expander–focuser system shown in Fig. 2 can be written as
A
B
C
D
×
=
1 L1 0
1 Z3 0
1
1
1
0
−1/FL3
1
1
0
1/FL1
1
1 Z0 0
1 L2 0
1
1
1
−1/FL2
1
(1)
1
The transformation matrix for the beam expander which is a subset of the optical system can be written as
A
B
C
D
=
1
1
−1/FL2
1
1 L1 0
1
1
0
1/FL1
1
(2)
By multiplying the matrices the beam expander transformation matrix can be reduced to
A
B
C
D
=
1 − L1/FL1
L1
− (FL1 + FL2 − L1) / (FL1 ∗ FL2)
1 − L1/FL2
(3)
The Galilean beam expander matrix represents afocal system, if the optical power of the optical system is zero, this leads to L1–FL1–FL2 = 0. Thus, the Galilean beam expander can be reduced to
A
B
C
D
=
−FL2/FL1
FL1 + FL2
0
−FL1/FL2
(4)
From Eq. (4) the Galilean beam expander waist magnification, ˇ, can be derived as
ˇ = ω1 /ω0 = FL2/FL1 > 1
(5)
A Gaussian laser beam can be characterized by a complex beam parameter q (z), ω and R(z). An expression for each of the terms is shown below 1 1 i = , − q (z) R (z) n ω2
and
R (z) = z 1 +
ω2 (z) = ω02 1 +
z 2 R
z
z 2 zR
(6)
where ω is the beam waist radius at any plane z along the beam propagation direction, R(z) is the radius of curvature of the spherical wave front at position z, is the wavelength, n is the medium refractive index, and z and zR (beam Rayleigh range) can be written as z=
ω2 n
M2
(7)
where M2 is the beam quality parameter. zR =
ω02 n
M2
193.00 mm
637 nm Laser
Mirror
Front View For the 5x4 Fiber-Optic Bundle Interface
200 μ Core Diameter Optical Fiber
Fig. 1. Schematic illustrating the Galilean beam expander–focuser and the 2-D galvanometer arrangement for coupling 532 ± 1 nm (20 mW) and 637 ± 3 nm (20 mW) laser beams into the excitation fiber optic bundle interface which consist of twenty two 200 m core diameter step-index multimode fibers arranged in a 5 × 4 array (two spare fibers are placed on the array two sides).
Fig. 2. Schematic depicting the Galilean laser beam expander–focuser optical components and the input and output laser beam symbols.
Please cite this article in press as: S.D. Alaruri, Application of the ABCD ray matrix method for the design of a refractive 2.5× beam expander–focuser used for coupling laser beams into a 5 × 4 fiber-optic array: Comparison with Zemax Gaussian and geometrical optics calculations, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.076
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Beam Radius (mm)
1 0.8 0.6
FL= 60.00 mm FL= 50.00 mm
0.4
FL= 40.00 mm
0.2 0
0 10 20 30 40 50 60 70 80 90 100 110 120 Beam Posion (mm)
Fig. 3. Plot showing the focused beam radius size, ω2, as a function of the beam radius location, Z3, calculated for three focusing lens focal lengths, namely, 40 mm, 50 mm and 60 mm.
qout =
(A ∗ qin + B) (C ∗ qin + D)
8
Setting the real part of qout = 0 and the imaginary part of qout = Z3, one can calculate the size of the beam waist radius, ω2, and the location of ω2. Here, it is worth noting that the collimation distance, L1, between the negative and the positive lenses can be calculated for different ω0 and z0 values by setting the D term in Eq. (3) equal to zero [7]. 4. Results and discussion The first step in the design of the beam expander–focuser was to select the focusing lens focal length. As shown in Fig. 3, 40 mm, 50 mm and 60 mm plano-convex focal length lenses were examined and a 60 mm focal length focusing lens was selected due to the small variation in the size of the focused laser beam as a function of beam position. Consequently, error contributions due to the size of the focused beam as function of distance such as opto-mechanical mounts (decenter, tilt and de-space), laser beam pointing stability, galvanometer drift and galvanometer stability can be minimized during assembly and the focused laser beam remains contained within the fiber 200 m core diameter under all system operating conditions. Furthermore, the plano-convex lens aberrations were minimized by orienting the lens flat surface toward the input end of the fiber-optic bundle. The second step in the design procedure was to determine the minimum beam diameter entering the focusing lens, 2*ω1, which produces a focused beam diameter 2*ω2 < 100 m in diameter. The system design budget allocated 100 m for the focused beam diameter, 20 m as a safety margin and 80 m for the optical system errors (i.e., galvanometer drift and repeatability, thermal system effects, laser beam pointing stability and opto-mechanical components stackup errors). This allocation was intended to contain the laser beam within the core diameter of the 200 m excitation fiber for the worst condition of beam drift with respect to the center of the fiber core. Fig. 4 is a schematic depicting the focused 532 nm spot at the center of the 200 m core diameter fiber and the shift in the center of the focused spot due to error contributions caused by the laser beam pointing stability, galvanometer repeatability and galvanometer drift. The third step in the design procedure was to calculate the beam diameter, 2*ω1, needed to achieve a focused beam diameter, 2*ω2, less than 100 m. Using the ABCD ray trace method and Zemax the 2*ω2 values were calculated as a function of 2*ω1 in the range between 0.20 mm to 2.00 mm for a 60 mm FL lens. A summary for
Fig. 4. Schematic depicting the beam size error budget which includes error contributions due to the laser beam pointing stability, and galvanometer drift and stability for a focused 532 nm laser beam coupled into a 200 m core diameter fiber using a 60 mm FL focusing lens. In the case of the 637 nm laser the calculate focused beam diameter is ∼44.73 m (worst case), laser beam pointing stability, galvanometer repeatability and galvanometer drift error contributions are ∼28.0 m, 6.0 m and 10.5 m, respectively.
