### Fix some small typos.

parent e2ddde43
 ... ... @@ -65,7 +65,7 @@ Our basic setup consists of: - furthermore, the reference wavelength of |Finesse| is :math:\lambda = 1064 nm. For these parameters Eq. :eq:eq_ex10_Delta gives the following value for For these parameters Eq. :eq:eq_ex10_Delta gives the following value for dimensionless :math:\Delta .. math:: ... ... @@ -135,7 +135,7 @@ respectively. # Set a 1-D CCD at n1, measure around peak at 0.8 at 600 and 1000m ccdline ccd1 node=n1.p1.i xlim=[0.5,1.0] npts=200 # Also put a full 2-D CCD at the same position ccd ccd2 node=n1.p1.i xlim=[-3,3] ylim=[-3,3] npts=200 ccd ccd2 node=n1.p1.i xlim=[-3,3] ylim=[-3,3] npts=200 series( noxaxis(name="S600"), ... ... @@ -161,8 +161,8 @@ Plotting the output from the two 1-dimensional :kat:element:ccdline detectors ax[i].set_xlabel("x/w0") ax[i].set_ylabel("intensity") we see that the Gaussian bundle over distance becomes lower and wider, but remains its peak at the expected position :math:x/w0 = 0.8, confirming we are we see that the Gaussian profile over distance becomes lower and wider, but retains its peak at the expected position :math:x/w0 = 0.8, confirming we are simulating a parallel but shifted beam. For reference we also plot the 2-dimensional beam cross section at 600 meters as ... ... @@ -184,7 +184,7 @@ We next measure the phase shift of the outgoing beam resulting from the tilting of the beamsplitters. For this we use the following simulation script (again in addition to the :ref:basic experimental setup ), using a :kat:element:fline detector measuring the phase along the beam cross section at a distance 600 meter after the second beamsplitter. at a distance 600 meter after the second beamsplitter. For reference, we also run the same simulation for :math:\beta=0. We expect the full result to be shifted over :math:0.8 w_0 and to have a phase that is larger by about :math:27.1^\circ. ... ... @@ -197,14 +197,14 @@ that is larger by about :math:27.1^\circ. # Calculate expected Delta and delta phi Delta = s2*np.sin(2*xbeta)/w0 dphi = 2*s2*np.sin(xbeta)**2/basekat.lambda0*360 kat2 = finesse.Model() kat2.parse(basescript) script2 = f""" fline fl1 node=n1.p1.i xlim=[{Delta}-{dx},{Delta}+{dx}] npts=200 fline fl2 node=n1.p1.i xlim=[-{dx},{dx}] npts=40 series( noxaxis(name='full'), change(bs1.xbeta=0), ... ... @@ -227,7 +227,7 @@ perfectly. .. jupyter-execute:: f,ax = plt.subplots(ncols=1, figsize=(8, 5)) ax.plot(kat2.fl1.xdata, np.angle(out2['full']['fl1'], deg=True), 'r', label="measured") ... ... @@ -251,13 +251,13 @@ experimental setup ) kat3 = finesse.Model() kat3.parse(basescript) script3 = f""" # Note: important to specify n,m amplitude_detector ad1 node=n1.p1.i f=0 n=0 m=0 xaxis(bs1.xbeta, lin, -1.5e-5, 1.5e-5, 40) """ kat3.parse(script3) out3 = kat3.run() ... ...
 ... ... @@ -22,7 +22,7 @@ of incidence of both beamsplitters (or mirrors). We will derive here the shift in the position of the beam as a function of (among others) this angle :math:\beta. In addition we will also derive the effect of this parameter on the phase of the beam, resulting from the changed pathlength between the laser and the detector. Both formulae will be verified path length between the laser and the detector. Both formulae will be verified using |Finesse| simulations in example :ref:example10. In the next section we will discuss the required number of higher order ... ... @@ -185,7 +185,7 @@ We work out the numerator: &= 2 \sin\alpha \sin\beta ( 1 - \cos^2\beta) - 2 \cos\alpha \cos\beta \sin^2\beta \\ &= 2 (\sin\alpha \sin\beta - \cos\alpha \cos\beta) \sin^2\beta \\ &= -2 \cos(\alpha + \beta) \sin^2 \beta \end{aligned} ... ... @@ -219,8 +219,8 @@ Summary typically small but not per se compared to the wavelength :math:\lambda, leading to a significant phase shift. Effect finite size beam ^^^^^^^^^^^^^^^^^^^^^^^ Effect on a finite size beam ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The above calculation was done for an infinitesimally narrow beam hitting the beamsplitter exactly in the midpoint (its rotation point). We show here that the ... ...
