## Cylindrical quasi-cavity waveguide for static wide angle pattern projection

Optics Express, Vol. 15, Issue 6, pp. 3023-3030 (2007)

http://dx.doi.org/10.1364/OE.15.003023

Acrobat PDF (206 KB)

### Abstract

Beam deflection methods such as rotary mirrors and motorized turning optical heads suffer from a variety of electro-mechanical related problems which affect laser scanning performance. These include wobble, jitter, wear, windage and synchronization issues. A novel optical structure consisting of two concentric and cylindrical interfaces with unique optical coating properties for the static projection of a laser spot array over a wide angle is demonstrated. The resulting ray trajectory through the waveguide is modeled using linear equations. Spot size growth is modeled using previously defined ray transfer matrices for tilted optical elements. The model is validated by comparison with experimental spot size measurements for 20 transmitted beams. This novel form of spot projection can be used as the projection unit in optical sensing devices which range to multiple laser footprints.

© 2007 Optical Society of America

## 1. Introduction

1. J. Besl “Active, Optical Range Imaging Sensors,” Mach. Vision and Appl. **1**,127–152 (1988). [CrossRef]

2. F. Blais, “Review of 20 Years of Range Sensor Development,” Electronic Imaging **13**,231–240 (2004). [CrossRef]

## 2. Methods and materials

### 2.1 Quasi-cavity properties

6. B. E. A. Saleh and M. C. Teich, *Fundamentals Of Photonics* (Wiley-Interscience, 1991). [CrossRef]

7. H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. **10**,1550–1567 (1966). [CrossRef]

*R*and

_{1}*R*, respectively, separated by a BK-7 glass medium of thickness

_{2}*d*=

*R*-

_{2}*R*, and entrance and exit windows. Light transmission is achieved by depositing nano-layered thin film coatings on both interfaces. The rear side is deposited with a highly reflective coating (R≳99%) and the front side with a partial transmission coating (T≤13%), both effective over the 600 - 900nm waveband. Hence, at every reflection with the outer interface, a fraction of the light is transmitted through the cavity thus generating a laser spot. The reflected power undergoes further reflections within the cavity to generate subsequent laser spots as described in the next subsection.

_{1}### 2.2 Ray tracing

*m*is the gradient and

*b*is the y-intercept. The two interfaces are plotted as arcs where

*R*is the interface radius and

*n*corresponds to interface 1 or 2.

*z*-

*y*coordinate plane illustrating the ray trace for a 90°-cavity are shown in Figs. 2 and 3 for

*R*=0.25m and

_{1}*R*=0.263m.

_{2}### 2.3 Gaussian modeling

*q*, in conjunction with ABCD transfer matrices [6

6. B. E. A. Saleh and M. C. Teich, *Fundamentals Of Photonics* (Wiley-Interscience, 1991). [CrossRef]

*R*, and the spot radius of the beam,

*W*, are functions of the propagation distance

*z*and the Rayleigh range zR. These distances are described by the complex variable

*q*, given by

*q*must be derived at the laser aperture, at the rear interface entrance window and after refraction through the entrance window. The output beam radius

*W*, divergence angle θ

_{out}_{0}and wavelength λ of the laser producing the incident beam is used to calculate the propagation distance at the laser output aperture,

*z*. First we obtain the waist radius

_{R}*W*using λ and θ

_{0}_{0}.

*z*, using the previously derived

_{R}*W*.

_{0}6. B. E. A. Saleh and M. C. Teich, *Fundamentals Of Photonics* (Wiley-Interscience, 1991). [CrossRef]

*ABCD*[6

6. B. E. A. Saleh and M. C. Teich, *Fundamentals Of Photonics* (Wiley-Interscience, 1991). [CrossRef]

*q*at the entrance window and after refraction through it, respectively.

*q*values at the laser output aperture, at the cavity window and after refraction through the cavity window are derived, calculation of

*q*after

_{n}*n*reflections is carried out through an iterative two-step process. Firstly, the transfer matrix for a beam propagating through a uniform medium length is used to evaluate the

*q*value at an interface. Secondly, either the matrix for reflection or refraction is used to calculate the

*q*values for the transmitted or reflected rays, respectively. This iterative process is described in Fig. 4.

*ABCD*matrices which incorporate coordinate transformation for reflection and refraction of a Gaussian laser beams at an off-axis ellipsoidal surface have been used as reported in [5

5. G. A. Massey and A. E. Siegman, “Reflection and Refraction of Gaussian Light Beams at Tilted Ellipsoidal Surfaces,” Appl. Opt. **8**,975–978 (1969). [CrossRef] [PubMed]

^{th}beam.