0.250 0.200 2*ω2 (mm)
Defining the q-parameter in terms of the beam input (output) plane and the A, B, C and D elements of the transformation ray matrix representing the optical system given in Eq. (1), yields
532 nm-ABCD
0.150
0.051x-0.85
y= R² = 0.979
0.100
532 nm-Zemax 637 nm-Zemax
0.050 0.000 0.000
637 nm-ABCD
0.500
1.000
1.500
2.000
2*ω1 (mm) Fig. 5. Graph showing the variation in the focused beam diameter, 2*ω2, as a function of the incident beam diameter, 2*ω1, for the 60 mm FL focusing lens. The calculations were performed for the 532 nm and 637 nm wavelengths using the ABCD ray matrix method and Zemax (Physical Optics/Paraxial Gaussian Beam Feature).
the obtained results is provided in Fig. 5. From Fig. 5 it can be concluded that in order to achieve 2*ω2 ≤ 0.10 mm the beam diameter incident on the focusing lens, 2*ω1, needs to be >0.454 mm. Further, from Fig. 5 it can be seen that the diameter of the focused laser spot is inversely proportional to the diameter of the laser beam entering the focusing lens and an increase in the size of 2*ω1 leads to a decrease in the beam diameter 2*ω2. The fourth design step was to select the focal lengths for the Galilean telescope negative and positive lenses based on the beam diameter incident on the focusing lens, 2*ω1, which leads to a beam spot size 2*ω2 < 100 m in diameter. Since the beam diameter produced by the 637 nm laser, 2*ω0 , is greater than the calculated beam diameter incident on the focusing lens, 2*ω1, (i.e., 1.250 mm > 0.454 mm), then a Galilean telescope is not required for expanding the 637 nm laser beam. Utilizing Eq. (5), one can calculate the minimum magnification required for the 532 nm laser beam: ˇ = 2*ω1 /2*ω0 = 0.454 mm/0.320 mm = 1.42. Thus, the minimum telescope magnification needed to achieve a focused laser beam diameter (2*ω2) < 100 m is 1.42. This result means that the needed ratio of FL2/FL1 should be >1.42. Due to commercial availability of lenses, focal lengths equal to −50 mm (planoconcave) and 125 mm (planoconvex) were selected for the Galilean telescope negative and positive lenses, respectively, (|FL2/FL1| = |125/−50| = 2.5). In the fifth step of the design the ABCD ray matrix method and Zemax were utilized for calculating the focused laser beam
Please cite this article in press as: S.D. Alaruri, Application of the ABCD ray matrix method for the design of a refractive 2.5× beam expander–focuser used for coupling laser beams into a 5 × 4 fiber-optic array: Comparison with Zemax Gaussian and geometrical optics calculations, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.076
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Table 1 Key laser beam specifications for the 532 nm and 637 nm lasers provided by the lasers manufacturer. Wavelength (nm)
Beam waist diameter 2ω0 (mm) (at 1/e2 )
Beam waist location with respect to laser front (mm)
Beam quality factor M2
Beam divergence (full angle at 1/e2 diameter) (mrad)
532.0 ± 1.0 637.0 ± 3.0
0.320 ± 0.048 <1.25
42.0 ± 75.0 0 ± 1000
<1.1 <1.2
<3 <1
Table 2 Summary for the average beam spot diameter measurements obtained with the profilemeter for the 532 nm and 637 nm lasers. nm
Measured average incident beam dia. on focusing lens (m)
Max calculated focused beam dia. at 61.2 mm (m)
Measured average focused beam dia. at 61.2 mm (m)
Measured max/min focused beam dia. at 61.2 mm
Measured beam ellipticity
532 637
829.01 ± 15.42 1374.37 ± 20.2
73.3 44.73
71.26 ± 3.18 90.32 ± 4.28
74.20/63.50 95.00/85.60
0.85 ± 0.03 0.89 ± 0.06
diameter using the 532 nm and 637 nm laser beam specifications listed in Table 1. Based upon the obtained results from both methods, the maximum calculated focused beam diameter for the 532 nm and 637 nm laser beams were 73.3 m and 44.73 m, respectively, at 61.2 mm from the focusing lens back surface. Here, it is worth noting that the beam diameters were calculated for all possible input beam waist diameters and locations entering the optical system. The location of the focused beam spot size ranged between 56.58 mm and 60.00 mm for all possible iterations of input laser beam parameters. Utilizing Eq. (3) the Galilean telescope collimation distance was calculated for the 532 nm input laser beam parameters listed in Table 1. The ABCD ray matrix method calculations were performed using Mathematica [13]. An average telescope collimation distance, L1, equal to 77.10 mm was obtained for all possible input beam waist sizes and waist locations. The maximum and minimum calculated telescope