Chapter Two

1
Chapter Two Optical Power Measurement Contents
1. Power Meters with Thermal Detectors 2. Power
Meters with Photodetectors 3. LED-Power
Measurement 4. High-Power Measurement 5.
Uncertainties in Power Measurement 6.
Responsivity Calibration 7. Linearity Calibration
2
Introduction Two types of power measurements
Absolute power measurement Needed in
conjunction with optical sources, detectors and
receiver Relative power measurement Important
for the measurement of attenuation, gain and
return loss Two main groups of optical power
meters Power meters with thermal detectors the
temperature rise caused by optical
radiation Power meters with photodetectors the
incident photons generate electron-hole pairs
Question How to measure the power with a thermal
detector?
3
Introduction Comparison of thermal power meters
and photodetector power meters Although
photodetector-type power meters suffer from a
relatively small wavelength coverage and the need
for absolute calibration, their astounding
sensitivity usually makes them the preferred
choice. Nevertheless, power meters with thermal
detectors are sometimes preferred in the
calibration laboratories because of their wide
and flat wavelength characteristics. In addition,
thermal detectors can be directly traceable to
electrical power measurements. Altogether, there
is good reason for the existence of both types of
power meters.
Characteristics Power meters with thermal detectors Power meters with photodetectors
Wavelength dependence wavelength-independent wide wavelength range - wavelength dependence - wavelength range 21
Self-calibration available - not available (calibration indispensable)
Sensitivity - very low (typically 10 µW) very high (down to less than 1 pW)
Accuracy 1 depending on calibration method 2 depending on calibration method
4
Power Meters with Thermal Detectors
Example thermal detector Substitution radiometry (Self-calibration method) Substitution radiometry First exposed to the optical radiation Then the radiation is switched off (with a shutter of chopper) and replaced by electrically generated power Important elements of a thermal detector with electrical substitution An absorptive layer collects the incident light A heating resistor perform substitution, thermally well coupled to the absorptive layer An isolated sheet of silver equalizing any temperature differences, coated with black paint A thermopile (a series connection of thermocouples) measures temperature rise, in close proximity to the silver
Questions What is the parameter to be controlled
to achieve time-independent temperature? What
provides the basis for the accuracy of this
method?
5
Power Meters with Thermal Detectors
The figure shows the thermal detector with electrical substitution
Question What is the series of apertures for?
6
Power Meters with Thermal Detectors
For highest accuracy Reference plane on large thermal mass to maintain constant temperature during the relatively long measurement times Blocking of background radiation and stray light using a jacket with thermal isolation Optimization of heat flow a negligible thermal resistance between the absorptive layer and the heater High absorptance reduce reflection, which does not contribute to the temperature rise Accurate measurement of the electrical power eliminate heat dissipation from the resistor lead Alternative operation Continuously heated by electrical power which is slightly larger than the optical power to be measured The sensor voltage is recorded without the optical power applied Then the sensor is exposed to the optical power, and a feedback loop reduces the electrical power until the sensor voltage is the same as before (on-line calibration) The desired optical power measurement result is simply the difference of electrical powers between the two steps
7
Power Meters with Thermal Detectors
Principal problems Low sensitivity The correspondent long measurement time Possible improvement Replacing pyroelectric sensors or thermopiles with semiconductor material Typical characteristics of thermal power meters Sensitivity down to 1 µW Uncertainty as low as 1 Spectral range from ultraviolet to far infrared Time constant of several seconds to minutes depending on the detector size
Question Why does the thermal power meters need a
long measurement time?
Question Using the alternative measurement
procedure mentioned in the previous slide, is it
possible to have initial electrical power lower
than the optical power to be measured? Explain
why.
8
Power Meters with Thermal Detectors
Cryoradiometer A thermal detector that is placed into vacuum and cooled to approximately 6 K using liquid helium The most precise optical power meters due to At 6 K, the thermal mass (the energy needed to raise the temperature by 1 K) of the absorbing material is drastically reduced Heat loss due to radiation is virtually eliminated because the radiated energy is proportional to T4 (T in K) Heat contributions from the resistor leads can be eliminated by making them superconducting Convection losses are eliminated
Questions What is the consequence of having low
thermal mass? How do the superconducting resistor
leads help? How do the convection losses
eliminated?
