1. Field of the Invention
The present invention relates to optical cavity mode mismatch sensors and, more particularly, to a mode matching sensor telescope.
2. Description of the Related Art
Higher order mode-sensing techniques currently utilize CCD cameras, clipped photodiode arrays or bullseye photodiodes. These sensors provide feedback error signals for correcting either the beam waist size or waist location, but also have drawbacks. Some of the drawbacks include slow signal acquisition for CCD sensors, 50 percent reduction in sensing capabilities for clipped arrays, and expensive custom parts that are difficult to setup for bullseye photodiodes. For example, because a clipped photodiode array requires a split in the power between the diodes, sensing capabilities are reduced by 50 percent. Similarly, while bullseye photodiodes offer a very convenient geometry to sense Laguerre-Gauss (LG) modes, they are not very commonly manufactured, difficult to setup and thus not very cost effective.
The present invention is a new approach for sensing optical cavity mode mismatch by the use of a cylindrical lens mode converter telescope, radio frequency quadrant photodiodes (RFQPDs), and a heterodyne detection scheme. A cylindrical lens mode converter telescope allows the conversion of beam profiles from the LG basis to the Hermite-Gauss (HG) basis, which can be easily measured with QPDs. The present invention transforms mature alignment sensors into equally mature mode matching sensors. By applying the mode converter in reverse, the LG10 mode turns into an HG11 mode, which is shaped perfectly for a quadrant photodiode. After the quadrant photodiode, well-known heterodyne detection methods may be used to extract a robust mode matching error signal. Converting to the HG basis is thus performed optically, and mode mismatched signals are measured using widely produced RFQPDs to obtain a feedback error signal with heterodyne detection.
The present invention will be more fully understood and appreciated by reading the following Detailed Description in conjunction with the accompanying drawings, in which:
Referring to the figures, wherein like numerals refer to like parts throughout, there is seen in
Referring to
In order to implement a mode converter into an actively controlled optical system, it is necessary to derive an error signal from the output that is linearly proportional to the waist position or size of the optical cavity. This is done by applying a mask to the output images as seen in
Typical optical cavity alignment sensing requires the ability to measure the HG01 and HG10 modes with a quadrant photodiode. Mode mismatch manifests itself as concentrically symmetric LG modes. Since the whole beam will be affected by the mode converter, it is worthwhile to examine what happens to the alignment signals after they pass through. Passing a well aligned LG mode through the mode converter requires no specific rotation angle since the LG mode is radially symmetric. The cylindrical lenses will always produce a HG mode that is rotated 45 degrees from the cylindrical lens focusing axis in
Passing HG01 or HG10 alignment signals through the mode converter does require a specific rotation angle. The alignment signals must enter at 0 degrees with respect to the focusing axis to maintain their shape. If this condition is not respected then mixing of the modes will occur, as seen in
The mode converter does not affect the misalignment signals in a notable way as long as the alignment modes enter at 0 degrees with respect to the focusing axis.
Computer simulation provided a quick method for testing our prediction before performing the experiment. A combination of FINESSE, which mainly uses ABCD matrix math at its core, and MATLAB was selected for the experiment. FINESSE was chosen because it had previously been used to generate mode mismatching signals from an optical cavity that were sensed with bullseye photodiodes and thus provides a good baseline for simulation comparison. MATLAB was used as a means to process our transverse electric field expression using Fourier Optics. FINESSE was used to produce the mode mismatched cavity in
Appendix
The complex beam parameter of a Gaussian beam with Rayleigh range zR is defined as:
q=+iR. (1)
Beam size w and phase front radius of curvature R are then given by
where λ=2π/k is the wave length of the light. It allows expressing the Gaussian beam in a simple form:
where A is a complex constant (amplitude). It can be helpful to introduce the field amplitude on the optical axis, ψ=A/q, which now evolves along the z-axis due to the Gouy phase evolution. Thus, for any given location on the optical axis z, the Gaussian beam is completely described by the two complex parameters ψ and q. The main advantage of this formalism becomes apparent when using ray-transfer matrices M defined in geometric optics (e.g. Saleh, Teich) to represent the action of a full optical system. The two complex parameters after the system (qf, ψf), are given in terms of the initial parameters (qi, ψi) by
and the change of the Gouy phase through the system, Δϕ, is given by
This expression is consistent with the usual definition of Gouy phase for a Gaussian beam as ϕ=arctan z/zR, and can be proven by verifying it for a pure free-space propagation and a pure lens.
