====== Optimum length ======

===== Introduction =====
Different properties of ET scale differently depending on the length - some even in opposite directions - so there is in principle an optimum length. Apart from the obvious ones like noise and sensitivity, the length also has a big impact on the cost. Longer arms need bigger beams in order to avoid having highly divergent beams, too. This is an attempt to list the constraints on length to allow us to arrive at the optimum.

  * Beam size limited by maximum coating size? What can we actually do now/in the future?
  * Are we at the optimal point in parameter space?

===== Factors that influence cavity parameters ======

==== Infrastructure ====
Split into fixed costs (caverns, entrance tunnels, etc.) + cost per length. Tunnel diameter set by beam size, number of interferometers / filter cavities, etc.

For LIGO Cosmic Explorer the cost scaling by length is different as it involves excavating a non-linear amount of land as a function of length. Mike Zucker's work [[http://events.ego-gw.it/indico/getFile.py/access?contribId=12&sessionId=4&resId=0&materialId=slides&confId=34|presented by Matt Evans in the 2016 Florence ET/CE meeting]] projected some rough costs and found that 40km is significantly more expensive per unit length than 20km or less.

For ET, though, the underground facility will have a more linear construction cost, so it's arguable whether the optimal length for above ground vs below ground is the same. P311 in design study has costs: vacuum €200M, main caverns etc. €200M, tunnels cost €400M.

=== Use of lenses in arms ===
To reduce construction cost, it has been proposed to use thinner diameter beam tubes alongside lenses that periodically refocus the beam, meaning the beam can still be made large at the end mirrors (to benefit from coating thermal noise). This was explored for example in [[https://dcc.ligo.org/G1400956|G1400956]]. The thermooptic noise of the substrate appears to create significant noise, and seismic noise requires a cavity mirror style suspension for each lens, so this option is unlikely to work.

That said, cost savings in ET for thinner tubes would likely be significantly more than for LIGO CE...
==== Beam size ====
Required beam size determined by wavelength, cavity stability and loss requirements. For a given cavity length, there is a minimum beam size determined by the beam divergence.

Beam size on the mirrors depends on the waist size. For ET-LF it is 29.0mm and for ET-HF it's 25.1mm (defined as the radius of the beam at its smallest point, i.e. in middle of cavity at z = 5000m). This leads to a Rayleigh range (z_r) of 1704.57m (ET-LF) and 1860.18m (ET-HF), and therefore a beam size on the mirrors (assuming a symmetric cavity) of 89.87mm (ET-LF) and 71.98mm (ET-HF).

The stability factor of a cavity determines how degenerate the alignment signals from the cavity mirrors get. Graef et al (2012) calculated the cavity g-factor for different interferometers (note it did not include ET-D parameters so these are calculated below):
^ Interferometer ^ Cavity length (m) ^ RoC ITM (m) ^ RoC ETM (m) ^ g-factor ^
| Advanced LIGO  | 3996              | 1934        | 2245        | 0.832    |
| Advanced Virgo | 3000              | 1420        | 1683        | 0.871    |
| ET-D LF        | 10000             | 5580        | 5580        | 0.627    |
| ET-D HF        | 10000             | 5690        | 5690        | 0.574    |

ET-D g-factors for each mirror are equal, i.e. g1 = g2 = (1 - L / RoC). Cavity g-factor is g1 * g2.

=== Scaling with length ===
With all other factors held constant, beam radius scales as the square root of length, giving a reduction in coating thermal noise by the same factor.

=== Alignment signals ===
Longer cavities naturally have more stringent requirements on alignment due to geometry. One potential problem is that, with ET's low g-factors, the alignment signals are small. Towards instability, misalignments produce stronger 01/10 modes, though these are more degenerate. This is somewhat mitigated since the alignment requirement in stable cavities are more relaxed - we'd get smaller error signals, but we'd also have more relaxed control requirements.
==== Mirror size ====
Manufacturing capabilities sets the maximum mirror size. Some ideas exist to make composite masses, but these still need to be coated. LMA recently coated the Virgo beam splitter, which had a 55cm coating, so we might assume this to be the order of magnitude coating size we can produce. Assuming modest improvement in coating sizes, ET's coatings could be made ~50% larger than now, which in theory allows the length to be extended by 2.