9
Power Meters with Photodetectors
Principal advantages Great sensitivity can measure power levels down to less than 1 pW (-90 dBm) High modulation frequency response fast measurement time Ease of use Principal disadvantages Exhibits a relatively strong wavelength dependence No self-calibration Categories Small-area power meters only measures power from a fibre Large-area power meters open beams and fibre applications Important elements in a large-area power meter Antireflective coating on the connector adapter Pinhole and angled position of the detector
10
Power Meters with Photodetectors
The figure shows a cross-section through a commercial large-area optical sensor head based on a photodetector Operations Temperature stabilization using a thermoelectric cooler ensures stable measurement results The photodetector is operated at zero-bias voltage in order to eliminate any offset currents
Question Why is the photodetector angled
positioned?
11
Power Meters with Photodetectors
The most important contributions to accurate power measurements are Individual correction of wavelength dependence Temperature stabilization Wide power range with good linearity Good spatial homogeneity Low polarization dependence Low reflections Compatibility with different types of fibre PIN diode The figure shows a cross-sectional view of a planar InGaAs PIN diode
12
Power Meters with Photodetectors
PIN diode Operation Each incident photon is absorbed in the intrinsic (i-) layer An electron-hole pair is created the photon energy bandgap energy (material-dependent The holes and electrons are swept out of the i-region by the large built-in electric field photocurrent Terms which describe the conversion efficiency Quantum efficiency ?, defined as the number of electrons per photon Responsivity r, defined as the photocurrent per unit of optical power
Question What is the value for ? in the ideal
case?
13
Power Meters with Photodetectors
PIN diode From the definition, the responsivity r is given by, Each photon represents the energy Eph, where h Plancks constant, ? optical frequency, and c speed of light in vacuum The optical power which corresponds to one photon is, The correspondent electrical current is one electron charge q per time span ?t, The linear spectral responsivity of an ideal photodetector with ? 1,
14
Power Meters with Photodetectors
PIN diode Practical photodetectors deviate from this ideal wavelength dependence in several ways A long wavelength limit (cutoff wavelength) photon energy becomes lower than the bandgap energy, determined by the detector material At short wavelength absorption outside of the i-region reduces the number of electron-hole pairs The responsivity may also be reduced by recombination when the electrons recombine with the holes before they reach the electrodes Reflections from the detector surface can produce substantial inaccuracies in optical power and insertion loss measurement A periodic structure of the responsivity may be observed due to optical interference in the diode Antireflective coatings Single-layer, quarter-wavelength coating are most often used Multi-layer coating low reflectivity over a wider wavelength range
Question Pure InGaAs has a refractive index of
3.5. Calculate the reflectivity (or Fresnel
reflection) at the detector surface. Answer
31
15
Power Meters with Photodetectors
Spectral Responsivity The figure shows typical responsivity measurement results for three types of photodetectors Short wavelength range (500 -1000 nm) silicon Long wavelength region both Germanium and InGaAs
16
Power Meters with Photodetectors
Spectral Responsivity Germanium Lower cost solution Recommended when the sources to be measured are spectrally narrow and the wavelength is well known (around 1550 nm) 1 error when the power meters wavelength setting is incorrect by 1 nm InGaAs Flat around 1550 nm Better than 0.1 per nm wavelength error Well suited for optical amplifier (EDFA) applications More expensive technology
Question What is the property that makes InGaAs
detectors suitable for EDFA applications?
17
Power Meters with Photodetectors
Temperature Stabilization Temperature-stabilized detectors generate reproducible measurement results The figure shows the responsivity of a germanium detector It exhibits a relatively small temperature dependence for most of the wavelength range There is substantial change beyond the cutoff wavelength shift of cutoff wavelength, 1nm/K
18
Power Meters with Photodetectors
Spatial Homogeneity The responsivity of photodetectors can vary across the detector surface The figure shows the relative responsivity of an InGaAs photodetector at 1550 nm Inhomogeneous photodetector surfaces create measurement uncertainties the position and diameter of the incident beam cannot be perfectly controlled
19
Power Meters with Photodetectors
Spatial Homogeneity Multimode fibre A dark and light speckle pattern is formed if illuminated with a narrow spectral-width optical source This will cause the power distribution in the fibre cross-section to fluctuate
Questions How is the speckle pattern formed? What
will happen if a wide spectral source is used?