 ... ... @@ -38,7 +38,7 @@ layout, providing the largest valid range of :math:\beta. Once we have found the optimal :math:z_0, we will use it to study the valid :math:\beta range as a function of the distance between the two beamsplitters, looking in particular at the scaling under changes of the waist size :math:w_0 and linking it to the diffraction angle and Rayleigh range. linking it to the diffraction angle and Rayleigh range. Base script for the different simulations """"""""""""""""""""""""""""""""""""""""" ... ... @@ -86,7 +86,7 @@ same as that used in :ref:example10. basekat = finesse.Model() basekat.parse(basescript) Tilt angle dependency for various maxtem Tilt angle dependency for various maxtem """""""""""""""""""""""""""""""""""""""" We add an appropriate :kat:analysis:xaxis action to the model to measure the ... ... @@ -103,7 +103,7 @@ total energy as a function :math:\beta for 4 different values of maxtem: kat1 = finesse.Model() kat1.parse(basescript) kat1.parse("xaxis(bs1.xbeta, lin, 0, 4e-5, 40)") out1a = [] for maxtem_val in maxtem_vals: kat1.modes(maxtem=maxtem_val) ... ... @@ -123,7 +123,7 @@ and 1000 meter: # Reset maxtem to its default kat1.modes(maxtem=maxtem) out1b = [] for s3_val in s3_vals: kat1.spaces.s3.L = s3_val ... ... @@ -154,7 +154,7 @@ From the left figure, it is clear that we can simulate larger and larger :math:\beta if we include more and more higher order modes in the simulation, i.e. by increasing the maxtem value. The valid range of :math:\beta scales roughly linearly with maxtem, meaning the number of modes to be includes roughly scales quadratically (note that for maxtem is :math:N there are roughly scales quadratically (note that for maxtem = :math:N there are :math:\frac{1}{2}(N+1)(N+2) modes). .. todo:: do we expect convergence to break down at very high beta? ... ... @@ -192,7 +192,7 @@ reference to the other (see :ref:expressions). # Make sure zy equals zx kat2.g1.zy = kat2.g1.zx.ref # vary beta and z0 kat2.parse(f""" series( ... ... @@ -220,8 +220,8 @@ plot the close-up range around the beamsplitters. We see that the optimal location for the waist is around 1 km right of the laser, i.e. at the location of the first beamsplitter. Effect waist size """"""""""""""""" Effect of waist size """""""""""""""""""" Now that we know the optimal location for the Gaussian waist, we will fix it that position and then study the effect of changing the distance between ... ... @@ -244,23 +244,23 @@ both :math:w_0 and :math:z_0 are changing simultaneusly (see w0a = 10 beta_a = (1e-5, 4e-5) La = 1000 # w0, beta-range and L for 12mm run w0b = 12 beta_b = (1e-5/1.2, 4e-5/1.2) Lb = 1000*1.2**2 # Define model kat3 = finesse.Model() kat3.parse(basescript) # could set w0x and w0y separately, easier to use ref for y kat3.g1.w0y = kat3.g1.w0x.ref # Position gauss at optimal z0: i.e. position bs1 kat3.g1.zx = -1000 kat3.g1.zy = kat3.g1.zx.ref # run both subsimulations using series() kat3.parse(f""" series( ... ... @@ -275,7 +275,7 @@ both :math:w_0 and :math:z_0 are changing simultaneusly (see And we plot the results .. jupyter-execute:: # Plot results for both waist sizes f,ax = plt.subplots(ncols=2, figsize=(11.8, 5)) for (i, name, w0val) in ([0, 'a', w0a], [1, 'b', w0b]): ... ... @@ -293,7 +293,7 @@ And we plot the results out3['a'].x.min()*1.2**2,out3['a'].x.max()*1.2**2) ax.contour(out3['a']['pd1'].T, colors='green', linestyles='dotted', extent = pxy_extent_scaled); Looking first at the left picture, we see that the usable range of :math:\beta roughly remains constant till a distance of about 200-300 meter and then quickly decreases for larger distances. This distance of 200-300 meter corresponds ... ... @@ -306,7 +306,7 @@ roughly to the Rayleigh range :math:z_R which for the parameters in the left plot (:math:w_0 = 10\textrm{mm}, \lambda = 1.064 \mu\textrm{m}) is 295 meter. It would seem that our default value of 400 meter is not ideal, but we will discuss this further below when looking at the maximum attainable shift :math:\Delta. maximum attainable shift :math:\Delta. Although in general the largest usable :math:\beta increases as a function of maxtem` we expect that it will be proportional to the diffraction angle ... ...
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