collimation distance was 84.04 mm and 71.27 mm, respectively, for all possible laser beam input parameters. As such, L1 was set to 84.20 mm to ensure that the 532 nm beam is expanded for all possible scenarios of input beam parameters. Lastly, the optical system shown in Fig. 1 was constructed and the beam spot size calculations were validated using a spinning knife-edge beam profilemeter (Thorlabs, Model BP 104-VIS) with ten randomly selected 532 nm and 637 nm lasers. The tilt, de-space and de-centering opto-mechanical tolerances for all components were maintained at ±0.5◦ , ±0.25 mm and ±0.1 mm, respectively. To ensure that the 532 nm laser beam coming out of the Galilean telescope is expanded for each selected laser, the beam diameter was measured at 0.4 m and 0.6 m from the back surface of the telescope positive lens. The measured average beam diameters at 0.4 m and 0.6 m were 890.75 ± 28.94 m and 965.51 ± 29.22 m. Table 2 summarizes the average measurements obtained for the beam diameter in the focusing lens plane, the focused beam diameter at 61.2 mm from the focusing lens back surface, the maximum and minimum focused beam diameters at 61.2 mm from the focusing lens back surface and the measured focused beam ellipticity for both lasers. Finally, the geometric ray-trace fiber coupling efficiency [13] for the laser beam expander focuser into a 200 m core diameter multimode fiber was calculated using Zemax for uncoated lenses. The calculated fiber coupling efficiency was 100% when Fresnel reflections and bulk absorption effects were not accounted for;
and 92.599% when Fresnel and material absorption effects were accounted for. Thus, the estimated contribution due to reflections and bulk absorption is approximately 7%. Further, the geometric ray-trace Zemax fiber coupling efficiency calculations for the multimode fiber are in good agreement with the experiment measurements reported earlier in this paper (i.e., 92.54 + 3.78%). 5. Conclusions A 2.5× Galilean beam expander–focuser for coupling a 532 nm laser beam into 200 m core diameter 5 × 4 fiber array has been design, constructed and validated. Experimental measurements collected with the profilemeter indicate that the 532 nm laser beam can be focused down to an average spot of 71.26 ± 3.18 m in diameter. Furthermore, the 60 mm focal length focusing lens which is part of the optical system is used to focus a 637 nm laser beam into an average spot 90.32 ± 4.28 m in diameter. The optical system incorporates a galvanometer which can be used for coupling light into the fibers asynchronously for 50 ms periods of time. Such an optical switching scheme is used for exciting multiple PCR cells in a DNA/RNA amplification system. References [1] L. Lévesque, Divergence of far-infrared laser beam and collimation for Galilean and Keplerian system designs, Opt. Laser Technol. 41 (2009) 557–561. [2] Robb, Paul N., Laser Beam Expanders, US Patent No. 5532,880, July 2, 1996. [3] Laikin Milton, Lens Design, fourth ed., CRC, Press, Boca Raton, FL, 2006. [4] Kingslake Rudolf, R. Barry Johnson, Lens Design Fundamentals, second ed., Elsevier, Inc, Burlington, MA, 2010. [5] Amnon Yariv, Optical Electronics, third ed., Holt, Rinehart and Winston, New York, NY, 1985. [6] A. Gerrard, J.M. Burch, Introduction to Matrix Methods in Optics, John Wiley and Sons, Ltd, New York, NY, 1975. [7] Gerhard Kloos, Martix Methods for Optical Layout, vol. TT77, SPIE Press, Bellingham, WA USA, 2007. [8] Alexander Hornberg, Propagation of Gaussian Beams, Laser Tech. J. 2 (2) (2007) 75–80. [9] Radiant Zemax, Version 12, Redmond, WA, 2013, https://www.radiantzemax. com/en. [10] Wang, Jianhua, Zuyun Fang, Jian J. Chen, Fiber Optic Switch Using Galvanometer-Driven X–Y Scanning, US Patent No. US 6721,474 B2, April 13, 2004. [11] Chande, Tushar S., Marshall G. Jones, Angel L. Ortiz, John L. August, Laser Beam Directing System, US Patent No. US 4,838,631, June 13, 1989. [12] S. Alaruri, et al. System and Method Including Thermal Cycler Modules, US patent US8962308 B2, Feburary 24, 2015. [13] Wolfram Research, Mathematica 2015, version 9.0, Champaign, IL. http://www.wolfram.com/mathematica/.
Please cite this article in press as: S.D. Alaruri, Application of the ABCD ray matrix method for the design of a refractive 2.5× beam expander–focuser used for coupling laser beams into a 5 × 4 fiber-optic array: Comparison with Zemax Gaussian and geometrical optics calculations, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.05.076