20
Power Meters with Photodetectors
Power Range and Nonlinearity Sources of nonlinearity Photodetector nonlinearity, electronic nonlinearity The photodetector nonlinearity Noise at low power levels Supralinearity at medium power levels Saturation at high power levels The electronic nonlinearity In-range nonlinearity of the analogue amplifier Ranging discontinuity non-matching amplifier gains The nonlinearity is defined as Where r(P) is the power meters responsivity at an arbitrary power level, and r(P0) is the responsivity at the reference level (usually 10 µm)
Question What causes nonlinearity in the analogue
amplifier at high power levels?
21
Power Meters with Photodetectors
Power Range and Nonlinearity The nonlinearity is usually wavelength dependent wavelength-dependent photodetector The figure illustrates the possible nonlinearity effects of an optical power meter Dark current Limits the low end of the power range Depends on the active area and on the semiconductor material The shot noise current is given by,
22
Power Meters with Photodetectors
Power Range and Nonlinearity Total short noise where r responsivity and Popt received optical power Noise equivalent power (NEP) Signal-to-noise ratio (SNR) SNR improvements Reduce dark current either by cooling or by reducing the detectors active area (the dark current is proportional to the active area) Longer averaging time
Question What will happen to the SNR as the Popt
increases?
23
Power Meters with Photodetectors
Power Range and Nonlinearity The figure shows the power dependence of the SNR
Question Why is the SNR linear for low Popt?
24
Power Meters with Photodetectors
Power Range and Nonlinearity Range discontinuity The power meter does not display exactly the same power level when switching between power ranges Caused by the necessity to switch the gain of the electronic amplifier, depending on the input power level Supralinearity An increase in responsivity typically starting at power levels 100 µW Due to traps in the semiconductor material causing increased recombination at low power levels When the power reaches higher levels, then these traps become saturated, the recombination decreases, and the responsivity increases Saturation Caused by reduction of the electric field across the pn-junction along with recombination in the active region
25
Power Meters with Photodetectors
Polarization Dependence Causes for polarization dependence Crystalline structure in the semiconductor material and in the photodetectors coating Mechanical stress in the detector Tilting against the beam axis (to reduce multiple reflections) A relatively strong wavelength dependence of the polarization characteristics can also be observed and is usually caused by the quality of the antireflective coating
26
Power Meters with Photodetectors
Optical Reflectivity and Interference Effects Without antireflective coating, optical detectors exhibit reflectivities up to 30 - cause multiple reflection and optical interference problems Examples of antireflective coating Silica on silicon detectors Silicon nitride on InGaAs detectors The figure shows the measured reflectance of an InGaAs photodetector with a single-layer antireflective coating made from silicon nitride with a thickness of a quarter wavelength
27
Power Meters with Photodetectors
Optical Reflectivity and Interference Effects Silicon nitride has a refractive index n 1.95 and acts as an impedance transformer matching the refractive index of InP (n 3.2) with air (n 1) The quarter-wavelength layer is responsible for the overall minimum around 1250 nm for this specific diode The additional ripple is caused by the upper InP layer which forms an additional resonator due to the fact that InP has a refractive index of 3.2, in contrast to the refractive index of 3.52 fro the intrinsic InGaAs layer. The reflectance varies substantially from detector to detector A slight thickness change of the InP layer shifts the pattern to different wavelength The detector surface or the glass cap may cause reflections If the detector is sufficiently large, the unwanted power fraction on the detector the photodetector reflectance x the reflectance of the optical interface To reduce this problems, the adapter is coated with an antireflective coating on the inside, and a pinhole shields the highly reflective connector end
Question Why is the pattern shifted to different
wavelength when the thickness changes?
28
Power Meters with Photodetectors
Compatibility with Different Fibres Compatibility with Single-mode Fibres The far-field power density (irradiance) from a single-mode fibre, H(z) is usually described by a gaussian beam where z distance from the source on the beam axis, w radius of the beam waist at which the power has dropped to 1/e2, at the distance z, and r radial distance from the optical axis The numerical aperture (NA) of the fibre is defined by the 5 angle of the far field. If the detector diameter coincides with the circle created by the numerical aperture, then the detector misses 5 of the total beam power The corresponding 95 detector radius is Generally, when the power density at the detector radius has decayed to x, then there is x of the total power outside the detector. This is a property of the gaussian beam
29
Power Meters with Photodetectors
Compatibility with Different Fibres Compatibility with Single-mode Fibres The coupling efficiency is given by It is advisable to replace w, the 1/e2 beam radius, by the 5 beam radius which corresponds to the fibres numerical aperture. The gaussian beam profile yields, Then the coupling efficiency can be expressed on the basis of the numerical aperture
Question If the detector radius is 2.5 mm, the
distance between the fibre end and the detector
is 8 mm, and the numerical aperture of the
single-mode fibre is 0.3, calculate the coupling
efficiency. Answer 96
30
Power Meters with Photodetectors
Compatibility with Different Fibres Compatibility with Angled Fibre Ends Angled fibre ends are aimed at reducing reflections The figure shows a single-mode fibre, both with straight and angled fibre end Assume that, in the case of the straight fibre end, the detector captures the beam fully, and that in the angled case the detector misses a part of the beam
31
Power Meters with Photodetectors
Compatibility with Different Fibres Compatibility with Angled Fibre Ends In both cases, the numerical aperture of the fibre is defined by the 5 angle ? of the far field In the angled case, the tilt of the beam axis ß can be calculated using Snells law, The effective numerical aperture for the angled case is To capture the beam fully A shorter distance to the detector would be needed Tilting the fibre, so that the beam axis is realigned to hit the centre of the detector Using a lens to reduce the effective beam diameter
Question What is the effective numerical aperture
for the straight case?