*R*= 1.053m and

_{1}*R*= 1.063m, only an increase from 1.21mm to 1.64mm is seen over the 20 spots. This drop in spot size growth is the direct result of larger cavity radii.

_{2}### 2.4. Experimental setup for spot size measurements

*m*rad divergence was used to produce a beam at an incident angle of 29° relative to the z-axis. The cavity radii were

*R*= 0.25m and

_{1}*R*= 0.263m. This generated 24 spots with the fabricated quasi-cavity described previously.

_{2}_{00}beam.

## 3. Measurement results and discussion

### 3.1 Spot size measurements

*ABCD*non-paraxial matrices, which is presented in section 2.C. Note that the beam width growth over the first 20 spots is considerable for the fabricated cavity, where the 20th spot measures double the diameter of the first spot. This is due to the non-focusing and compounded nature of the spot size growth where every change in spot size depends on the previous Gaussian complex parameter.

### 3.2 Ray trajectory measurement

*R*= 0.25m and

_{1}*R*= 0.263m. Experimentally, a screen was placed in the same position relative to the cavity. The projected spot positions were marked on the screen and then measured with a ruler. The recorded y coordinates for the 20 intersections is compared to the measured positions. Figure 9 shows the modeled and measured positions along the screen. Position on screen refers to the distance from the spot to screen end (277,0).

_{2}^{th}spot. This is due to an increasingly larger angle each beam makes with the screen which results in a wider spot. Hence approximation of the spot center by hand becomes more difficult. However, Fig. 9 shows close conformity of the modeled and the approximately measured positions, therefore confirming the accuracy of the ray tracing method described previously.

## 4. Conclusions and future work

^{th}spot sizes for a cavity of radii

*R*= 0.25m and

_{1}*R*= 0.263m have been 1.2mm and 2.8mm respectively. However, simulated results have shown that the 20th spot size can be reduced to only 1.6mm by increasing the cavity radii to

_{2}*R*= 1.05m and

_{1}*R*= 1.063m, thus improving the scanning quality.

_{2}## References and links

1. | J. Besl “Active, Optical Range Imaging Sensors,” Mach. Vision and Appl. |

2. | F. Blais, “Review of 20 Years of Range Sensor Development,” Electronic Imaging |

3. | G. Stutz, “Guiding Light,” SPIE OE Magazine |

4. | A. B. Colquhoun, D. W. Cowan, and J. Shepherd “Trade-offs in rotary mirror scanner design,” Proc. SPIE |

5. | G. A. Massey and A. E. Siegman, “Reflection and Refraction of Gaussian Light Beams at Tilted Ellipsoidal Surfaces,” Appl. Opt. |

6. | B. E. A. Saleh and M. C. Teich, |

7. | H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. |

**OCIS Codes**

(080.2730) Geometric optics : Matrix methods in paraxial optics

(080.2740) Geometric optics : Geometric optical design

(280.3420) Remote sensing and sensors : Laser sensors

**History**

Original Manuscript: February 7, 2007

Revised Manuscript: March 8, 2007

Manuscript Accepted: March 9, 2007

Published: March 19, 2007

**Citation**

Kaveh Sahba, Kamal E. Alameh, Clifton L. Smith, and Arie Paap, "Cylindrical quasi-cavity waveguide for static wide angle pattern projection," Opt. Express **15**, 3023-3030 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-6-3023

Sort: Year | Journal | Reset

### References

- J. Besl "Active, Optical Range Imaging Sensors," Mach. Vision and Appl. 1, 127-152 (1988). [CrossRef]
- F. Blais, "Review of 20 Years of Range Sensor Development," Electronic Imaging 13,231-240 (2004). [CrossRef]
- G. Stutz, "Guiding Light," SPIE OE Magazine 5,25-27 (2005).
- A. B. Colquhoun, D. W. Cowan, and J. Shepherd "Trade-offs in rotary mirror scanner design," Proc. SPIE 1454,12-19 (1991).
- G. A. Massey and A. E. Siegman, "Reflection and Refraction of Gaussian Light Beams at Tilted Ellipsoidal Surfaces," Appl. Opt. 8,975-978 (1969). [CrossRef] [PubMed]
- B. E. A. Saleh and M. C. Teich, Fundamentals Of Photonics (Wiley-Interscience, 1991). [CrossRef]
- H. Kogelnik and T. Li, "Laser Beams and Resonators," Appl. Opt. 10,1550-1567 (1966). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.