If astigmatism is introduced, either intentionally with cylindrical lenses or accidentally through imperfections, cylindrical symmetry around the beam axis will be lost. As long as we introduce this astigmatism along a pre-determined axis (say the x-axis), we can simply proceed by introducing separate q-parameters for the x- and y-axis, qx and qy. Since ray-transfer matrices are introduced with only 1 transverse axis, the propagation of qx and qy is done with ray-transfer matrices defined for the corresponding transverse axis. Thus we now have a separately-defined beam size wx, wy, phase front radius of curvature Rx, Ry, Rayleigh range zRx, zRy and Gouy phase ϕx and ϕy for each of the two transverse directions. The corresponding fundamental Gaussian beam is given by
where A is again a complex amplitude. Next we introduce the Hermite-Gaussian basis set corresponding to the fundamental Gaussian beam. In the literature this is typically done only relative to a single q-parameter, but directly generalizes to the case with separate qx and qy parameters:
Here, we redefined the overall amplitude N such that the total power P in a mode is simply given by P=∫|Ψnm|2dxdy=|N|2. That equation (8) is of the same form can be seen using the identity izR/q=eiϕw0/w. Furthermore, ψξ was defined in analog to the field amplitude ψ introduced after equation (4), that is the field amplitude on the optical axis of the fundamental mode. It thus evolves, together with qξ, according to equations (5) and (6). Note though that there is an extra Gouy phase term for the higher order modes that is explicitly excluded from the definition of ψξ. As a result, the overall Gouy phase evolution of Ψnm({umlaut over (x)}, y, qx, qy) is proportional to ei(n+1/2)ϕ
As expected, these modes still solves the paraxial Helmholtz equation
exactly.
Referring to
For the present invention, some design considerations are worth noting. First, after the mode converter the signals for pitch and yaw will be in different Gouy phases. For ideal sensing of all 6 degrees of freedom, 3 QPDs are needed at 0°, 45° and 90° Gouy phase separation. However, in practice a setup with 2 QPDs 67.5° apart may be sufficient. The beam should be focused so that the beam waist is in the middle of the mode converter. The cylindrical lens separation must be f√{square root over (2)}, where f is the focal length of each of the cylindrical lenses. This constrains the Rayleigh range of the reference beam and therefore its beam size w0=−√{square root over ((1+1/√{square root over (2)})fλ/π)}.
The benefits of mode converter with quadrant photodiodes according to the present invention include the ability to measure mode mismatch using existing QPD. The mode converter also preserves cavity misalignment signals. For the present invention, no new electronics or channels are needed and QPDs are easier to align than BPDs. Compared to BPDs, QPDs are off-the-shelf and have a much better matched quadrant capacities on optical gains. Only one mode converter, followed by normal Gouy phase telescopes for each sensor is needed to sense all four alignment and two mode-match degrees of freedom. The only optical component critically sensitive to beam size is the mode converter itself. It is much easier to change than the segment size of (multiple) BPD.
As described above, the present invention may be a system, a method, and/or a computer program associated therewith and is described herein with reference to flowcharts and block diagrams of methods and systems. The flowchart and block diagrams illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer programs of the present invention. It should be understood that each block of the flowcharts and block diagrams can be implemented by computer readable program instructions in software, firmware, or dedicated analog or digital circuits. These computer readable program instructions may be implemented on the processor of a general purpose computer, a special purpose computer, or other programmable data processing apparatus to produce a machine that implements a part or all of any of the blocks in the flowcharts and block diagrams. Each block in the flowchart or block diagrams may represent a module, segment, or portion of instructions, which comprises one or more executable instructions for implementing the specified logical functions. It should also be noted that each block of the block diagrams and flowchart illustrations, or combinations of blocks in the block diagrams and flowcharts, can be implemented by special purpose hardware-based systems that perform the specified functions or acts or carry out combinations of special purpose hardware and computer instructions.
The present application claims priority to U.S. Provisional No. 62/525,432, filed on Jun. 27, 2017.
This invention was made with government support under Grant No. PHY-1352511 awarded by the National Science Foundation (NSF). The government has certain rights in the invention.
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Number | Date | Country | |
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20180374967 A1 | Dec 2018 | US |
Number | Date | Country | |
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62525432 | Jun 2017 | US |