32
Power Meters with Photodetectors
Compatibility with Different Fibres Compatibility with Fibres of High NA In situations with high numerical aperture, a power meter may not present the same responsivity to all parts of the beam Solution 1 Decreasing the distance between fibre end and photodetector Problems Reflection becomes significant The photodetectors responsivity is lower for those parts of the beam that hit the detector at larger angles
Question Why is the reflection problems may occur
if the distance between fibre end and
photodetector decreases?
33
Power Meters with Photodetectors
Compatibility with Different Fibres Compatibility with Fibres of High NA Solution 2 Using a lens with high numerical aperture in order to collimate the beam Problems Light emitted at larger angles will be more strongly reflected off the lens than the on-axis beams The figure shows the power meter with a lens inserted into the beam path
Question Suggest one improvement to this
solution?
34
Power Meters with Photodetectors
Compatibility with Different Fibres Compatibility with Fibres of High NA Solution 3 Using an integrating sphere in combination with the photodetector Ideally, the integrating sphere should perfectly scatter all incident light The detector should not be exposed to either direct beams from the source or to beams after only one reflection Beams forming large angle (high numerical aperture) against the connector axis go through different attenuations than the near axis beams In addition, some of the materials used to scatter the beam inside the integrating sphere tend to absorb moisture, so that the scattering characteristics change with the relative humidity
35
Power Meters with Photodetectors
Compatibility with Different Fibres Compatibility with Multi-mode Fibres A narrow linewidth source will generate irregular far-field patterns (speckle patterns), which are caused by optical interference between the different fibre modes Speckle patterns go through rapid changes when the fibre is moved, because changing the path lengths of the individual modes by only fractions of the wavelength creates a different speckle pattern Speckle patterns create additional uncertainties because the photocurrent is a convolution of the speckle pattern with the detectors spatial homogeneity
36
LED Power Measurement
LED power is difficult to measure because LEDs wide spectral width The photodetectors responsivity changes within the spectral range Correction Possible if the detectors spectral responsivity and the LEDs spectral power density are known The figure shows the situation for a 1550 nm and a germanium detector ?0 arbitrarily chosen wavelength (preferably the LED peak wavelength) for which the power meter is corrected rrel(?) responsivity relative to ?0 , where rrel(?0) 1 p0 spectral power density of the LED at the wavelength ?0, in watts/nm f(?) factor describing the LEDs spectral emission, where f(?0) 1
37
LED Power Measurement
The correct LED power is The uncorrected measurement result is A correction factor can be calculated to be
Question What if the LED spectrum is symmetrical
and the detectors responsivity is linearly
changing with respect to wavelength?
38
LED Power Measurement
The following measurement procedure is suggested Determine the LEDs centre wavelength, for example, from its data sheet Set the power meter to the LEDs wavelength ?0 and measure the LED power If the LED spectrum is essentially symmetrical and the photodetectors responsivity is nearly linear within the LEDs spectral band, use the measured power as the result If one of the above condition is not met, Calculate the correction factor as in the previous slide Multiply the measured power with the correction factor to obtain the correct power
Question Is the correction factor needed for
laser power measurement? Explain why.