No obvious limiting factor in suspension design for heavier mirrors - though perhaps tensile strength of silicon an issue?

Newtonian noise is slightly reduced for bigger mirrors due to inertia.
==== Suspensions ====
Vertical to horizontal coupling gets worse than in Advanced LIGO as the local gravity is different, and this allows vertical suspension thermal noise to contaminate the strain readout. Advanced LIGO has a requirement of less than 1:1000 coupling.

Coupling scales linearly with length by (length of arm) / (diameter of the earth). Diameter of earth is ~12e6 m, so for 10km that gives 0.0008, i.e. just within the Advanced LIGO requirement. Longer arms would exceed existing coupling requirements and would necessitate longer suspension stages to lower the suspension thermal noise in the detection band.

==== Thermal distortion / lensing ====
These effects are more or less independent of cavity length, instead depending on circulating power.
===== Optimisation =====

==== Scaling "laws" ====
^ Noise            ^ Length scaling ^ Notes ^	 
| Sensing            | 1              | |	 
| Radiation pressure | 1 / L <nowiki>^</nowiki> (1 + a)  | Longer arms means bigger beams, hence bigger mirrors, hence lower radiation pressure |	 
| Seismic            | 1 / L          | |	 
| Newtonian          | 1 / L <nowiki>^</nowiki> (1 + a)  | As with radiation pressure noise, heavier mirrors have more inertia |	 
| Suspension thermal | 1 / L + aL     | Displacement noise averaging, but local gravity misaligns end mirrors leading to greater vertical suspension thermal noise coupling |	 
| Coating thermal    | 1 / L <nowiki>^</nowiki> (3/2) | Beam radius scales by sqrt(L), hence noise reduced by this factor, plus displacement noise averaging |

==== "Conclusions" ====

  * Energy stored in suspensions a risk when fibres break - no fundamental limit there, it's an (tough) engineering problem!
  * Suspension thermal noise coupling seems to allow longer arms - 20km just about meets Advanced LIGO spec; but since suspension resonances will be lower it might not be a big problem
    * ET's small cavity g-factors relax alignment requirements, potentially allowing existing suspension designs to be used in some form
  * Not sure about ability to make mirrors with required radii of curvature and surface flatness for even longer arms (loss from contrast defect and scattering) - going from RoC of ~2km to >5km.
  * Beam size limited by available optical material. Nothing much has changed there. Longer arms need bigger beams.
    * Seems reasonable to expect coating facilities could make 50% larger coatings than currently, which would allow 20km arms
    * Wider pipes needed to support bigger beams - more expensive.
    * Lenses in arms would allow thinner pipes, but have problem of thermooptic noise of lens substrates (jointly limit sensitivity in LIGO CE).
  * When underground, construction costs scale roughly linearly beyond costs of caverns etc. - if facility was shared e.g. with CERN, costs could be shared which could allow for longer tunnels
    * Above ground, it gets more interesting - LIGO CE excavation vs interferometer costs seem balanced at ~20km

==== Next steps ====

  * Concrete numbers behind above statements.
  * This work can be revisited when the site selection and material choices become more clear.
===== References =====
  * [[http://events.ego-gw.it/indico/materialDisplay.py?contribId=12&sessionId=4&materialId=slides&confId=34|Matt Evans talk from Florence ET/CE meeting, 2016]]
  * [[http://iopscience.iop.org/article/10.1088/0264-9381/29/7/075003/meta|Graef et al. (2012)]]
  * [[https://journals.aps.org/prd/abstract/10.1103/PhysRevD.91.082001|Dwyer et al. (2015)]]
  * [[https://dcc.ligo.org/G1400956|G1400956]]
  * [[https://wiki.ligo.org/pub/LSC/LIGOworkshop2017/WebHome/Dawn2017Evan.pdf|Evan Hall talk from LIGO Dawn Meeting, 2017]]