39
High Power Measurement
Optical power meters based on photodetectors can measure maximum power levels of a few milliwatts Beyond this power level, the photodetector goes into saturation Possible output power exceeding the measurement range of conventional power meters The amplifier pump lasers, which produce 100 mW All optical amplifiers, which may exceed 1 watt (except for preamplifiers a few milliwatts) The figure shows a commercial high-power optical head with a 5 mm InGaAs detector and a window made from absorbing glass, to reduce the incident optical power to a suitable level
40
High Power Measurement
Local overheating of the absorber Occurs when the incident optical power levels exceed 100 mW Prevention create a spot diameter of not less than 3 mm on the detector (measured at the 5 points) Solutions for high-power measurement Inserting a scattering filter between the fibre end and the detector Wide wavelength range High-power capability Scattering introduces depolarization reduces the polarization dependence of the optical head Different beam geometries (fibre types) will cause different attenuations this technique splits power away from the detector
Question At the given distance of 8 mm between
the end of a standard single-mode fibre and the
detector, determine whether it is adequate to
prevent the local overheating of the absorber
when the numerical aperture of the fibre is 0.1.
Answer No
41
High Power Measurement
Solutions for high-power measurement Inserting a mesh-type filter consisting of thin wires between the fibre end and the detector Wide wavelength range High-power capability increased wire temperature will not influence the attenuation Splitting some power away before the measurement Limited to certain fibre types because the coupler fibres must be of the same types as the fibre to be measure Inserting an integrating sphere between the fibre end and the detector Usually an expensive solution Some angle dependence dependence on relative humidity Several of these techniques can be combined A collimating lens may have to be inserted before these filters to ensure that beam diameter remains smaller than the detector diameter
42
High Power Measurement
Common to all of these techniques is the need for calibrating the filter attenuation The figure shows the calibration setup An optical attenuator may have to be inserted between the source and the detector to ensure stable output power
43
High Power Measurement
The calibration is for the specific fibre and wavelength Procedure Set a power level that can be handled by the unattenuated sensor Measure the power P1 Attach the filter Measure the power again P2 The desired filter attenuation is the ratio of the two power levels
44
Uncertainties in Absolute Power Measurement
Random uncertainty due to power instability Power instabilities could be inherent to the source or caused by external reflection travelling back to the source Systematic uncertainty due to power-meter calibration It is assumed that the power meter is regularly calibrated following the manufacturers recommendations and that the wavelength correction is set to the wavelength of the source The absolute uncertainty and the conditions for which this uncertainty applied should be obtained from the power meters data sheet Systematic uncertainty due to the spectral width of the source Laser diodes measurement negligible LED measurement No error if the spectrum is symmmetrical about the centre wavelength and the power meters responsivity is linear within the wavelength range of interest Otherwise, a correction factor or an uncertainty can be calculated
45
Uncertainties in Absolute Power Measurement
Systematic uncertainty due to wavelength The wavelength of the source (centre wavelength) should be accurately known Otherwise, the partial uncertainties will be the wavelength uncertainty multiplied by the power meters responsivity versus wavelength slope (/nm) at that wavelength Systematic uncertainty due to beam geometry In the best case, the beam is centred on the detector and the beam diameter is about 2/3 of the detector diameter If this is not the case, then an appropriate uncertainty may have to be calculated Particularly, problems can be expected when the fibre end is angled and the beam partly misses the detector
46
Uncertainties in Absolute Power Measurement
Systematic uncertainty due to power level Optical power meters have extremely wide power ranges of up to 100 dB Uncertainties due to power level can be expected when the actual power approaches the noise level, or when it exceeds the high end of the specified power range Systematic (and random) uncertainty due to reflections Commercial power meters are often calibrated with an open beam In the actual measurement with a fibre, the fibre is held by a connector and connector adapter In this case doubly reflected power may strike the detector, causing an increase of the power reading Reflections can also cause power stability problems
47
Responsivity Calibration
The most important criterion in conjunction with accurate measurement of absolute power Generally, all power meters are calibrated through comparison A test meter and a power measurement standard are exposed to a suitable radiation source, either sequentially or in parallel If a calibration in fine-wavelength steps over a wide wavelength range is desired, then the source should be a halogen white-light source which is spectrally filtered with a monochromator A power level of approximately 10 µW and a spectral width of up to 5 nm are desirable The figure shows a typical monochromator-type calibration setup
48
Responsivity Calibration
Two types of standard sensors Thermal detector Photodetector sensors A monochromator-based calibration setup is expensive and difficult to operate and maintain A more affordable setup is shown in the figure
49
Responsivity Calibration
Dual-wavelength calibration A dual-wavelength source (FP laser) generates precisely known wavelengths around 1300 and 1550 nm The attenuator is used to isolate the source and to set the appropriate power level The coupler is used to split the power and to provide power monitoring A specially calibrated optical head is used as the standard A blank adapter serves as a spacer, to enlarge the spot diameter on the detector to approximately 2.4 mm (at the 5 points)
Question How to perform absolute power
calibration over wavelength using the above
configuration?
50
Responsivity Calibration
Switching the two coupler arms between the standard and the test meter (DUT) can be used to determine both the split ratio and the correction factor P the correct power levels from the standard D the displayed power of the DUT
51
Responsivity Calibration
Coupling ratio Correction factor The correction factor can either be used to correct the test meter or, without correction, as a test result for the calibration certificate
Question What will happen to the coupling ratio
and the correction factor if there is a drift of
the source power?
52
Linearity Calibration
Power meter linearity calibration is necessary because of two reasons To extend the calibration of absolute power to the whole power range To prepare the basis for high-accuracy loss and gain measurements The linearity is expected to be almost wavelength-independent sufficient to calibrate at only one or two wavelengths within the detectors spectral responsivity region Photodetectors provide excellent linearity from the noise level to approximately 1 mW often the specifiable linearity is limited by the performance of the linearity calibration setup Linearity Calibration based on Comparison Procedure Measure an arbitrary attenuation with both the test meter and a standard meter Compare the two attenuation results
Question Is the specifiable linearity limited by
the linearity of the detector?
53
Linearity Calibration
A possible measurement setup is shown in the figure The first attenuator is used to set the power level, to generate additional fixed attenuations and to split the power (a power-splitter is built into this specific attenuator model) The second attenuator is used to increase the measurement range for very high power levels - the second attenuator reduces the power level to the usable range for the standard sensor For very low power levels, the two sensors can be switched and the second attenuator produces the low power levels
54
Linearity Calibration
Any difference between the two measured attenuations indicates nonlinearity The nonlinearity of an optical power meter is internationally defined so that it represents directly the correspondent error in a loss measurement where A is the true power ratio, Am is the measured power ratio, Dx/D0 is the displayed power ratio (of the test meter), Px/P0 are the true power ratio (of the standard meter) The calibration procedure is as follows, Set the desired reference power on the test meter, D0. Record the powers P0 (standard meter) and D0 Increase (decrease) the attenuation of the first attenuator and record the powers P1 (P2, ..) and D1 (D2, ..) Calculate the nonlinearity for the power D1 (D2, ..) using the above equation. In these calculations, the reference level is changing from step to step, which is why these nonlinearities are termed partial
Question What is the value of nonlinearity at D0?
55
Linearity Calibration
The calibration procedure is as follows, Increase the attenuation further by repeating the first two steps, until the low (high) end of the power range is reached. It is advisable to measure the nonlinearity due to range discontinuities by simply changing the power range and recording the measurement results in both ranges Decrease the attenuation to obtain the power levels above P0 and to obtain the correspondent nonlinearity results
56
Linearity Calibration
Linearity Calibration based on Superposition This is a self-calibrating method which does not need a standard meter A possible measurement setup is shown in the figure below Procedure In the beginning, the two attenuators are both set to high attenuation and so that each beam separately gives rise to the same powers at the DUT, Da Db
57
Linearity Calibration
Linearity Calibration based on Superposition Procedure Each attenuator is equipped with a shutter. The shutter of the respective other attenuator remains closed Then the beams are combined by opening both shutters at the same time. This reading should now be the sum of the two preceding individual readings Any deviation indicates nonlinearity. Accordingly, the nonlinearity for the first power Dc is The next cycle starts by generating the combined power separately with each of the attenuators, before combining them again At the end of the measurement, the partial nonlinearities for all steps will be determined
58
Linearity Calibration
Linearity Calibration based on Superposition The figure shows the power superposition used in linearity calibration
59
Linearity Calibration
Linearity Calibration based on Superposition Finally, the total nonlinearity can be calculated, in other words, the nonlinearity with respect to a fixed reference level Start by choosing a reference level, at which the total nonlinearity is zero by definition Then use the following equation for power levels lower than the reference level where n -1, -2, etc. indicates the power level number below the reference point and N is the partial nonlinearity for the i-th step (i 0 for the step between the reference power and the next-higher power). For power levels higher than the reference level, the total nonlinearity is where n 1, 2, etc. The final result is a list of total linearities for the whole power range in 3 dB steps (because the power is doubled in each step)